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no imply d(x,, xy) < 4 2 ; on the other hand, by (3.13.11) there is a p o 2 no such that d(b, x p o )< 4 2 ; by the triangle inequality, it follows that d(b, x,) < E for any n 2 n o . A metric space E is called complete if any Cauchy sequence in E is convergent (to a point of E, of course).
(3.14.3)The real line R is a complete metric space. Let (x,) be a Cauchy sequence of real numbers. Define the sequence (n,) of integers by induction in the following way: no = 1 and n k + l is the sma~~estinteger > n, such that, f o r p 3 and q 3 n k + l , Ix, - xyl < I / 2 k + Z ; the possibility of the definition follows from the fact that (x,) is a Cauchy sequence. Let I, be the closed interval [x,, - 2 - , , x,, 2 - k ] ; we have I , + ~c r,, for IX,~ - xnL+,l < 2 - k - ' ,. on the other hand, for m 2 t ? k , x, E 1, by definition. Now from axiom (IV) (Section 2.1) it follows that the nested intervals I, have a nonempty intersection; let a be in I, for all k . Then it is clear that la - x,I d 2 - h + ' for all m 3 n k r hence a = lim x,.
+
n+ m
(3.14.4)Ij'a subspace F of a metric space E is complete, F is closed in E. Indeed, any point a E F is the limit of a sequence (x,) of points of F by (3.13.13). The sequence (x,) is a Cauchy sequence by (3.14.1), hence by assumption converges t o a point b in F; but by (3.13.3) b = a, hence a E F ; this shows F = F. Q.E.D.
(3.14.5)In a complete metric space E, any closed subset F is a complete subspace. For a Cauchy sequence (x,) of points of F converges by assumption to a point a E E, and as the x, belong to F, a E F = F by (3.13.7). Theorems (3.14.4) and (3.14.5) immediately enable one to give examples both of complete and of noncomplete spaces, starting from the fact that the real line is complete.
Ill
54
METRIC SPACES
The fundamental importance of complete spaces lies in the fact that to prove a sequence is convergent in such a space, one needs only prove it is a Cauchy sequence (one also says that such a sequence satisfies the Cauchy criterion); the main difference between application of that test and of the definition of a convergent sequence is that in the Cauchy criterion one does not need to know in advance the oalue of the limit. We have already mentioned that on a same set E, two distances d , , d , may be topologically equivalent, but the identity mapping of E, into E, (El, E, being the corresponding metric spaces) may fail to be uniformly continuous. This is the case, for instance, if we take E = R, d,(x, y ) = Jx- pJ, d l ( x , y ) being the distance in the extended real line, restricted to R; E, is then complete and not El since El is not closed in 8. When two distances d,, d, are such that the identity mapping of El into E, is uniformly continuous as well as the inverse mapping, d, and d, are said to be unijornily equbalent. Cauchy sequences are then the same for both distances. For instance, if there exist two real numbers a > 0, fl > 0 such that, for any pair of points x,y in E, adl(x, y ) < d,(x, y ) < fldl(x,y ) , then (1, and d, are uniformly equivalent distances. Let E, E’ be two metric spaces, A a subset of E , f a mapping of A into E’; the oscillatioti of,f in A is by definition the diameter 6 ( j ( A ) )(which may be a).Let a be a cluster point of A ; the oscillation o j f at the point a \\tit11 respect to A is Q ( a ; f ) = infii(,f( V n A)), where V runs over the set of
+
V
neighborhoods of a (or merely a fundamental system of neighborhoods). (3.14.6) Suppose E’ is a complete metric space; in order that
lim f(x) %*a, x E A
exist, a necessary arid su-cient condition is that the oscillation of ,f at the point a, with respect to A, be 0. The condition is necessary by (3.13.2). Suppose conversely that it is satisfied, and let (x,) be a sequence of points of A converging to a ; then it follows from the assumption that the sequence ( f ( x , ) ) is a Cauchy sequence in E‘, for, given any E > 0, there is a neighborhood V of a such that d ‘ ( f ( x ) , f ( y ) )< E for any two points x, y in V n A , and we have x,,E V n A except for a finite number of indices. Hence the sequence ( f ( x , ) ) has a limit a’. Moreover, for any other sequence (y,) of points of A , converging to a, the limits of ( f ( x , ) ) and of (f(y,,)) are the same since d’(f(x,),f(y,)) < E as soon as x, and yn are both in V n A. Hence lim f ( x ) = a‘ from the definition of the limit and from (3.13.14).
x-a, x E A
15
ELEMENTARY EXTENSION THEOREMS 55
PROBLEMS 1. (a) Let E be an rdtvametvic space (Section 3.8, Problem 4). In order that a
sequence (x,) in E be a Cauchy sequence, show that it is necessary and sufficient that lim d ( x , , ~ , , + ~ ) = O . n- m
(b) Let X be an arbitrary set, E the set of all infinite sequences x = (x,) of elements of X. For any two distinct elements x = (xJ, y = (y.) of E, let k ( x , y ) be the smallest integer 12 such that x, # yn ; let d(x, y ) = I/k(x,y ) if x # y , d ( x , x) = 0. Prove that d is an ultrametric distance on E, and that the metric space E defined by d is complete. 2. Let 'p be an increasing real valued function defined in the interval 0 < u < +a,and such that q(0) = 0, q(u)> 0 if u > 0, and y(ir u ) < p(ir) q ( u ) . Let c/(x, y ) be a distance on a set E ; then d,(x, y ) = q(d(x, y ) ) is another distance on E. (a) Show that if q is continuous at the point [I = 0, the distances d a n d d, are uniformly equivalent. Conversely, if, for the distanced, there is a point xo E E which is not isolated in E (3.10.10),and if ciand d, are topologically equivalent, then q is continuous at the point I I = 0. (b) Prove that the functions
+
I!'
(0 < r
< l),
log(1
+
I/),
u/(I
+
+
ti),
inf(1, u )
satisfy the preceding conditions. Using the last two, it is thus seen that for any distance on E, there is a uniformly equivalent distance which is borinded. 3. On the real line, let d(x, y ) = Ix - y / be the usual distance, d'(x, y ) = / x 3 - y31 ; show that these two distances are topologically equivalent and that the Cauchy sequences are the same for both, but that they are not uniformly equivalent. 4. Let E be a complete metric space, d the distance on E, A the intersection of a sequence (U,) of open subsets of E ; let F, = E - U,, and for every pair of points x , y of A, write
d h , Y ) =L(x, y)l(l + f ; d x , Y ) ) , and d'(x, Y ) = d(x, Y )
+ C d d x , y)/2". Show that on m
Il=O
the subspace A of E, d' is a distance which is topologically equivalent to d , and that for the distance d',A is a complete metric space. (Note that a Cauchy sequence for d' is also a Cauchy sequence for d, but that its limit in E may not belong to any of the F, .) Apply to the subspace I of R consisting of all irrational numbers.
15. EL EM E N T A R Y E X T E N S I 0 N TH E OREMS
(3.15.1) Lei f and g be two continuous mappings of a metric space E into a metric space E'. The set A of the points x E E such that f ( x ) = g(x) is closed iti E. It is equivalent to prove the set E - A open. Let a E E - A, then f ( a ) # g(a); let d'( f (a), g(a)) = a > 0. By continuity o f f , g at a and from (3.6.3) it follows that there is a neighborhood V of a in E such that for
56
Ill METRIC SPACES
x E V, d’( f (a), f ( x ) ) < 4 2 and d’(g(a),g(x)) < a/2. Then for x E V, f ( x ) # g(x), otherwise we would have d‘(f(a),g(a)) < c( by the triangle inequality. (“ Principle of extension of identities ”) Let f, g be two contin(3 .I5.2) uous mappings of a metric space E into a metric space E’; if f ( x ) = g ( x ) for all points x of a dense subset A in E, then f = g.
For the set of points x where f ( x ) = g(x) is closed by (3.15.1) and contains A. (3.15.3) Let f, g be two continuous mappings of a metric space E into the The set P of the points x E E such that f ( x ) < g ( x ) is extended real line closed in E.
w.
We prove again E - P is open. Supposef(a) > g(a), and let p E R be such that f ( a ) > p > g(a) (cf. (2.2.16) and the definition of R in Section 3.3). The inverse image V b y j o f the open interval ]p, + 001 is a neighborhood of a by (3.11 .I); so is the inverse image W by g of the open interval [ - 00, p[. Hence V n W is a neighborhood of a by (3.6.3), and for x E V n W, f(x) > p > g(x). Q.E.D. (3.15.4) (“ Principle of extension of inequalities”) L e t f , g be two continuous mappings of a metric space E into the extended real line R; i f f ( x ) < g(x) for all points x of a dense subset A of E, then f ( x ) < g(x)for all x E E.
The proof follows from (3.1 5.3) as (3.1 5.2) from (3.1 5.1). (3.15.5) Let A be a dense subset of a metric space E, and f a mapping of A into a metric space E‘. In order that there exist a continuous mapping f of E into E’, coinciding with f in A, a necessary and suficient condition is that, for any x E E, the limit lim f ( y ) exist in E’; the continuous mapping f is Y+X,Y EA then unique.
As any x E E belongs to A, we must have f ( x ) = lim f ( y ) by (3.13.5), Y-X,YEA
hencef(x) = lirn f ( y ) ; this shows the necessity of the condition and the Y+X, Y E A
fact that if the continuous mapping J exists, it is unique (this follows also
16 COMPACT SPACES
57
from (3.1 5.2)). Conversely, suppose the condition satisfied, and let us prove that the mappingfdefined byf(x) = lim f ( y ) is a solution of the extension Y-x, Y FA
problem. First of all, if x E A, the existence of the limit implies by definition f(x) = f(x), hence f extends f, and it remains to see that f is continuous. Let x E E, V' a neighborhood off(x) in E'; there is a closed ball B' of center f (x) contained in V'. By assumption, there is an open neighborhood V of x in E such thatf(V n A) c B' (by (3.13.1)). For any y E V,f(y) is the limit of f a t the pointy with respect to A, hence also with respect to V n A, by (3.13.5); hence, it follows from (3.13.7) that f(y) E f(V n A), and therefore f ( y ) E B' since B' is closed. Q.E.D.
(3.15.6) Let A be a dense subset of a metric space E, and f a uniformly continuous mapping of A into a complete metric space E'. Then there exists a continuous mapping f of E into E' coinciding with f i n A; moreover, f is uniformly continuous.
To prove the existence off, it follows from (3.15.5) and (3.14.6) that we have to show the oscillation of f at any point x E E, with respect to A, is 0. Now, for any E > 0, there is 6 > 0 such that d(y, z) < 6 implies d'(f(y), f(z)) < 4 3 (y, z in A). Hence, the diameter of f ( A n B(x; 612)) is at most ~ / 3 which , proves our assertion. Consider now any two points s, t in E such that d(s, t ) c 612. There is a Y E A such that d(s, y) < 614 and d ' ( f ( s ) ,f(y)) c ~ / 3 and , a z E A such that d(t, z) < 614 and d'Cf(t),f(z)) < 4 3 . From the triangle inequality it follows that d(y, z) < 6, and as y , z are in A, d'(f (y),f (z)) < 4 3 ; hence, by the triangle inequality, d'(f(s),3(t))< E ; this proves that f is uniformly continuous.
PROBLEM
Let n + r. be a bijection of N onto the set A of all rational numbers x such that 0 < x < 1 (2.2.15). We define a function in E = [0, I ] by puttingf(x) = 1/2", the infinite sum being ," 0, there is aJinite covering of E by sets of diameter < E . This
is immediately equivalent to the following property: .for afiy E > 0, there is a finite subset F of E such that d(x, F) < cfor every x E E. In the theory of metric spaces, these notions are a substitute for the notion of “finiteness” in pure set theory; they express that the metric space is, so t o speak, “ approximately finite.” Note that, from the definition, it follows that compactness is a topological notion, but precompactness is not (see remark after (3.17.6)).
(3.16.1) For a metric space E, the following three conditioiis are equivalent: (a) E is compact; (b) any infinite sequence in E has at least a cluster value; (c) E is precompact and coniplete.
(a) * (b): Let (x,) be an infinite sequence in the compact space E, and let F, be the closure of the set {x,, x , + ~ ., . . , x,,+,,, . . .}. We prove there is a point belonging to all F,. Otherwise, the open sets U, = E - F, would form a covering of E, hence there would exist a finite number of them, UnI,. . . , U,,, forming a covering of E; this would mean that F,, n F,, n ... n F,, = 0; but this is absurd, since if n is greater than max(n,, . . . , nk), F, (which is not empty by definition) is contained in all the F,,,(1 < i < k). Hence the intersection
m
n= 1
F, contains at least a point a. By (3.13.11) and the definition of a
cluster point, a is a cluster value of (x,,). (b) * (c): First any Cauchy sequence in E has a cluster value, hence is convergent by (3.14.2), and therefore E is complete. Suppose E were not precompact, i.e. there exists a number c( > 0 such that E has no finite covering by balls of radius a. Then we can define by induction a sequence (x,) in the following way: x1 is an arbitrary point of E; supposing that d(x,, x,) 2 ct for i Zj,1 < i < n - I , I < ,j < n - 1, the union of the balls of center x i (1 < i < n - I) and radius ct is not the whole space, hence there is x, such that d ( x i ,x,) 2 c( for i < 17. The sequence (x,) cannot have a cluster value, for if a were such a value, there would be a subsequence (x,J converging t o a, hence we would have d(a, x n k )< 4 2 for k > ko , and therefore d(x,, , x,J < ct for h 3 k, , k 3 k , , 12 # k , contrary t o the definition of (x,,). (c) * (a): Suppose we have an open covering (U,JiELof E such that no finite subfamily is a covering of E. We define by induction a sequence (B,) of
16 COMPACT SPACES 59
balls in the following way: from the assumption it follows that the diameter of E is finite, and by multiplying the distance on E by a constant, we may assume that 6(E) < 1/2,hence E is a ball B, of radius 1. Suppose the B, have been defined for 0 < k < n - 1, and that for these values of k , B, has a radius equal to 1/2,,and there is no finite subfamily of (U,),eL which is a covering of B, . Then we consider a finite covering (vk),,,,, of E by balls of radius 1/2”; among the balls vk which have a nonempty intersection with B,,-,, there is one at least B, for which no finite subfamily of (U,) is a covering; otherwise, as these V , form a covering of B,-,, there would be a finite subfamily of (U,) which would be a covering of B,-l ; the induction can thus proceed indefinitely. Let x, be the center of B,; as B,-l and B, have a common point, the triangle inequality shows that
x,) Hence, if n
< 1 p - 1 + 1/2” < 1/2,-2
< p < q, we have
This proves that (x,,) is a Cauchy sequence in E, hence converges to a point a. Let 1, be an index such that a E Un0;there is an a > 0 such that B ( a ; a ) c U,, . From the definition of a, it follows there exists an integer n such that d(a, x,) < 4 2 , and l/2”< 42. The triangle inequality then shows that B, c B ( a ; a ) c Ulo. But this is a contradiction since no finite subfamily of (U,) is supposed to be a covering of B,, .
(3.16.2) Any precompact metric space is separable. If E is precompact, for any n there is, by definition, a finite subset A, of E such that for every x E E, d(x, A,,) < l/n. Let A = A,; A is at most denumerable, and for each x E E, d(x, A) d(x, A) = 0, E =A.
< d(x, A,)
u n
< l/n for any n, hence
(3.16.3) Let E be a metric space. A n y two of the following properties imply the third: (a) E is compact. (b) E is discrete (more precisely, homeomorphic to a discrete space). (c) E isfinite.
60
Ill METRIC SPACES
(a) and (b) imply (c), for each one-point set {x} is open, hence the family of sets {x} is an open covering of E, and a finite subfamily can only be a covering of E if E is finite. On the other hand, (c) implies both (a) and (b), for each one point set being closed, every subset of E is closed as finite union of closed sets, hence every subset of E is open, and therefore E is homeomorphic to a discrete space. Finally, as there is only a finite number of open sets, E is compact.
(3.16.4) In a compact metric space E, any infinite sequence (x,) which has only one cluster value a converges to a. Suppose a is not the limit of (x,); then there would exist a number a > 0 such that there would be an infinite subsequence (x,,) of (x,) whose points belong to E - B(a; a). By assumption, this subsequence has a cluster value b , and as E - B(a; a ) is closed, b belongs to E - B(a; a ) by (3.13.7). The sequence (x,) would thus have two distinct cluster values, contrary to assumption.
(3.16.5) Any continuous mapping f of a compact metric space E into a metric space E‘ is uniforndy continuous. Suppose the contrary; there would then be a number a > 0 and two sequences (x,) and (y,) of points of E such that d(x,, y,) < I/n and d’(f(x,),f ( y , ) ) 3 a. We can find a subsequence (x,) converging to a point a, and as d(x,,, y,,) < l/nk,it follows from the triangle inequality that the sequence (ynL)also converges to a. But f is continuous at the point a, hence there is a 6 > 0 such that d ’ ( f ( a ) , f ( x ) ) < a / 2 for d(a, x) < 6. Take k such that d(a, x,) < 6, d(a, y n k )< 6; then d’(f(x,,),f(y,,)) < a contrary to the definition of the sequences (x,) and (y,).
(3.16.6) Let E be a compact metric space, (U,),ELan open covering of E. There exists a number a > 0 such that any open ball of radius a is contained in at least one of the U, (“ Lebesgue’s property ”). For every x E E, there exists an open ball B(x; r,) contained in one of the sets U,. As the balls B(x; r,/2) form an open covering of E, there exist a finite number of points x i E E such that the balls B(x,; r J 2 ) form a covering of E. If a > 0 is the smallest of the numbers rxi/2,it satisfies the required property: indeed, every x E E belongs to a ball B(x,; rJ2) for some i, hence B(x; a) is contained in B(x,; rXi)since a < r x i / 2 ;but by construction B(x,; r X i ) is contained in some U,.
17 COMPACT SETS 61 PROBLEMS 1. Give an example of a precompact space in which the result of (3.16.6) fails to be true. 2. For a metric space E, show that the following properties are equivalent: (a) E is compact; (b) every denumerable open covering of E contains a finite subcovering; (c) every decreasing sequence (F,) of nonempty closed sets of E has a nonempty intersection; (d) for any infinite open covering ( U A ) A eof, ~E, there is a subset H c L, distinct from L and such that (UA),." is still a covering of E; (e) every poititwisefinire open covering (U,) of E (i.e. such that for any point x E E, x E UA only for a finite subset of indices) contains a finite subcovering; (f) every infinite subspace of E which is discrete is not closed. (Using (3.16.1), show that (f) implies (a), and that (d) and (e) imply (f).) 3. Let E be a metric space, d the distance on E, %(E) c $(E) the set of all closed nonempty subsets of E. We may suppose that the distance on E is bounded (Section 3.14, Problem 2). For any two elements A, B of S(E), let p(A, B) = sup d(x, B), h(A, B) = X€A sup(p(A, B), p(B, A)). (a) Show that, on g(E), h is a distance (the "Hausdorff distance"). (b) Show that for any four elements A, B, C, D of g(E), one has
h(A u B, C u D) < max(h(A, C), h(B, D)). (c) Show that if E is complete, 8(E) is complete. (Let (X.) be a Cauchy sequence in s(E); for each n, let Y , be the closure of the union of the sets X,,, such that p 2 0; consider the intersection of the decreasing sequence (Y.) in E.) (d) Show that if E is precompact, %(E) is precompact (use the problem in Section 1.1). Therefore, if E is compact, 8(E) is compact. 4. Let E be a compact metric space. For every E > 0, let N,(E) be the smallest integer n such that there exists a covering of E by n sets of diameter < 2.5; let M,(E) be the largest integer m such that there exists a finite sequence of m points of E for which the distance of any two (distinct) of these points is > 8 . The number H,(E) = log N,(E) is called the &-entropy of E, the number C,.(E) = log MJE) the &-capacity of E. (a) Show that M,,(E) < N,(E) < M,(E), hence C,,:(E) < H,(E) < C,(E). (b) Show that the functions N,(E) and MJE) of E , defined for 8 > 0, are decreasing and continuous on the right (to prove the continuity of N,(E) on the right, use contradiction, and apply problem 3(d)). (c) If A and B are closed nonempty subsets of E, show that NdA
LJ
M A LJ
+N O ) , B) < HAA) + H,(B), B) < N A N
M,(A
LJ
B) < MAA)
CAA LJ B) < CAA)
+ MAB)
+ CdB).
(d) If E is a closed interval of R of length I , show that NJE) = M2,(E) = 1/28 if / / 2 is~ an integer, and N,(E) = M,,(E) = [//2.5] 1 (where [ I ] is the largest integer < t for t > 0) if / / 2 ~is not an integer.
+
17. COMPACT SETS
A compact (resp. precompact) set in a metric space E is a subset A such that the subspace A of E be compact (resp. precompact).
62
111 METRIC SPACES
(3 .I 7.1)
Any precompact set is bounded.
This follows from the fact that a finite union of bounded sets is bounded (3.4.4). The converse of (3.17.1) does not hold in general, for any distance is equivalent to a bounded distance (Section 3.14, Problem 2) (but see (3.17.6)).
(3.17.2)
Any compact set in a metric space is closed.
Indeed, such a subspace is complete by (3.16.1), and we need only apply (3.14.4).
(3 .I 7.3) In a compact space E, every closed subset is compact.
For such a set is obviously precompact, and it is a complete subspace by (3.14.5).
A relatively compact set in a metric space E is a subset A such that the closure A is compact. (3.17.4) Any subset of a relatively compact (resp. precompact) set is relatioely compact (resp. precompact).
This follows at once from the definitions and (3.17.3).
(3.17.5) A relatively compact set is precompact. In a conlplete space, a precompact set is relatively compact.
The first assertion is immediate by (3.17.4). Suppose next E is complete and A c E precompact. For any E > 0, there is a covering of A by a finite number of sets c, of A having a diameter < ~ / 2 each ; ck is contained in a closed ball D, (in E) of radius 4 2 . We have therefore A c D,, the set
u D, k
u
being closed, and each D, has a diameter
k
< E . On the other hand, A is a
complete subspace by (3.14.5), whence the result. A precompact space E which is not complete gives an example of a precompact set which is not relatively compact in E.
17 COMPACT
SETS 63
(3.17.6) (Borel-Lebesgue theorem). In order that a subset of the real line be relatively compact, a necessary and suficient condition is that it be bounded. In view of (3.17.1), (3.17.4), and (3.17.5), all we have to do is to prove any closed interval [a, b] is precompact. For each integer n, let xk = a k(b - a)/n (0 < k d I ? ) ; then the open intervals of center xk and length 2/n form a covering of [a, b]. Q.E.D.
+
Remark. If, on the real line, we consider the two distances d,, d, defined in Section 3.14, it follows from (3.17.1) that E, is not precompact, whereas El is precompact, since the extended real line R, being homeomorphic to the closed interval [- 1, + I ] of R (3.12), is compact by (3.17.6). (3.17.7) A necessary and sufJjcient condition that a subset A of a metric space E be relatively compact is that every sequence of points of A have a cluster value in E. The condition is obviously necessary, by (3.16.1). Conversely, let us suppose it is satisfied, and let us prove that every sequence (x,) of points of A has a cluster value in E (which will therefore be in A by (3.13.7)), and hence that A is compact by (3.16.1). For each n, it follows from the definition of closure that there exists y , E A such that d(x,, y,) d Ijn. By assumption, there is a subsequence (y,J which converges to a point a ; from the triangle inequality it follows that (x,,) converges also to a. Q.E.D. (3.17.8) The union of two relatiuely compact sets is relatively compact. From (3.8.8) it follows that we need only prove that the union of two compact sets A, B is compact. Let (UJdELbe an open covering of the subspace A u B ; each U, can be written (A u B) n V,, where V, is open in E, by (3.10.1). By assumption, there is a finite subset H (resp. K) of L such that the subfamily(A n V,)neb,(resp. (6 n VA)AeK)is a covering of A (resp. B). It is then clear that the family ((A n B) n V,), E H is a covering of A u B. (3.17.9) Let f be a continuous mapping of a metric space E into a metric space E‘. For every compact (resp. relatiuely compact) subset A of E, ,f(A) is compact, hence closed in E’ (resp. relatively cornpact in El).
64
Ill
METRIC SPACES
It is enough t o prove that f ( A ) is compact when A is compact. Let ( U l ) l E Lbe an open covering of the subspace ,f(A) of E'; then the sets A nf -'(UA) form an open covering of the subspace A by (3.11.4); by assumption, there is a finite subset H of L such that the sets A n , f - ' ( U L ) for I E: H still form a covering of A ; then the sets U, =f(A n f - ' ( U L ) ) for ;IE H will form a covering of,f(A). Q.E.D.
(3.17.10) Let E be a nonetiipty compact metric space, f a continuous mapping of E into R; tliett,f(E) is bounded, and there exist tlt'o points a , h in E such that . f @ ) = inf,fW,f(b) = sup.f(.4. xeE
XE
E
The first assertion follows from (3.17.9) and (3.17.1). On the other hand, f(E) is closed in R by (3.17.2), hence supf(E) and inff(E), which are cluster points off(E), belong to,f(E).
(3.17.1 1) Let A be a conipact subset in a metric space E. Then the sets V,(A) (see Section 3.6) f o r m a fundatiiental system of neigliborhoods of A. Let U be a neighborhood of A ; the real function x + d(x,E - U) is > 0 and continuous in A by (3.11.8), hence there is a point xo E A such that d(xo,E - U) = inf d(x, E - U) by (3.17.10). But d(xo,E - U) = r > 0; X E A
hence V,(A) c U.
(3.17.12) I f E is a compact nietric space, f a continuous injectii'e tilapping of E into a metric space E', then f is a lrotiieotiiorphism of' E onto f (E). All we need to prove is that for every closed set A c E, ,f(A) is closed inf(E) (by (3-11.4)); but this follows from (3.17.3) and (3.17.9).
PROBLEMS
1. Let f be a uniformly continuous mapping of a metric space E into a metric space E'. Show that for any precoinpact subset A of E,f(A) is precompact. 3. In a metric space E, let A be a compact subset, B a closed subset such that A n B = 0. Show that d(A, B) > 0.
18 LOCALLY COMPACT SPACES 65 3. Let E be a compact ultrametric space (Section 3.8, Problem 4), d the distance on E. Show that for every xo E E, the image of E by the mapping .Y -+cf(xo, x ) is an at most denumerable subset of the interval [0, +a[in which every point (with the possible exception of 0) is isolated (3.10.10).(For any I’ = d(xo,x ) > 0, consider the 1.u.b. of cl(xo, y ) on the set of points y such that d(x,,, y ) < 1’, and the g.1.b. of d(xo,z) on the set of points z such that c/(x,, z) > r; use Section 3.8,Problem 4). 4. Let E be a compact metric space, d the distance on E,fa mapping of E into E such that, for any pair (x, y ) of points of E, d(f(x),f(y)) 2 d(x, y). Show thatfis an isometry ofE onto E. (Let a, b be any two points of E ; put fn =fn-, o f , a, =fn(a), b, =f,(b); show that for any E > 0 there exists an index k such that d(a, ak) C E and d(b, bk) < E (consider a cluster value of the sequence (a“)), and conclude that f(E) is dense in E and that 4 f ( a ) , f ( b ) )= 4 0 , b).) 5. Let E, E’ be two metric spaces,fa mapping of E into E’. Show that if the restriction off to any compact subspace of E is continuous, thenfis continuous in E (use (3.13.14)). 6 . Let E,E’ be two metric spaces, f a continuous mapping of E into E , K a compact subset of E. Suppose the restriction f l K off is injective and that for every x E K, there is a neighborhood V, of x in E such that the restriction f 1 V, o f f is injective. Show that there exists a neighborhood U of K in E such that the restriction f 1 U is injective (use contradiction and (3-17.11)).
18. LOCALLY COMPACT SPACES
A metric space E is said to be locally conipact if for every point x E E, there exists a compact neighborhood of x in E. Any discrete space is locally compact, but not compact unless it is finite (3.16.3).
(3.18.1) The real line R is locally conipact hut not compact. This follows immediately from the Borel-Lebesgue theorem (3.17.6).
(3.18.2) Let A be a conipact set in a locally compact metric space E. Then there exists an r > 0 such that V, (A) (see Section 3.6) is relatitlely conzpact in E. For cach x E A, there is a compact neighborhood V, of x ; the 9, form an open covering of A, hence there is a finite subset {xl, . . . , x,} in A such that the
TX,(1 < i 6 n) form an open covering of A. The set U =
u n
i= 1
V,: is com-
pact by (3.17.8) and is a neighborhood of A; hence the result, by applying (3.17.11).
66
Ill METRIC SPACES
(3.18.3) Let E be a locally compact metric space. The following properties are equivalent: (a) there exists an increasing sequence (U,) of open relatively conipact c Un+l for every 11, and E = U,; sets in E such that
u
n,
n
(b) E is a denurnerable uriion of compact subsets; (c) E is separable.
u,
It is clear that (a) implies (b), since is compact. If E is the union of a sequence (K,) of compact sets, each subspace K, is separable (by (3.16.2)); if D, is an at most denumerable set in K,, dense with respect to K,,, then D = D, is at most denumerable, and dense in E, since E =
u n
K, c
u u 15, n
c
15; hence (b) implies (c). Let us suppose finally that E
fl
is separable, and let (T,) be an at most denumerable basis for the open sets of E (see (3.9.4)). For every x E E, there is a compact neighborhood W, of x, hence, by (3.9.3), an index M ( X ) such that x E Tnc,) c W,. It follows that those of the T, which are relatively compact already constitute a basis for the open sets of E. We can therefore suppose that all the T, are relatively compact. We then define U, by induction in the following way: U1 = TI, U n + , is the union of Tntl and of V,(U,), where r > 0 has been taken such that V,(u,) is relatively compact (which is possible by (3.18.2)); it is then clear that the sequence (U,) verifies property (a). (3.18.4) In a locally compact metric space E, eriery opeii subspace atid every closed subspace is locally compact. Suppose A is open in E; for every a E E, there is a closed ball B’(a; r ) which is compact (from the definition of a locally compact space and (3.17.3)). On the other hand, there is r‘ Q r such that the ball B’(a; r’) is contained in A ; as it is compact by (3.17.3), A is locally compact. Suppose A is closed in E, and let a E A ; then, if V is a compact neighborhood of a in E, V n A is a neighborhood of a in A by (3.10.4), and is compact by (3.17.3); this proves A is locally compact.
PROBLEMS 1. If A is a locally compact subspace of a metric space E, show that A is locally closed (Section 3.10, Problem 3) in E. The converse is true if E is locally compact (use (3.18.4)). 2. (a) Show that in a locally compact metric space, the intersection of two locally com-
pact subspaces is locally compact (cf. Problem I).
19 CONNECTED SPACES AND CONNECTED SETS 67
(b) In the real line, give an example of two locally compact subspaces whose union is not locally compact, and an example of a locally compact subspace whose complement is not locally compact. 3. (a) Give an example of a locally compact metric space which is not complete. (b) Let E be a metric space such that there exists a number r > 0 having the property that each closed ball B’(x; r ) ( x E E) is compact. Show that E is complete and that for any relatively compact subset A of E, the set V:,,(A) of the points x E E such that d(x, A) < rj2 is compact.
19. CONNECTED SPACES A N D CONNECTED SETS
A metric space E is said to be connected if the only subsets of E which are both open and closed are the empty set /zr and the set E itself. An equivalent formulation is that there does not exist a pair of open nonempty subsets A, B of E such that A u B = E, A n B = /zr. A space reduced to a single point is connected. A subset F of a metric space E is connected if the subspace F of E is connected. A metric space E is said to be locally connected if, for every x E E, there is a fundamental system of connected neighborhoods of+.
(3.19.1) In order that a subset A of the real line R be connected, a necessary and suficient condition is that A be an interval (bounded or not). The real line is a connected and locally connected space. The second assertion obviously follows from the first. Suppose A is connected; if A is reduced to a single point, A is an interval. Suppose A contains two distinct points a < 6. We prove every x such that a < x < b belongs to A. Otherwise, A would be the union of the nonempty sets B = A n ] - c o , x [ and C = A n ] x , +a[, both of which are open in A and such that B n C = From this property, we deduce that A is necessarily an interval. Indeed, let c E A, and let p , q be the g.1.b. and 1.u.b. of A in R ; if p = -a,then for every x < c, there is y < x belonging to A ; hence x E A, so ] - co, c] is contained in A; if p is finite and p < c, for every x such that p x < c there is y E A such that p < y < x, hence again x E A, so that A contains the interval ] p , c]. Similarly, one shows that A contains [c,q[ if q > c ; it follows that in any case A contains the interval ]p,q [ , and therefore must be one of the four intervals in R of extremities p , q (of course, if p = - 00 (resp. q = 00) p (resp. q ) does not belong to A). Conversely, suppose A is a nonempty interval of origin a and extremity b in R (the possibilities a = - co, n 4 A, b = + co, b 4 A being included).
a.
-=
+
68
I l l METRIC SPACES
Suppose A = B LJ C, with B, C nonempty open sets in A and B n C = 0; suppose for instance x E B, y E C, and x < y . Let z be the 1.u.b. of the bounded set B n [x, y ] ; if z E B, then z < y and there is by assumption an interval [ z , z + h [ contained in [x,y ] and in B, which contradicts the definition of z ; if on the other hand z E C, then x < z , and there is similarly an interval ] z - h, z ] c C n [x,y ] , which again contradicts the definition of z (see (2.3.4));hence z cannot belong to B nor to C, which is absurd since the closed set [x, y ] is contained in A. Hence A is connected.
(3.19.2) rf' A is a connected set in a nietvic space E, then any set B such that A c B c A is connected.
For suppose X, Y are two nonempty open sets in B such that X u Y = B, X n Y = @; as A is dense in B, X n A and Y n A are not empty, open in A, and wewouldhave(X n A) LJ (Y n A) = A , ( X n A) n (Y n A) = 0, a contradiction.
(3.19.3) 111 a metric space E, let (A,), be a family of connected sets haoing a nonempty igtersection; theti A = A, is connected.
u
,EL
Let a be a point of
n A,, and suppose A
=B
v C, where B, C are non-
LEL
empty open sets in A such that B n C = @. Suppose for instance a E 0 ; by assumption there is at least one A such that C n A, # 0; then B n A, and C n A, are open in A, and such that (B n A,) u (C n A,) = A,, (B n A,) n (C n A,) = 0, a contradiction since B n A, # @.
(3.19.4) Let ( A i ) l f i s nbeasequence of coniiected sets such that A i n
0for 1 < i d n - 1;
then
u Ai n
#
is connected.
i= 1
This follows at once from (3.19.3)by induction on n. From (3.19.3)it follows that the union C ( x )of allconnected subsets of E containing a point x E E is connected, hence the largest connected set containing x; it is called the connected component qf x in E. It is clear that for any y E C(x), we have C ( y ) = C(x), and if y $ C(x), then C(x) n C ( y ) = 0; moreover, it follows from (3.19.2)that C(x) is closed in E. For any subset A of E, the connected components (in the subspace A) of the points of the subspace A are called the connected coniponents of A ; if every connected component of A is reduced to a single point, A is said to be totally disconnected.
19 CONNECTED SPACES AND CONNECTED SETS 69
A discrete space is totally disconnected; the set of rational numbers and the set of irrational numbers are totally disconnected, by (2.2.16) and (3.19.1). (3.19.5) In order that a nietric space E be locally connected, a necessary and suflcient condition is that the connected coniponents of the open sets in E be open. The condition is sufficient, for if V is any open neighborhood of a point E, the connected component of x in the subspace V is a connected neighborhood of x contained in V, hence E is locally connected. Thecondition is necessary, for if E is locally connected and A is an open set in E, B a connected component of A, then for any x E B, there is by assumption a connected neighborhood V of x contained in A, hence V c B by definition of B, and therefore B is a neighborhood of every one of its points, hence an open set. x
E
(3.19.6) Any nonernpty opeti set A in the real line R is the union of an at most denumerable .family of' open interoals, ti0 two of which have common points. From (3.19.1) and (3.19.5) it follows that the connected components of A are intervals and open sets, hence open intervals. The intersection A n Q of A with the set Q of rational numbers is denumerable, and each component of A contains points of A n Q by (2.2.16); the mapping Y -+ C(r) of A n Q into the set 0.of the connected components of A is thus surjective, and therefore, by (1.9.2), 0.is at most denumerable. (3.19.7) Let f be a continuous niapping of E into E'; for any connected E, f(A) is connected.
subset A of
Suppose f ( A ) = M u N, where M and N are nonempty subsets of f(A), then, by (3.11.4), A n f -'(M) open in f(A) and such that M n N = 0; and A n f -'(N) would be nonempty sets, open in A and such that A = (A nf -'(M))u(A n f-'(N))and(A n f - ' ( M ) ) n ( A n f-'(N)) = contrary to assumption.
a,
(3.19.8) (Bolzano's theorem) Let E be a connected space, f a continuous niapping of E into the real line R. Suppose a, b are two points of f(E) such
70
Ill
METRIC SPACES
tliat a < h. Theti, f o r
anj’
c such tliat a < c < h tkere exists x
E
E such that
f ( x ) = c.
Forf(E) is connected in R by (3.19.7), hence an interval by (3.19.1). (3.19.9) Let A be a subset of a metric space E. If‘ B is a coiinecred siibset of E such that both A n B arid (E - A) n B are not empty, theii (fr(A)) n B is not empty. In particular, if E is connected, any subset A o j E u‘istiiict f i w r r E atid 0has at least oiie froiitier point. Suppose (fr(A)) n B = 0; let A‘ = E - A ; as E is the union of A, and fr(A), B would be the union of U = A n B and V = A’ n B, both of which are open in B and not empty by assumption (for a point of A n B must belong to A n B since fr(A) n B = 0, and similarly for A‘ n B); as U n V 0, =this would be contrary t o the assumption that B is connected.
A’
Remark. If we agree to call “curve” the image of an interval of R by a continuous mapping (see Section 4.2, Problem 5), (3.19.7) shows that a “curve” is connected, and (3.19.9) that a “curve” linking a point of A and a point of E - A meets fr(A), which corresponds to the intuitive idea of connectedness” (see Problem 3 and Section 5.1, Problem 4). “
PROBLEMS
1. Let E be a connected metric space, in which the distance is not bounded. Show that in
E every sphere is nonenipty. (a) Let E be a compact metric space such that in E, the closure of any open ball B(u: r ) is t h e closed ball B’(a; r ) . Show that in E any open ball B(a; 1.) is connected. (Suppose B(a; r ) is the union C D of two noncmpty sets which are open in B(u: r ) and if ci E C , consider a point x F D such that the distance I . Show that E is a locally compact subspace of E, which is not locally connected; the connected components of E are B, C and the A,, 0 7 3 I ) , but the intersection of all open and closed sets containing a point of B is B u C. Let E be a locally compact metric space. (a) Let C be a connected component of E which is compact. Show that C is the intersection of all open and closed neighborhoods of C. (Reduce the problem to the case in which E is compact, using (3.18.2). Suppose the intersection B of all open and closed neighborhoods of C is different from C; B is the union of two closed sets M 3 C and N without common points. Consider in E two open sets U 3 M and V 3 N without common points (Section 3.11, Problem 3). and take the intersections of E - (M u N) with the complements of the open and closed neighborhoods of C.) (b) Suppose E is connected, and let A be a relatively compact open subset of E. Show that every connected component of A has at least a cluster point in A (if not, apply (a) to such a component, and get a contradiction). (c) Deduce from (b) that for every compact subset K of E, the intersection of a connected component of K with E - K is not empty.
c
7. 8.
9.
PRODUCT OF TWO METRIC SPACES 71
c
c
c
c
-_.
20. P R O D U C T O F TWO M E T R I C SPACES
Let El, E, be two metric spaces, d,, dz the distances on El and E, . For any pair of points x = (xl, x,), y = (yl, yz) in E = El x E, , let
4 x ,v>= max(d,(x,, A),4(xZ, yZ)>.
72
111
METRIC SPACES
It is immediately verified that this function satisfies the axioms (I) to (1V) in Section 3.1, in other words, it is a distance on E ; the metric space obtained by taking d as a distance on E is called the product of the two metric spaces El, E, . The mapping (x,,x,) 4 (x, , sl) of El x E, onto E, x El is an isometry. We observe that the two functions d’, d” defined by
d‘(x,Y>= 4(.%Y l )
+ dz(x2
d ” k v) = ((dl(X1, Y l V
9
Yz)
+ (CJ2(x,
9
Y2)>2>”2
are also distances on E, as is easily verified, and are unifornily equioalerit to d (see Section 3.14),for we have
d(x,I’) d d”(x, y ) < d’(x, 4’) < 2 4 % Y ) . For all questions dealing with topological properties (or Cauchy sequences and uniformly continuous functions) it is therefore equivalent to take on E any one of the distances d, d’, d”. When nothing is said to the contrary, we will consider on E the distance d. Open (resp. closed) balls for the distances d, d,, d, will be respectively written B, B,, B, (resp. B’, B’,, B;) instead of the uniform notation B (resp. B’) used up to now.
(3.20.1) For any point a = ( a l , a,) E E a i d arty r > 0, we hare B(a; r ) = Bl(al ; r ) x B,(a, ; r ) and B’(a; r ) = B;(a, ; r ) x B;(a, ; r ) . This follows at once from the definition of d.
(3.20.2) I f A , is an open set in El, A , an open set in E,, then A, x A, is open in E, x E, . For if a = (a,, a , ) E A l x A,, there exists rl > 0 and r, > 0 such that B,(a,; rl) c Al, B,(a,; r z ) c A,; take r = min(r,, r , ) ; then by (3.20.1), B(a; r ) c A, x A,,
(3.20.3) For any pair of sets A, c El, A, c E,, A, x A, = A , x A,; in particular, in order that A, x A , be closed in E, a necessary and suficient condition is that A, be closed in El and A, closed in Ez . If a = (a,, a,) EA, x A,, for any&> 0 there is, by assumption, an x1 E A , and an x, E A, such that d,(a,,x,)< E , d z ( a z ,x,) < E ; hence if x = (x,,x,),
20
PRODUCT OF TWO METRIC SPACES 73
d(a, x) < E . On the other hand, if (a,, a,) $A, x A, then either a, $ A, o r a, # A,; in the first case, the set (El - A,) x E, is open in E by (3.20.2), contains a and has a n empty intersection with A, x A,, hence a 4 A, x A,; the other case is treated similarly.
(3.20.4) Let z f ( z ) = (f,(z),f,(z)) be a mapping of a metric space F into E = El x E,; in order that f be continuous at a point z o , it is necessary and --f
suflcient that both f l and f 2 be continuous at zo . Let xo = ( f i ( z o ) f2(z0)); , then we have
by (3.20.1),and the result follows from (3.11 .I) and (3.6.3).
(3.20.5) Let f = ( , f l , f , ) be a mapping of a subspace A of a metric space F into El x E,, and let a EA; in order that f haoe a limit at the point a
with respect to A, a necessary and suflcient condition is that both limits b, = lirn f,(z), b , = lirn f2(z) exist, and then the limit of f is =-+a,z
z-a, z E A
b = (h,b2).
E
A
This follows at once from (3.20.4)and the definition of a limit. In particular:
(3.20.6) In order that a sequence of points z, = (x,,, y,) in E = El x E, be confiergent, a necessary and suflcient condition is that both limits a = lirn x,, b = lim y,, exist and then lim z, = ( a , b). n+
m
?I-+ m
n+ m
Note that for cluster values of sequences, if (a, b) is a cluster value of a is a cluster value of (x,,) and b a cluster value of (y,,), as follows from (3.20.6)and the definition of cluster values; but it may happen that ((x,, y,)) has no cluster value, although both (x,,) and (y,,) have one: for instance, in the plane R2, take x,, = l / n , y,, = n, x,,+~= n, y2n+l= l/n. However, if (x,) has a limit a, and b is a cluster value of(y,,), then (a, b) is a cluster value of ( ( x , , y,)), as follows from (3.20.6). ( ( x , , y,)),
74
I l l METRIC SPACES
(3.20.7) In order that a sequence of points z, = (x,, y,) in El x E, be a Cauchy sequence, a necessary and sufficient condition is that each of the sequences (x,), (y,) be a Cauchy sequence. This follows at once from the definition of the distance in El x E2 and the definition of a Cauchy sequence.
(3.20.8) Let z -+f(z) = ( f i ( t ) ,f2(z)) be a mapping of a metric space F into El x E,; in order that f be uniformly continuous, it is necessary and sufficient that both fland f2 be uniformly continuous. This follows immediately from the definitions.
(3.20.9) I f E is a metric space, d the distance on E, the mapping d of E x E into R is uniformly continuous. For Id(x, y ) - d(x’, y’)l
< d(x,x’) + d(y, y’) by the triangle inequality.
(3.20.10) Theprojectionspr, andpr,areuniformlycontinuous in E = El x Ez . Apply (3.20.8)to the identity mapping of E.
(3.20.11) For any a, E E2 (resp. a, E El), the mapping x1 +(xl, a,) (resp. x, 4(al, x,)) is an isometry of El (resp. E,) on the closed subspace El x {a,} (resp. {all x E2) of El x E, * This is an obvious consequence of the definition of the distance in El x E,, and of (3.20.3).
(3.20.12) For any open (resp. closed) set A in El x E, , and any point a , E El, rhe cross section A(al) = pr2(A n ({al} x E,)) is open (resp. closed) in E, .
By (3.20.11)it is enough to prove that the set A
n ({a,} x E,) is open
(resp. closed) in {al} x E z , which follows from (3.10.1)and (3.10.5).
(3.20.13) For any open set A in El x E,, pr, A (resp. pr, A) is open in El (resp. Ez).
20 PRODUCT OF TWO METRIC SPACES 75
Indeed, we can write pr, A =
u
XI
A(x,), and the result follows from
EEI
(3.20.12) and (3.5.2). Note that if A is closed in El x E,, pr, A needs not be closed in El. For instance, in the plane R2,the hyperbola of equation x y = 1 is a closed set, but its projections are both equal to the complement of (0) in R, which is
not closed. Let f be a mapping of E = El x E, into a metric space F. I f f is continuous at a point (al,a,) (resp. uniformly continuous), then the mapping f ( . , a,): x, --f f (x,, a,) is continuous at a, (resp. uniformly continuous). (3.20.14)
That mapping can be written x1 4(x,, a,) -+f ( x l ,a,), hence the result follows from (3.20.11), (3.11.5), and (3.11.9). The converse to (3.20.14) does not hold in general. A classical counterexample is the function f defined in R2 by f (x, y ) = xy/(x2 y 2 ) if ( x ,y ) # (0,O)andJ(0,O) = 0;f is not continuous at (0, 0), forf ( x ,x ) = 1/2 for x # 0.
+
Let El, E 2 , F,, F, befour metric spaces,f , (resp. f,)a mapping of El into F1(resp. of E, into F,). In order that the mappingf:(xL,x 2 )+ (f,(x,),f,(x,)) of El x E, into F, x F, be continuous at a point (a,,az) (resp. uniformly continuous), it is necessary and su$cient that f , be continuous at a, and f , at a, (resp. that both f, and f , be uniformly continuous). (3.20.15)
The mapping (x,, x,) +fi(xl) can be written f i 0 prl, hence the sufficiency of the conditions follow from (3.20.4), (3.20.8), and (3.20.10). On the other hand, the mapping fi can be written x, -+ pr,( f ( x , , a,)) and the necessity of the conditions follows from (3.20.14) and (3.20.10). (3.20.16) Let El, E, be two nonempty metric spaces. In order that E = El x E,
be a space of one of the following types: (i) discrete; (ii) bounded; (iii) separable; (iv) complete; (v) compact; (vi) precompact; (vii) locally compact; (viii) Connected; (ix) locally connected; it is necessary and suficient that both El and E2 be of the same type.
76
I l l METRIC SPACES
The necessity part of the proofs follows a general pattern for properties (i) to (vii): from (3.20.11) it follows that El and E, are isometric to closed subspaces of El x E,; and then we remark that properties (i) to (vii) are “inherited” by closed subspaces (obvious for (i) and (ii), and proved for properties (iii) to (vii) in (3.10.9), (3.14.5), (3.17.3), (3.17.4), (3.18.4)). For property (viii), the necessity follows from (3.19.7) applied to the projections pr, and pr,; similarly, if E is locally connected, for any (a,, a 2 )E E and any neighborhood V, of a, in El, V, x E, is a neighborhood of (a,, u2), hence contains a connected neighborhood W of (a,, a,); but then pr, W is a connected neighborhood of a, contained in V,, by (3.19.7) and (3.20.13). The suficiency of the condition for (i) and (ii) is an obvious consequence of the definition of the distance in E, x E, . For (iii), if D,, D, are at most denumerable and dense in E,, E, respectively, then D, x D, is at most denumerable by (1.9.3), and is dense in E by (3.20.3). For (iv), if (z,) is a Cauchy sequence in E, then (pr, z,) and (pr, z,) are Cauchy sequences in El and E, respectively by (3.20.7), hence they converge to a,, a2 respectively, and therefore (z,) converges to (a,, a*) by (3.20.6). For (vi), if (A,) (resp. (B,)) is a finite covering of El (resp. E2) by sets of diameter < c , then ( A i x B,) is a finite covering of El x E, by sets of diameter < E ; and by (3.16.1), the sufficiency of the condition for (iv) and (vi) proves it also for (v). The proof for (v) yields a proof for (vii) if one remembers the definition of neighborhoods in El x E,. For (viii), let (a,, a2), (b,, b 2 ) be any two points of E; by (3.20.11) and the assumption, the sets { a , } x E, and El x {b,} are connected and have a common point (a,, b,). Hence their union is connected by (3.19.3), and it contains both (a,, a,) and (b,, b 2 ) ;therefore, the connected component of (al, a 2 ) in E is E itself. The same argument proves the sufficiency of the condition for (ix), remembering the definition of the neighborhoods in E. (3.20.17) I n order that asubset A of El x E, be relatively conlpacr, a )iecessary and suficient condition is that prl A ai7d pr, A be relatioely compact ii7 El ai7d E, respectively. The necessity follows from (3.17.9) applied to pr, and pr,; the sufficiency follows from (3.20.16), (3.20.3) and (3.17.4).
All definitions and theorems discussed in this section are extended at once to a finite product of metric spaces.
20
PRODUCT OF TWO METRIC SPACES
77
PROBLEMS
1. Let E, F be two metric spaces, A a subset of E, B a subset of F ; show that fr(A x B) = (fr(A) x B) u (A x fr(B)). 2. Let E, F be two connected metric spaces, A # E a subset of E, B # F a subset of F; show that in E x F the complement of A x B is connected. 3. (a) Let E, F be two metric spaces, A (resp. B) a compact subset of E (resp. F). If W is any neighborhood of A x B in E Y F, show that there exists a neighborhood U of A in E and a neighborhood V of B in F such that U x V c W (consider first the case in which B is reduced to one point). (b) Let E be a compact metric space, F a metric space, A a closed subset of E x F. Show that the projection of A into F is a closed set (use (a) to prove the complement of prz A is open). (c) Conversely, let E be a metric space such that for every metric space F and every closed subset A of E x F, the projection of A into F is closed in F. Show that E is compact. (If not, there would exist in E a sequence (x,) without a cluster value. Take for F the subspace of R consisting of 0 and of the points l / n (n integer 2 1) and consider in E Y F the set of the points ( x n , lit?).) 4. Let E be a compact metric space, F a metric space, A a closed subset of E x F, B the (closed) projection of A into F. Let yo E B and let C be thecross section A-’(yo) = { x E E I ( x , y o ) E A]. Show that for any neighborhood V of C in E, there is a neighborhood W of yo in F such that the relation y c W implies A-’(y) C V (“continuity of the “roots” of an equation depending on a parameter”). (Use Problem 3(a).) 5. (a) Let f be a mapping of a metric space E into a metric space F, and let G be the graph off’in E x F. Show that iff is continuous, G is closed in E x F, and the restriction of pr, to G is a homeomorphism of G onto E. (b) Conversely, if F is compact and G is closed in E x F, then f is continuous (use Problem 3(b)). (c) Let F be a metric space such that for any metric space E, any mapping of E into F whose graph is closed in E x F is continuous. Show that F is compact (use the construction of Problem 3(c)). 6. Let E, F, G be three metric spaces, A a subset of E x F, B a subset of F x G, C = B 0 A = {(x, z ) E E x G j3y E F such that (x, y ) E A and (y, z) E B}. Suppose bothA and B are closed and the projection of A into F is relatively compact; showthat C is closed in E x G (use Problem 3(b)). 7. Let (En) ( n 2 I ) be an infinite sequence of nonempty metric spaces, and suppose that for each ti, the distance d,, on En is such that the diameter of E. is < 1 (see Section 3.14,
Problem 2(b)). Let E be the infinite product
PIXI
En. m
(a) Show that on E the function d((xn),( y o ) )=
“=I
d,(x., y.)/2” is a distance.
(b) For any x = (x.) E E, any integer m >, 1 and any number r > 0, let VJx; r ) be the set of all y = (y.) E E such that & ( x k , yk) < r for k < m. Show that the sets V,,(x;r ) (for all m and v) form a fundamental system of neighborhoods of x in E. (c) Let (x‘”’))be a sequence of points x(‘”)= (x,!“’)).~~ of E; in order that ( x c m )converge ) to a = (a,) in E (resp. be a Cauchy sequence in E), it is necessary and sufficient that for each 12 the sequence (x!””).,~I converge to a. in En (resp. be a Cauchy sequence in En). In order that E be a complete space, it is necessary and sufficient that each En be complete.
78
Ill
METRIC SPACES r
(d) For each n , let A, be a subset of E, ; show that the closure in E of A equal to
fi A,.
=
IT A,, is
"=I
(e) In order that E be preconipact (resp. compact), i t is necessary and sufficient that each En be preconipact (resp. compact). (f) In order that E be locally compact, i t is necessary and sufficient that each En be locally compact, and that all En, with the exception of a finite number at most, be compact. ( 8 ) In order that E be connected, i t is necessary and sufficient that each E. be connected. (11) In order that E be locally connected, it is necessary and sufficient that each E,, be locally connected and that all Em,with the exception of a finite number at most, be connected.
CHAPTER I V
ADDITIONAL PROPERTIES OF THE REAL LINE
Many of the properties of the real line have been mentioned in Chapter 111, in connection with the various topological notions developed in that chapter. The properties gathered under Chapter IV, most of which are elementary and classical, have no such direct connection, and are really those which give to the real line its unique status among more general spaces. The introduction of the logarithm and exponential functions has been made in a slightly unorthodox way, starting with the logarithm instead of the exponential; this has the technical advantage of making it unnecessary to define first amin(117, n integers > 0) as a separate stepping stone toward the definition of ax for any x. The Tietze-Urysohn theorem (Section 4.5) now occupies a very central position both in functional analysis and in algebraic topology. It can be considered as the first step in the study of the general problem of exfendirlg a continuous mapping of a closed subset A of a space E into a space F, to a continuous mapping of the whole space E into F; this general problem naturally leads to the most important and most actively studied questions of modern algebraic topology.
1. C O N T I N U I T Y OF ALGEBRAIC OPERATIONS
+ y of R x R into R is uniformly continuous. This follows at once from the inequality
(4.1.I) The mapping (x,y ) -+ x
+ < Ix’ - XI + ly’ - yl
I(x’ + y’) - ( x y)l and the definitions.
79
IV ADDITIONAL PROPERTIES OF THE REAL LINE
80
(4.1.2) The mapping (x, y ) -+ xy of R x R into R is continuous; for any a E R, the mapping x -+ a x of R into R is uniformly continuous. Continuity of xy at a point (xo, yo) follows from the identity XY - XOYO = Xo(Y
- Y o ) + (x - X 0 ) Y O
+ (x - X O ) ( Y
- Yo).
Given any E > 0, take 6 such that 0 < 6 < 1 and 6(1x01 + lyol + 1) < E ; then the relations Ix - xo( < 6, ly - yoJ < 6 imply [xy - ?coyo[< c. Uniform continuity of x -+ax is immediate, since [ax' - ax1 = J a J. Jx'- XI.
(4.1.3) Any continuous mapping f of R irito itsdf such that f(x +f ( y ) is of type x -+ cx, with c E R.
+ y) =
f(x)
Indeed, for each integer n > 0, we have, by induction on n, f ( n x ) = nf(x); on the other handf(0 + x) =f ( 0 ) + f ( x ) , hencef(0) = 0, and f ( x + (-x)) = f ( x ) f ( - x ) = f ( 0 ) = 0, hence , f ( - x ) = -f(s). From that it follows that for any integer n > 0, f ( x / n ) = f ( x ) / n , hence for any pair of integers p , q such that q > 0, f ( p x / q ) =!f'(x)/q; in other words, f ( r x ) = rf(x) for any rational number r. But any real number t is limit of a sequence (r,) of rational numbers (by (2.2.16) and (3.13.13)),hence, fromtheassumptionon,fand (4.1.2) f ( t x ) = f(lim r,x) = limf(r,x) = lim r,f'(.x) = f ( x ) . lim r, = t f ( x ) . Let then,
+
n-1 a0
n-tm
n-tm
c = . f ( l ) , and we obtainf(x) = c x for every
(4.1.4)
x E R.
n+
cc
The mapping x -+ l/x is continuous at ecerj, poitit xo # 0 in R.
For given any E > 0, take 6 > 0 such that 6 < min(lxol/2, ~Ix,[~/2); then the relation Jx-xoJ < 6 first implies 1x1 > Jxol- 6 > 1x01/2, and then Il/x - l/xol = 1x0 - xl/lxx,I Q 21x0 - xl/lxo[2< E .
Any rational function (xl, . . . , x,) -+ P(xl, . . . , x,)/Q(x,, . . . , x,) where Q are polynomials with real coeficients, is continuous at each point (al, . . . , a,,) qf R" where Q(a,, . . . , a,) # 0. (4.1.5)
P
and
The continuity of a monomial in R" is proved from (4.1.2) by induction on its degree, then the continuity of P and Q is proved from (4.1.1) by induction on their numbers of terms; the final result follows from (4.1.4).
1
(4.1.6) The inappings
continuous in R x R.
CONTINUITY OF ALGEBRAIC OPERATIONS
(x,y ) + sup(x, y ) and
81
(x, y ) --+ inf(x, y ) are unifornzly
+ +
Ix - y ( ) / 2 and inf(x, y ) = (x As sup(x, y ) = (x y the result follows from (4.1.I) and (3.20.9).
+ y - (x - y ( ) / 2 ,
(4.1.7) All open intervals in R are homeomorphic to R.
From (4.1 .I .) and (4.1.2) it follows that any linear function x -+ a x + b, with a # 0, is a homeomorphism of R onto itself, for the inverse mapping x - i a - ' x - a-'b has the same form. Any two bounded open intervals ]LY, /?[, ] y , 6[ are images of one another by a mapping x --+ax+ b, hence are homeomorphic. Consider now the mapping x --t xi( I + 1x1) of R onto 1- 1, I[; the inverse mapping is x xi( 1 - 1x1) and both are continuous, since x --t 1x1 is. This proves R is homeomorphic to any bounded open interval; finally, under the preceding homeomorphism of R onto ] - I , + 1[, any unbounded open interval ]a, + a[or ] - co, a[ of R is mapped onto a bounded open interval contained in ] - I , + 1 [, hence these intervals are also homeomorphic to R .
+
--f
+
(4.1.8) With respect to R x R, the ,fuiiction (x, y ) x y has a lilllit at euerypoint (a, b) g f R x R, except at the points (- GO, + co) and ( + co, - 00); that litnit is equal to + cc (resp. - 00) if' one at least of tl7e coordinates a, b is +co (resp. -m). -+
Let us prove for instance that if a # - co, x + y has a limit equal to +co at the point (a, + co). Given c E R, the relations x > b, y > c - b imply x + y > c, and the intervals ]b, + co] and ]c - 6, + co] are respectively
neighborhoods of a and + GO if b is taken finite and < a ; hence our assertion. The other cases are treated similarly.
With respect to R x R, the function (x, y ) xy has a limit at every point (a, 6 ) qf R x 8, except at the points (0, +a),(0, -a),( f a , 0), (- co,0 ) ;that litnit is equal to + co (resp. - 00) if one at least of the coordinates a, b is injnite, and ifthey have the same sigii (resp. opposite signs). (4.1.9)
-+
Let us show for instance that if a > 0, xy has the limit + co at the point (a, co). Given c E R, the relations x > b, y > c/b, for b > 0, imply xy > c, and the intervals ]b, + co] and ]c/b, + GO] are neighborhoods of a and + 00, if b is taken finite and 0
lim
l/x = -co.
x+o,x - co, for any b < c, thereisx E Esuchthatb 0; from which it follows that f is strictly increasing, since if x < y , then y = zx with z > 1 andf(y) = f ( x ) + f ( z ) > f ( x ) . On the other hand, we have the following lemma: (4.3.1.2) For any integer n 2 1, there is a z > I such that Z" 6 a.
Remark that there is an x such that 1 < x < a, hence a = xy with y > I ; if z1 = min(x, y ) , we have z: 6 x y = a and z1 > 1. By induction define z, > I such that z: < z , , - ~ ,hence z:" 6 a, and a fortiori z l < a. The lemma shows that 0 - : x
then I f ( y ) -f(x)I < f ( z ) 6 l / n for ly - XI , ~(s,),~ , is called a series if the elements x, ,s, are linked by the relations s, = xo x1 * . x, for any n, or, what is equivalent, by xo = s o , x, = s, - snV1 for n b 1 ; x, is called the nth term and s, the nth partial sum of the series; the series will often be called the series of general term x,, or simply the series (x,)
+ +
(and even sometimes, by abuse of language, the series is said to converge to s if lim s, n-+ m
and written s = xo + ...
= s; s is
+ x, + . * . or
+
m
1x,).
The series
n=O
then called the sum of the series m
s=
1x,; r, = s - s,
is called the
n=O
nth remainder of the series; it is the sum of the series having as kth term x , + ~ ;by definition lim r, = 0. n+ m
(Cauchy's criterion) Ifthe series ofgeneral term x, is convergent, E > 0 there is an integer no such that, f o r n 3 no and p k 0, IIs,+,, - s,II = I I X , + ~ + X , , + ~ I I < E . Conversely, i f that condition is satis-ed and if the space E is complete, then the series of general term x, is convergent.
(5.2.1)
then f o r any
+
This is merely the application of Cauchy's criterion to the sequence (s,) (see Section 3.14). As an obvious consequence of (5.2.1)it follows that if the series (x,) is convergent, lim x, = lim (s, - s , , - ~ )= 0; but that necessary condition n-t m
n-+ m
is by no means sufficient.
96
V
NORMED SPACES
(5.2.2) I f the series (x,) and are convergent and have sunis s, s', then the series (x, + x@ converges to the siini s + s' and the series (Ax,) to the sum As f o r any scalar A. (XI,)
Follows at once from the definition and from (5.1.5).
(5.2.3) If (x,) and (x;) are ~ M ' Oseries such that xi = x, except f o r a finite number of indices, they are both concergent or both nonconcergent. For the series (x; - x,) is convergent, since all its terms are 0 except for a finite number of indices.
(5.2.4) Let (k,) be a strictly increasing sequence of integers 3 0 with ko = 0 ;
1 xp, then the series (y,) conrerges
kn+i-l
ifthe series (x,) coilverges to s , and ij-y, =
p=k,
also to s. This follows at once from the relation
n
k,, + I - I
i=O
j=O
1yi = 1 x j and from (3.13.10).
PROBLEMS
Let (a,) be an arbitrary sequence in a normed space E ; show that there exists a sequence 0, and a strictly increasing sequence (kJ of (x,) of points of E such that lim x,
+ + + "-0
integers such that (I= , xo x1 . . . x x nfor every n. Let u be a bijection of N onto itself, and for each n , let ~ ( nbe) the smallest number of intervals [a, 61 in N such that the union of these intervals is cs([O, n]) (a) Suppose p is boimcied in N. Let (x,) be a convergent series in a norined space E; show that the series (x0(J is convergent in E and that
m
"=a
m
x,
=
n=o
xSCn).
(b) Suppose p is unbounded in N. Define a series (x,) of real numbers which is convergent, but such that the series (rS(,,Jis not convergent in R . (Define by induction on k a strictly increasing sequence (mk)of integers having the following properties: (1) If n, is the largest element of u([O, i n x ] ) ,then [0, n k ] is contained in u([O, mk+,]). (2) p(md > k 1. Define then x,, for nk < n < n k + , such that x. = 0 except for 2k conveniently chosen values of n , at each of which x,, is alternately equal to Ilk or to - l / k . ) Let (x,) be a convergent series in a normed space E ; let u be a bijection of N onto itself, and let r(n) = lu(n) - nl . sup \\xn,\\.
+
man
3 ABSOLUTELY C O N V E R G E N T SERIES Show that if lirn r ( n ) = 0, the series (xO{"))is convergent in E and that n-
m
c x0("))- c xy for large n.)
m
97
c xO("). m
n=O
x, =
n=O
n
(Evaluate the difference
k=O
k=O
Let (x,J (m> 0, n > 0) be a double sequence of points of a normed space E. Suppose that: ( I ) for each m > 0, the series x , , ,-t ~ x , , , ~ . . . x,, . . . is convergent in E; .. . ; ( 2 ) for each n > 0, the series let y,, be its sum, and let Y,,,,= x,,,, + x ,,,," TO" YI" . . Y,,,, . . is convergent in E; let t . be its sum. x,, . is convergent; let (a) Show that for each n > 0, the series xOn xI, . z, be its sum.
+
+
+
+ +. + +.
(b) In order that
c
+
+ +. +
y,,, =
,n=0
(a) Show that theseries
+..
c z,, it is necessary and sufficient that lim t, 0. 1 is convergent and has a sum equal to c m2-n2
n=o
n-m
=
-
3/4m2
n31,n;fm
(decompose the rational fraction l/(m' - x')). 1 (b) Let u,"" = -if m # n, and N,,= 0; show that m' - nz
5(2
n = o U,")
n,=0
E( f
= - n=o
, = o U",") # 0.
I f f is a function defined in N x N, with values in a metric space, we denote by f ( m , n ) the limit o f f (when it exists) at the point (+to, to) of a x 8, with lirn
+
m-m."-m
respect to the subspace N x N (Section 3.13). Let (x,") be a double sequence of real numbers, and let s,, = C XIk. h 6 m , kQn
(a) If
lim s,,,,exists, then
m-m,n-a,
lirn
m-m,n-+m
xmn= 0. Give an example in which xnn,= x m n ,
~ , ~ , ~ ~ = - x , ~ , ~ ~ + ~ = - ~ , ~ + ~ , ~ ~ f o r m ~ 2 n +that 1 , x lirn ~ . , ~ sm,=O, .=O,such
+
+
+
+ +
+.
m-rm,n-tm
+
+.
and none of the series xmz0 xml . . . x,." . . ,xOn x l , f.. . x,,, . * is convergent. (b) Give an example in which xnvn= 0 except if m = n 1 , m = n or n = rn 1
c x,,,,, m
(hence all series m, n, but
lirn
",+ m , n-
n=O m
,,s
m
,n=o
x,,,, are convergent),
c x,, c m
m
=
"=O
,=O
+
=0
for all indices
does not exist.
3. A B S O L U T E L Y C O N V E R G E N T SERIES
(5.3.1) In order that a series (x,) of positive numbers be convergent it is necessary and sufJjcie17t that for a strictly increasing sequence (k,) of integers
1x, W
3 0, the sequerice (sk,,) of partial sums be majorized, and then the sum s = is equal to sup sk,.
n=O
n
The assumption x, 3 0 is equivalent to s , - ~ < s, follows at once from (4.2.1).
and then the result
98
NORMED SPACES
V
In a Banach space E, an absolutely convergent series (x,) is a series such that the series of general term llxnll is convergent.
(5.3.2) In a Banach space E , an absolutely convergent series (x,) is convergent,
By assumption, for any and anyp 2 0, IIxn+lII
+
E
> 0, there is an integer no such that for n 2 no
+ IIX,,+~II
0, let no be such that IIxn-lll + + I I X , , + ~ \ ~ < E for n 2 no and p 2 0; then if m, is the largest integer in a-'([O, no]), we have (Iyn+lll+ + Ilyn+pll< E for n 2 m,, p 2 0; furthermore the difference sko - s, is the sum of terms xi withj > n o , hence IIsko - sno\l< E ; therefore, for n 2 no and n 2 m,, 11s; - snII < 3 ~ , which proves that
c x, m
n=O
a,
= n=O
y, .
Let A be any denumerable set. We say that a family ( x , ) , ~ of ~ elements of a Banach space E is absolutely summable if, for a bijection cp of N onto A, the series (xlPcn,) is absolutely convergent; it follows from (5.3.3) that this property is independent of the particular bijection cp, and that we can define the sum of the family (x,),,~ as
c xlP(,,),which we also write c x u . As any m
n=O
asA
denumerable set S c E can be considered as a family (with S as the set of indices) we can also speak of an absolutely summable (denumerable) subset of E and of its sum.
3 ABSOLUTELY CONVERGENT SERIES
99
(5.3.4) In order that a denumerable family ( x , ) , , ~of elements of a Banach space E be absolutely summable, a necessary and sufJicient condition is that )Ix,JI (J c A andfinite) be bounded. Then, for any E > 0, the jinite sums
1
U E l
there exists a jinite subset H of A such that, for any jinite subset K of A for which H n K = 0, IIxull < E , and for any jinite subset L 3 H of A,
1
The first two assertions follow at once from the definition and from (5.3.1). Then, for any finite subset L 2 H, we can write L = H u K with H n K = 0, hence 1) x x U - E x a l l < E ; from the definition of the sum aeH
aeL
1xu it follows (after ordering A by an arbitrary
bijection of N onto A)
(5.3.5) Let ( x , ) , , ~be an absolutely summable family of elements of a Banach space E. Then,jor every subset B of A, the family(xa),eBisabsolutely summable, and C IIxaII G E IIxaII. UEA
U E B
If B is finite, the result immediately follows from the definition. If B is infinite, then llxall < 1 Ilx,Il for each finite subset J of B, and the
x
Or01
result follows from (5.3.4).
aeA
(5.3.6) Let (x,),,~be an absolutely summable family of elements of a Banach space E. Let (B,) be an injinite sequence of nonempty subsets of A, such that A= B,, and B, n B, = 0for p # q ; then, if z, = 1 x u , the series (z,)
u n
is absolutely convergent, and convergent series).
k
m
1z, = C x,
n=O
ueB,
(“associativity”
of absolutely
aeA
Given any E > 0 and any integer n, there exists, by (5.3.2), for each a finite subset J, of B, such that llZkI/ < llxull + E / ( H 1); if
J=
n
+
E
< n,
asJk
J k , we have therefore
1 Ilz,(l < 1llxull + n
k=O
k=O
aE l
E
G
1 IIxaII 4-
aeA
E;
(5.3.1)
then proves that the series (z,) is absolutely convergent. Moreover, let H be a finite subset of A such that, for any finite subset K of A such ( ( E , whence, for any finite subset L of A containthat H n K = 0, ( ( x a < ing H,
11 1xu aeA
c x,I aeK
aEL
< 2~ (see (5.3.4)). Let 170
be the largest integer such
100
V
NORMED SPACES
that H n B,, # @, and let n be an arbitrary integer 2 n o . For each k < n, let Jk be a finite subset of B, containing H n B,, and such that for any finite subset Lk of B, containing J,, we have llzk - C x,II < d ( n 1) (5.3.4). Then, if L =
u Lk,we have /I
from the definition of H that
I
1, let B. be the set of x E Bn-l such that IIx - yll < 6(B,-,)/2 for all y E Bn-l(&A) being the diameter of a set A). Show that 6(B,) < 6(B.-,)/2, and that the intersection of all the B. is reduced to (a b)/2. (b) Deduce from (a) that iffis an isometry of a real normed space E onto a real normed space F, thenf(x) = u(x) c, where u is a linear isometry, and c E F. 4. Let us call rectangle in N x N a product of two intervals of N; for any finite subset H of N x N, let $(H) be the smallest number of rectangles whose union is H. Let (H.) be an increasing sequence of finite subsets of N x N, whose union is N x N and such that the sequence ($(H.)) is bounded. Let E, F, G be three normed spaces, (x.) (resp. (y,)) a convergent series in E (resp. F),fa continuous bilinear mapping of E x F into G. Show that
+
+
(*)
5.
lim
n-m
C
0 such that: (1) the relations JIxJJ G 6, IIx'II Q 6 , IIx x'll Q 6 in E implyf(x x') = f ( x ) +f(x'); (2) the relations llxll 6 6, llhxll G 6 in E (with h E R) imply f(hx) = hf(x).
+
+
106
V
NORMED SPACES
(a) Show that iffsatisfies condition (1) and is continuous at the point 0, it is continuous in a neighborhood of 0 and is linear in a neighborhood of 0 (cf. Problem 1). (b) Let g be a mapping of E into F; in order that g be continuous at the point 0 and linear in a neighborhood of 0, a necessary and sufficient condition is that for any convergent series (x.) in E, the partial sums of the series (g(x.) be bounded in F. (To prove sufficiency, first observe that one must have g(0) = 0; if, for every n, there exist three elements u., v., w. of E such that llunll< 2-", llvnll < 2-", llwnll < 2-", u,, v. w. = 0 and g(uJ g(u,) +g(w.) # 0, form a series (x,) violating the assumption. If there are no such sequences u,, v., w., g verifies condition (1); show that it is necessarily continuous at 0.)
+ +
+
6. EQUIVALENT NORMS
Let E be a vector space (over the real or the complex field), llxlll and llxll, two norms on E; we say that llxlll isfiner than llxll, if the topology defined by (IxI(lis finer than the topology defined by IJxJI,(see Section 3.12); if we note El (resp. E,) the normed space determined by llxlll (resp. Ilxl12), this means that the identity mapping x + x of El into E, is continuous, hence, by (5.5.1), that condition is equivalent to the existence of a number a > 0 such that IIxIJ2< a - IJxJI1. We say that the two norms IIxII1, I(xI(, are equivalent if they define the same topology on E. The preceding remark yields at once: (5.6.1) In order that the two norms IIxlll, ((x((, on a vector space E be equivalent a necessary and sufficient condition is that there exist two constants a > 0,
b >O, such that allxll1
for any x E E.
< llxllz d bllxll1
The corresponding distances are then uniformly equivalent (Section 3.14). For instance, on the product El x E, of two normed spaces, the norms sup(IIxlII, IIx~II)~IIxlII + IIx211, (IIxlI12 + I I X ~ I I ~ ) are " ~ equivalent. On the space E = %?&), the norm l l f l l l defined in (5.1.4) is not equivalent to the norm Ilfll, = suplf(t)l (see Section 5.1, Problem 1). re1
7. SPACES O F C O N T I N U O U S MULTILINEAR MAPPINGS
Let E, F, be two normed spaces; the set Y(E; F) of all continuous linear mappings of E into F is a vector space, as follows from (5.1 S),(3.20.4), and (3.11.5).
7 SPACES OF CONTINUOUS MULTILINEAR MAPPINGS
107
For each u E Y(E; F), let IJuJI be the g.1.b. of all constants a > 0 which satisfy the relation IIu(x)II < a * llxll (see (5.5.1)) for all X. We can also write
(5.7.1) For by definition, for each a > IIuII, and llxll < I , IIu(x)I( 0, llxll 6 1
for any b such that 0 < b < IIuII, there is an x E E such that IIu(x)I( > bllxll; this implies x # 0, hence if z = x/llx/l, we still have IIu(z)II > b llzll = b, and as llzll = 1, this proves that b < sup IIu(x)I(,hence I(u(I< sup IIu(x)(I, 9
llxll 6 1
llxll 1
and (5.7.1)is proved. The same argument also shows that if E # (0)
(5.7.2) We now show that /lull is a norm on the vector space Y ( E ; F). For if u = 0, then IJuJJ= 0 by (5.7.1),and conversely if llull = 0, then u(x) = 0 for llxll ,< 1, hence, for any x # 0 in E, u(x) = llxllu(x/llxll)= 0. It also follows from (5.7.1) that IIAull = 111 * IJu/I; finally, if w = u u, we have II4dI IIu(x)II llu(x>II,hence llwll llull llull from (5.7.1).
1 ; then Ilx/aIl = (l/lctl)llx\l < r , andf(x/a) = 1, which contradicts the definition of V. By homogeneity and (5.5.1) it follows that f is continuous. If b $ H, we have x = g(x)b + y with y E H for each x E E, and g ( x ) = 0 is another equation of H ; hence g is continuous, and the mapping x + g ( x ) b of E into D = R b (resp. Cb) is therefore continuous, which proves the last part of (5.8.1) by (5.4.2).
+
(5.8.2)
For
In a normed space E, a hyperplane H is either closed or dense.
is a vector subspace (by (5.4.1)) which can only be E or H.
9 FINITE DIMENSIONAL NORMED SPACES
111
PROBLEMS
1. Let E be the (noncomplete) subspace of the space (co) of Banach, consisting of the sequences x = (6.) of real numbers, having only afinire number of terms different from 0. For any sequence (an)of real numbers, the mapping x -+ u(x) =
c m
En
GC. is
a linear
"=O
form on E, and all linear forms on E are obtained in that way; which of them are continuous (see (5.5.4) and Problem 1 of Section 5.7)? 2. (a) In a real normed space E, let H be the closed hyperplane of equation u(x) = 0, where u is a continuous linear form. Show that for any point a E E, the distance 4 a , n = 144 I illrrll. (b) In the space (co) of Banach, let H be the closed hyperplane of equation u(x) =
c 2-"& m
= 0;
if a # H, show that there is no point b E H such that d(a, H) = d(a, b).
"=O
3. In a real vector space E, the linear varieties of codimension 1 (Section 5.1, Problem 5) are again called hyperplanes; they are the sets defined by anequation ofthe type u(x) = a, where u is a linear form, a any real number; the hyperplanes considered in the text are those which contain 0, and are also called homogeneous hyperplanes; any hyperplane defined by an equation u ( x ) = a is said to be parallel to the homogeneous hyperplane defined by u(x) = 0. If A is a nonempty subset of E, a hyperplane of support of A is a hyperplane H defined by an equation u(x) = a , such that u(x) - a > 0 for all x E A, or u(x) - u < 0 for all x E A, and u ( x o ) = a for at least one point xo E A. (a) In a real normed space E, a hyperplane of support of a set containing a n interior point is closed (see (5.8.2)). (b) Let A be a compact subset of a real normed space E; show that for any homogeneous closed hyperplane H, defined by the equation u ( x ) = 0, there are two hyperplanes of support of A which are defined by equations of the form u(x) = a, and may eventually coincide; their distance is at most equal to the diameter of A. (c) In the space (c,) of Banach, consider the continuous linear form x + u ( x ) = m
2-"5, ; show that the closed ball B'(0; 1) has no hyperplane of support having an
"=O
equation of the form u ( x ) = a (cf. Problem 2(b)).
9. F I N I T E D I M E N S I O N A L N O R M E D SPACES
(5.9.1) Let E be an n-dimensional real (resp. complex) normed vector space; q(a,, . . . , a,) is a basis of E, the mapping
(51, of
* *
1
> Cn)
4
+
5 1 ~ 1
* * *
+ ("an
R" (resp. Cn) onto E is bicontinuous.
We use induction on n, and prove first the result for n = 1. We know by (5.1.5) that 5 + [al is continuous; as a, # 0 and IItalll = llalI1 * 151, we have 151 < (l/llalII) * \15al\\,which proves the continuity of 5al + 5, by (5.5.1).
112
V
NORMED SPACES
Suppose the theorem is proved for n - I , and let H be the hyperplane in E generated by a,, . . . , a,,-l; the inductive assumption implies that the norm on H (induced by that of E) is equivalent to the norm sup Iti]; I Ci 0
by (5.9.2)), take p.+, such that Ipn+,I* ~ ~ u n 0) is homeomorphic to the product space of the interval [a, b] and of the sphere S : Ilxll = 1; use then Riesz’s theorem (5.9.4).) 4. Let E be a real normed space of finite dimension n , and let A be a compact subset of E. With the notations of Section 3.16, Problem 4, show that there exists a constant a > 0 such that N,(A) < a . ( l / ~ ) ” if; in addition A has an interior point, show that there exists a constant h > O such that M,,(A)> b . ( I / & ) ” ; in that case, one has H,(A) , . C,(A) n log I / & . 5. Let E be a real normed space of infinite dimension, and let (En) be a strictly increasing sequence of vector subspaces of E of finite dimension. Show that there exists a sequence (x.) of points of E such that x , E En, ~ ~ =xI and n ~d(x,, ~ En-,) = 1. Let be a function > O defined for f 2 1 , strictly decreasing and such that lim ( @ ) = O , and let z,=
-
t-+m
2 9 ( n ) x , . If K is the compact subset of E consisting of 0 and the z , , show that M,(K) 2 # ( E ) , where # is the inverse function of rp (defined in a neighborhood of 0); M,(K) can thus increase arbitrarily fast when I/& tend to +a.
114
V
NORMED SPACES
10. SEPARABLE NORMED SPACES
(5.1 0.1) I f i n a normed space E there exists a total sequence (Section 5.4),E is separable. Conversely, in a separable normed space E, there exists a total sequence consisting of linearly independent vectors.
Suppose (a,,)is a total sequence, and let D be the set of all (finite) linear combinations rlal . rnanwith rational coefficients (when E is a complex vector space, by a "rational" scalar we mean a complex number c1 i j , with both c1 and fi rational). D is a denumerable set by (1.9.3) and (1.9.4). As by definition the set L of all linear combinations of the a,, is dense in E, all we have to prove is that D is dense in L, and as
+
+
+
this follows from (2.2.16). Suppose conversely E is separable; we can of course suppose E is infinite dimensional (otherwise any basis of E is already aJinite total subset). Let (a,,)be an infinite dense sequence of vectors of E. We define by induction a subsequence (a,")having the property that it consists of linearly independent vectors and that for any m < k, , a,,, is a linear combination of a k l ,. . . , ak,. To do this, we merely take for k , the first index for which a,, # 0, and for k,,+,the smallest index m > k, such that a,,, is not in the subspace V,, generated by a k l ,. .., a,"; such an index exists, otherwise, as V,, is closed by (5.9.2), V,, would contain the closure E of the set of all the a , , contrary to assumption. It is then clear that (a,") has the required properties, and is obviously by construction a total sequence.
PROBLEM
Show that the spaces (co) and I' of Banach (Section 5.3, Problem 5, and Section 5.7, Problem 1) are separable, but that the space I" (Section 5.7, Problem 1) is not separable. (Show that in I" there exists a nondenumerable family ( x J of points such that IlxA- x ujl = 1 for h # p , using Problem 2(b) of Section 4.2, and (2.2.17).)
CHAPTER VI
HILBERT SPACES
Hilbert spaces constitute at present the most important examples of Banach spaces, not only because they are the most natural and closest generalization, in the realm of “ infinite dimensions,” of our classical euclidean geometry, but chiefly for the fact that they have been, up to now, the most useful spaces in the applications to functional analysis. With the exception of (6.3.1),all the results easily follow from the definitions and from the fundamental Cauchy-Schwarz inequality (6.2.4).
1. H E R M I T I A N FORMS
For any real or complex number A, we write X for its complex conjugate (equal to A if A is real). A hermitiun form on a real (resp. complex) vector space E is a mapping f of E x E into R (resp. C) which has the following properties : (1)
f(x + x‘,Y ) =f(x, Y ) + f ( x ’ , Y ) ,
(W
+ Y’) = A x ,Y ) +m, Y’), f ( kr) = Af(x,Y ) ,
(W
f ( x , AY) = m x ,Y>,
(11)
(V)
f(x, Y
f ( Y >4
=f(% Y ) .
(Observe that (11) and (IV) follow from the other identities; (V) implies that f(x, x) is real.) When E is a real vector space, conditions (I) to (IV) express that f is bilinear and (V) boils down to f ( y , x) = f ( x , y), which 115
116
VI
HILBERT SPACES
expresses t h a t f i s symmetric. For any finite systems ( x i ) ,( y j ) , (ai), (Bj), we have (6.1 .I)
by induction on the number of elements of these systems. From (6.1.1) it follows that if E is finite dimensional and ( a i ) is a basis of E, f is entirely determined by its values a i j = f ( a i , a j ) , which are such that (by V)) aJ,l. = a I,J.’
(6.1.2)
Indeed we have then, for x = (6.1.3)
1 S i a i ,y = i
S ( X .,Y>
viai I
C aij ti
i, i
iij
.
Conversely, for any system ( a i j ) of real (resp. complex) numbers satisfying (6.1.2), the right hand side of (6.1.3) defines on the real (resp. complex) finite dimensional vector space E a hermitian form. Example (6.1.4) Let D be a relatively compact open set in R2, and let E be the real (resp. complex) vector space of all real-valued (resp. complex-valued) bounded continuous functions in D, which have bounded continuous first derivatives in D. Then the mapping
(where a, h, c are continuous, bounded and real-valued in D) is a hermitian form on E. A pair of vectors x, y of a vector space E is orthogonal with respect to a hermitian form f on E if f ( x , y ) = 0 (it follows from (V) that the relation is symmetric in x, y ) ; a vector x which is orthogonal to itself (i.e.f(x, x) = 0) is isotropic with respect to$ For any subset M of E, the set of vectors y which are orthogonal to all vectors x E M is a vector subspace of E, which is said to be orthogorial to M (with respect t o f ) . It may happen that there exists a vector a # 0 which is orthogonal to the whole space E, in which case we say the formfis degenerate. On a finite dimensional space E, nondegenerate hermitian forms f defined by (6.1.3) are those for which the matrix ( a i j ) is invertible.
2
POSITIVE HERMITIAN FORMS
117
PROBLEMS
1. (a) Let f be a hermitian form on a vector space E. Show that if E is a real vector space, then 4f(x, Y> = f ( x
+ Y , x +v) -f(x
-Y, x
-Y)
and if E is a complex vector space
4fh, Y ) =f(x
+ y. x + y ) - f ( x
-Y ,x
-y)
+ i f ( x + iy, x + iy) - if(x - iy, x
- iy).
(b) Deduce from (a) that if f ( x , x ) = 0 for every vector in a subspace M of E, then f ( x , y ) = 0 for any pair of vectors x , y of M. (c) Give a proof of (b) without using the identities proved in (a). (Write that f(x hy, x +Xu) = O for any h.) 2. Let E be a complex vector space. Show that iffis a mapping of E x E into C satisfying conditions (I), (II), (III), (IV), and such that f ( x , x ) E R for every x E E, then f is a hermitian form on E.
+
2. POSITIVE H E R M I T I A N FORMS
We say a hermitian form f on a vector space E is positive iff(x, x) 2 0 for any x E E. For instance, the form q defined in example (6.1.4) is positive if a, b, c are 2 0 in D.
(6.2.1)
(Cauchy-Schwarz inequality)
I f f is a positive hermitian form,
then for any pair of vectors
X,
If(x, Y>12 < f ( x , X)f(Y, Y ) y in E.
Write a =f ( x , x), b = f ( x , y ) , c = f ( y , y ) and recall a and c are real and 2 0. Suppose first c # 0 and write that f ( x I y , x + l y ) 2 0 for any scalar 1, which gives a + b2 + bA + c3J 0 ; substituting I = -b/c yields the inequality. A similar argument applies when c = 0, a # 0 ; finally if a = c = 0, the substitution I = - h yields -266 2 0, i.e. b = 0.
+
(6.2.2) In order that a positii!e hermitian form f on E be nondegenerate a necessary and suficient condition is tliat there exist no isotropic vector for f otlier than 0 , i.e. that f ( x , x) > 0 for any x # 0 in E.
Indeed, f ( x , x) all y E E.
=0
implies, by Cauchy-Schwarz, that f (x, y ) = 0 for
118
(6.2.3)
VI
HILBERT SPACES
(Minkowski's inequality) I f f is a positi1.e hermitian form, tlien (f(x
+y, x +
d ( A x , x))''2 + ( f b ,
for any pair of vectors x, y in E.
+
As f ( x y , x equivalent to
+ y ) = f ( x , x) + f ( x , y ) + A x , y ) + f ( y , y ) , the inequality is
2,gf(x,Y)= f ( x , Y >+f(x,P,)d 2 ( f ( x , x)f(Y,Y))'12 which follows from Cauchy-Schwarz. The function x -+ ( f ( x , x))'12 therefore satisfies the conditions (I), (Ill), and (IV)of (Section 5.1); by (6.2.2), condition (11) of Section 5.1 is equivalent to the fact that the form f is nondegenerate. Therefore, when f is a nondegenerate positive hermitian form (also called a positive dejnite form), ( f ( x , x))'I2 is a norm on E. A preliilbert space is a vector space E with a given nondegenerate positive hermitian form on E; when no confusion arises, that form is written (x I y ) and its value is called the scalar product of x and y ; we always consider a prehilbert space E as a normed space, with the norm llxll = (x I x)''', and of course, such a space is always considered as a metric space for the corresponding distance Ijx - y ( / . With these notations, the Cauchy-Schwarz inequality is written
I(xlu)l d IIxIl
(6.2.4)
IIYII
and this proves, by (5.5.1), that for a real prehilbert space E, (x, y ) -+ (x 1 y ) is a continuous bilinear form on E x E (the argument of (5.5.1) can also be applied when E is a complex prehilbert space and proves again the continuity of (x, y ) (x I y ) , although this is not a bilinear form any more). We also have, as a particular case of (6.1 .I): --f
(6.2.5) (Pythagoras' theorem) onal vectors, IIX
In a prehilbert space E, i f x , y are orthog-
+ YIl2 = l/xll2+ llY/I2.
A n isomorphism of a prehilbert space E onto a prehilbert space E' is a linear bijection of E onto E' such that ( f ( x )If(y)) = (x Iy) for any pair of vectors x, y of E. It is clear that an isomorphism is a linear isonzetry of E onto E'. Let E be a prehilbert space; then, on any vector subspace F of E, the restriction of the scalar product is a positive nondegenerate hermitian form ; unless the contrary is stated, it is always that restriction which is meant when F is considered as a prehilbert space.
3 ORTHOGONAL PROJECTION ON A COMPLETE SUBSPACE
119
A Hilbert space is a prehilbert space which is coniplete. Any finite dimensional prehilbert space is a Hilbert space by (5.9.1); other examples of Hilbert spaces will be constructed in Section 6.4. If in example (6.1.4) we take a > 0, b 2 0 , c 3 0, it can be shown that the prehilbert space thus defined is not complete.
PROBLEMS
1. Prove the last statement in the case a = 1 , b = c = 0 (see Section 5.1, Problem 1). 2. (a) Let E be a real nornied space such that, for any two points ,Y, y of E, 1I.y
+ Yl12 + Ilx - Y1IZ
= 2(11x1I2
+ llvIl2).
Show that f ( x , y) = IIx i-V I -~ Ilxl/* ~ - /IyIl2 is a positive nondegenerate hermitian form on E. (In order to prove that f ( h x , y ) = h f ( x , y ) , use Problem I of Section 5.5.) (b) Let E be a complex nornied space, Eo the underlying real vector space. Suppose there exists on Eo a symmetric bilinear form f ( x ,y) such that f ( x , x) = l1x/l2for every x E E o . Show that there exists a herniitian form g(x, y ) on E such that f ( x , y ) = .jR(g(x, y ) ) , hence g(x, x) = 1/x1l2for x E E. (Using the first formula of Problem 1 of Section 6.1, prove that f ( i x , y ) = - f ( x , i-v).) (c) Let E be a complex normed space such that Ilx - Y1I2
+ Ilx + Y / I 2 < 2(11x1I2 + llvll’)
for any pair of points x , y of E. Show that there exists a nondegenerate positive hermitian fornif(x,y) on E such thatf(x, x) = XI/^. (Use (a) and (b)). 3. Letfbe a positive nondegenerate hermitian form. In order that both sides of (6.2.1) be equal, it is necessary and sufficient that x and y be linearly dependent. In order that both sides of (6.2.3) be equal, it is necessary and sufficient that x and ,y be linearly dependent, and, if both are # 0, that y = Ax, with real and > 0. 4. Let a, b, c, d be four points in a prehilbert space E. Show that
/(a- c ( / . / / b- d ( (< /la - bll . JJc- dll
5.
+ 116
- cii
. / / a- dll.
(Reduce the problem to the case a = 0, and consider in E the transformation x+x/llxl~’, defined for x # 0.)When are both sides of the inequality equal? be a finite sequence of points in a prehilbert space E. Show that Let ( x i ) ,
If /lxi - x,Jl > 2 for i # j , show that a ball containing all the x i has a radius >(2(n
-
1)jn)’”.
3. O R T H O G O N A L PROJECTION O N A COMPLETE SUBSPACE
(6.3.1) Let E be a prehilbert space, F a complete tvctor subspace of E (i.e. a Hilbert space). For any x E E, there is one and only one point y = P F ( x )E F
120
VI
HILBERT SPACES
such that I/x - J~II = d(x, F). Tlze point y = P,(x) is also the only point z E F such that x - z is orthogonal to F. The mapping x P,(x) of E onto F is linear, continuous, and of norm 1 if F # { O } ; its kernel F’ = PF1(0) is the subspace orthogonal to F, and E is the topological direct sum (see Section (5.4)) of F and F‘. Finally, F is the subspace orthogonal to F‘. --f
Let CI = d(x, F); by definition, there exists a sequence (y,) of points of F such that lim 1I.x - ynl/ = a ; we prove (y,,) is a Cauchy sequence. fl-+
co
Indeed, for any two points u, u of E, it follows from (6.1.1) that (6.3.1.1)
hence
112.4
llym
+
+ -
U(l2
1/24
PJ12= 2(llu112
+ 11z~112>
- yn/I2= 2(llx - ~ , ~ ,+l / IIx ~ - ~ , , l l ~-) 4llx - ! d Y m
+ y,)112.
But
+ y,,) E F, hence jlx - )(ym + y,,)lI2 > a 2 ;therefore, if no is such that for n 2 n o , ( / x- y , ( / 2< g2 + E , we have, form > no and n 3 n o , I/ym- Y,;;? < 4e, +(yJ,
which proves our contention. As F is complete, the sequence (y,) tends to a limit y E F, for which IJx- yll = d(x, F). Suppose y’ E F is also such that /Ix - y’ll = d(x, F); using again (6.3.11), we obtain / ( y- y’(I2= 4a2- 4 /Jx- ) ( y y’)1I2, and as j ( y y ’ ) E F, this implies (ly - y’1I2 < 0,i.e. y’ = y. Let now z # 0 be any point of F, and write that (lx - ( y Az)/l* > a2 for any real scalar A # 0; this, by (6.1 .I), gives
+
+
21B(x - y 1 z )
+
+ R 2 / ( 2 / ( 2> 0
and this would yield a contradiction if we had 9 ( x - y 1 z ) # 0, by a suitable choice of 1. Hence B(x - y 1 z ) = 0, and replacing z by iz (if E is a complex prehilbert space) shows that #(x - y I z ) = 0, hence (x - y I z ) = 0 in every case; in other words x - y is orthogonal to F. Let y’ E F be such that x -y’ is orthogonal to F; then, for any z # 0 in F, we have JIx- (y’ z)1I2 = /Jx- y‘/12 j/zIlz by Pythagoras’ theorem, and this proves that y’ = y by the previous characterization of y . This last characterization of y = P,(x) proves that P , is linear, for if x - y and x’- y’ are orthogonal to F, then Ax - Ay is orthogonal to F and so is (x + x’)- ( y + y’) = (x - y ) (x’ - y ’ ) ; as y y’ E F and Ay E F, this shows that y y’ = P F ( x x’)and Ay = P , ( ~ x ) .By Pythagoras’ theorem, we have
+
+
+
(6.3.1.2)
+
llX/12
=
+
IIpF(x)112
+
+ (Ix
- pF(x)l/2
and this proves that IlPF(x)l1< I/x/J,hence (5.5.1) P, is continuous and has norm d 1 ; but as P F ( x )= x for x E F, we have /lP,I/ = 1 if F is not reduced to 0. The definition of P, implies that F’ = PF’(0) consists of the vectors x orthogonal to F; as x = P,(x) + (X - PF(,y)) and x - P F ( x )E F’ for any x E E, we have E = F F’; moreover, if x E F n F‘, x is isotropic, hence
+
3 ORTHOGONAL PROJECTION ON A COMPLETE SUBSPACE
x
= 0, and
121
this shows that the sum F + F' is direct. Furthermore, the mapping
x -iP F ( x )being continuous, E is the topological direct sum of F and F' (5.4.2). Finally, if x E E is orthogonal to F', we have in particular (x I x - P F ( x ) )= 0 ; but we also have (PF(x) I x - PF(x))= 0, hence /Ix - PF(x)(I2= 0, i.e. x = PF(x) E F. Q.E.D. The linear mapping PF is called the orthogonal projection of E onto F, and its kernel F' the orthogonal supplement of F in E. Theorem (6.3.1) can be applied to any closed subspace F of a Hilbert space E (by (3.14.5)), or to any Jinite dimensional subspace F of a prehilbert space, by (5.9.1).
(6.3.2) Let E be a prehilbert space; then, for any a E E, x -i (x I a) is a continuous linear form of norm Ilall. Conversely, if E is a Hilbert space, for any continuous linear form u on E, there is a unique vector a E E such that u(x> = (x I a)for any x E E. ByCauchy-Schwarz,I(x I a)l < llall * IIxII, whichshows(by5.5.1))~+ (x la) is continuous and has a norm < llalj; on the other hand, if a # 0, then for x, = a / ~ ~ we a ~have ~ , (x, I a) = llall; as llxoII = 1, this shows the norm of x - i ( x l a ) is at least Ilall. Suppose now E is a Hilbert space; the existence of the vector a (=O) being obvious if u = 0, we can suppose u # 0. Then H = u-'(O) is a closed hyperplane in E; the orthogonal supplement H' of H is a one-dimensional subspace; let b # 0 be a point of H'. Then we have ( x I b) = 0 for any x E H. But any two equations of a hyperplane are proportional, hence there is a scalar L such that u(x) = L(x 16) = (xI a) with a = Xb (see Appendix) for all x E E. The uniqueness ofa follows from the fact that the form (x I y ) is nondegenerate.
PROBLEMS 1. Let B be the closed ball of center 0 and radius 1 in a prehilbert space E. Show that for each point x of the sphere of center 0 and radius I , there exists a unique hyperplane of support of B (Section 5.8, Problem 3) containing x . 2. Let E be a prehilbert space, A a compact subset of E, 6 its diameter. Show that there exist two points a , b of A such that /la- bll = 8 and that there are two parallel hyperplanesof support of A (Section 5.8, Problem 3)containingaand b respectively, and such that their distance is equal to 6. (Consider the ball of center a and radius 8 and apply the result of Problem 1 .)
3.
Let E be a Hilbert space, F a dense linear subspace of E, distinct from E (Section 5.9, Problem 2). Show that there exists in the prehilbert space F a closed hyperplane H such that there is no vector # 0 in F which is orthogonal to H.
VI
122
4.
HlLBERT SPACES
Let X be a set, E a vector subspace of Cx, on which is given a structure of complex Hilbert space. A mapping ( x , y ) K(x, y ) of X x X into C is called a reproducing kernel for E if it satisfies the two following conditions: ( I ) for every y E X, the function K(. , y ) : x --f K(x, y ) belongs to E; (2) for any function f~ E, and any y E X, f(y) = (fI K ( . , A). (a) Show that K is a mapping of positive fype of X x X into C, i.e., for any integer of points of X , the mapping n 3 1 and any finite sequence ( x i ) , --f
+x W x i , xi)&
((Ad, (pi))
pi
I. J
of C2" into C is a positiue hermitian form. This in particular implies K(x, x ) 3 0 for every x E X, K(y, x ) = K(x, y ) and IK(x, y)12 < K(x, x ) K ( y ,y ) for x, y in X. Show that f0rj.E E, one has lf(y)I < llfll . (K(y, y))"' for y E X. (b) Show that if (f.)is a sequence of functions of E which converges (for the Hilbert space structure) to f~ E, then, for every x E X , the sequence (f.(x)) converges to f ( x ) in C; the convergence is uniform in any subset of X where the function x + K(x, x ) is bounded. n a finite sequence of points of X, (al)lslsn a sequence of n com(c) Let ( x i ) l i 1 Bbe plex numbers. Suppose det(K(x, , x i ) ) # 0, so that the system of linear equations cj K(xi, x J )= al (1
< i < n) has a
J= 1
unique solution (ci). Show that among the func-
tions f~ E such that f(x1) = a , for 1 f i S n, the function fo
n
= cj J= I
K(. , x i ) has the
smallest norm. In particular, among all functions f~ E such that f ( x ) = 1 for a point E X where K(x, x ) # 0, the function K(. , x ) / K ( x ,x ) has the smallest norm. (d) If X is a topological space and if all the functionsf€ E are continu0u.s in X, then the functions K(.,x) (where x takes all values in X, or in a dense subset of X) form a total subset of E (show that there is no element h # 0 in E which is orthogonal to all the elements K(. , x ) ) . In particular, if X is a separable metric space, E is a separable Hilbert space. 5. (a) The notations being those of Problem 4, in order that there exist a reproducing kernel for E, a necessary and sufficient condition is that for every x E X, the linear form f + f ( x ) be continuous in E. The reproducing kernel is then unique. (b) Deduce from (a) that if there exists a reproducing kernel for E, there also exists a reproducing kernel for every closed vector subspace El of E. If K, is the reproducing kernel for E l , show that for every function f~ E, the function y --f (f1 K,(. ,y ) ) is the orthogonal projection o f f i n El. If E, is the orthogonal supplement of El and K2 the reproducing kernel for E, , then K I $- K 1 is the reproducing kernel for E. 6. Let X be a set, E a vector subspace of Cx, on which is given a structure of complex prehilbert space. In order that there should exist a Hilbert space c Cxcontaining E, such that the scalar product on E is the restriction of the scalar product on B, and that there exists a reproducing kernel for B, it is necessary and sufficient that E satisfy the two following conditions: ( I ) for every x f X, the linear form f + f ( x ) is continuous in E; (2) for any Cauchy sequence (f,)in E such that, for every x E X, lim f.(x) = 0, x
one has lim llfnll
n-m
= 0.
(To prove the conditions are sufficient, consider the subspace
f?
20'"
of Cx whose elements are the functionsffor which there exists a Cauchy sequence (fn) in E such that limfn(x) = f ( x ) for every x E X. Show that, for all Cauchy sequences (f.) n-r m
having that property, the number lim llfnll is the same, and if l i f l l is that number, this n-rm
defines on f a structure of normed space which is deduced from a structure of prehilbert
4 HILBERT SUM OF HILBERT SPACES
123
space which induces on E the given prehilbert structure; finally show that E is dense in 8 and that 8 is complete, hence a Hilbert space, and apply Problem 5(a) to 8.) 7. Let X be a set, f a mapping of X into a prehilbert space H; show that the mapping ( x , y ) + ( f ( x )1 f(y)) of X x X into C is of positive type (Problem 4(a)). 8 . Let X be a set, K a mapping of positive type of X x X into C (Problem 4(a)). (a) Let E be the set of mappings u : X -+ C such that there exists a real number a > 0 having the property that the mapping ( x , Y ) +aK(x, Y ) - 4x)UO
is of positive type; let m(u)be the smallest of all real numbersa 3 Ohavingthat property. Show that m(u) is also the smallest number c such that, for every finite sequence ( x i ) of elements of X, the inequality
holds for all complex numbers h, , p (use the Cauchy-Schwarz inequality). For every E X, show that Iu(x)12 < K(x, x)m(u). (b) Show that E is a vector subspace of Cx, that ( m ( ~ ) ) is ” ~a norm on E and that m(u u ) 4- m(u - u ) 2 2(m(u) m(u)).Conclude that there is a nondegenerate positive hermitian form g(u, u) on E x E such that g(u, u ) = m(u), and that for this form E is a Hilbert space; one writes g(u, u ) = (u I u). (Use Problem 2(c) of Section 6.2 to prove the existence of g ; to show that E is complete, use the last inequality proved in (a)). (c) For every x E X, show that the function K ( . , x ) belongs to E and that (K(. , x) I K ( . ,y ) ) = K(x, y ) for all ( x , y ) E X x X (use Cauchy-Schwarz inequality). Prove that if X is a topological space and if K is continuous in X x X, the mapping x + K(. , x) of X into E is continuous. (d) Deduce from (a) that the Hilbert space E defined in (b) has a reproducing kernel, and if F is the closed vector subspace of E generated by the functions K(. ,x), the reproducing kernel for F (Problem 5(b)) is equal to K. 9. Let E be a prehilbert space, N a finite dimensional vector subspace of E, M a vector subspace of E having infinite dimension, or finite dimension >dim(N). Show that there exists in M a point x # 0 such that llxll= d(x, N). (Consider the intersection of M and of the orthogonal supplement of N.) x
+
+
4. H I L B E R T SUM O F HILBERT SPACES
Let (En) be a sequence of Hilbert spaces; on each of the En, we write the scalar product as ( x , ) y n ) .Let E be the set of all sequences x = (xl, x 2 , . . . , x,, . . .) such that x, E En for each n , and the series ( I I x , ~ ~ ~ ) is convergent. We first define on E a structure of vector space: it is clear that if x = (x,) E E, then the sequence (Axl,.. . , Ax,,,. . .) also belong to E. On the other hand, if y = (y,) is a second sequence belonging to E, we observe that IIx, ynl12< 2(/1~,11~ liyn112) by (6.3.1.1), hence the series (llx, + y,I12) is convergent by (5.3.1), and therefore the sequence (xl + y , , ..., x, + y n r ...) belongs to E. We define x + y = (x,+y,,),
+
+
124
V
HILBERT SPACES
Ax = (Ax,), and the verification of the axioms of vector spaces is trivial. On the other hand, from the Cauchy-Schwarz inequality, we have
IIxnII *
I(xn IYn)I G
IIYnIt G
HIIXnII’
+ IIynII’).
Therefore, if x = (x,) and y = (y,) are in E, the series (of real or complex numbers) ((x, I y,)) is absolutely convergent. We define, for x = (x,) and
y = (y,) in E, the number (x 1 y ) =
00
,= 1
(x, 1 y,); it is immediately verified
that the mapping (x, y ) -.+ (x I y ) is a Hermitian form on E. Moreover we have (x Ix) =
1 llxn\1’, hence ( x l y ) is a positive nondegenerate
hermitian
n= 1
form and defines on E a structure of prehilbert space. We finally prove E is in fact a Hilbert space, in other words it is complete. Indeed, let (xc”’)), where x(”’)=(x$)), be a Cauchy sequence in E: this means that for any E > 0 there is an m, such that for p 2 m, and q >, m, , we have m
(6.4.1)
For each fixed n, this implies first
IIxp) - xP)1l2< E ,
( x ~ ~ ) ) ~ = ~is , a~ ,Cauchy ... sequence in
hence the sequence
En,and therefore converges to a
limit y , . From (6.4.1) we deduce that for any given N N
C IIxn(P) - xn(4)II2 < &
n= I
as soon as p and q are 2 m o , hence, from the continuity of the norm, we deduce that
N
n= 1
<E
1 1 ~ : ~ ) - y,II’
integers N, we have
m “ .= .
for p 2 m,, and as this is true for all
IIxp)- ynllZG E .
I .
This proves first that the
sequence (xp’ - y,) belongs to E, hence y = (y,) also belongs to E, and - y11’ < E for p 2 m,, which ends the proof by showing we have (Idp) that the sequence (x(”’)) converges to y in E. We say that the Hilbert space E thus defined is the Hilbert sum of the sequence of Hilbert spaces (En). We observe that we can map each of the En into E by associating to each x, E En the sequence j,(x,) E E equal to (0, . . . , 0 , x, 0,. . .) (all terms 0 except the nth equal to x,); it is readily verified that j , is an isomorphism of En onto a (necessarily closed) subspace EA of E; j , is called the natural injection of En into E. From the definition of the scalar product in E, it follows that for m # n, any vector in EL is orthogonal to any vector in EL; furthermore, from the definition of the norm in E, it follows that for any x = (x,) E E, the series (j,(x,)) is convergent
4
in E, and x =
m
HILBERT SUM OF HILBERT SPACES
125
j,,(x,) (observe that the series (j,(x,)) is not absolutely con-
n= 1
vergent in general). This proves that the (algebraic) sum of the subspaces EA of E (which is obviously direct) is dense in E, in other words that the smallest closed vector subspace containing all the EA is E itself. Conversely: (6.4.2) Let F be a Hilbert space, (F,) a sequence of closed subspaces such that: (1) for m # n, any vector of F , is orthogonal to any vector of F,; (2) the algebraic sum H of the subspaces F , is dense in F . Then, i f E is the Hilbert sum of the F,, there is a unique isomorphism of F onto E which on each F, coincides with the natural injection j , of F, into E.
Let F; = j,,(F,,), and let h, be the mapping of FA onto F,, inverse to j , . Let G be the algebraic sum of the FA in E; that sum being direct, we can define a linear mapping h of G into F by the condition that it coincides with h, on each FA. I claim that h is an isomorphism of G onto the prehilbert space H (which, incidentally, will prove that the (algebraic) sum of the F, is direct in F ) ; from the definition of the scalar product in E, we have to check that
for Xk E Fk,yk E F k ; but by assumption (x, I yk) = 0 if h # k, and the result follows from the fact that each j k is an isomorphism. There is now a unique continuous extension T; of h which is a linear mapping of = E into = F, by (5.5.4); the principle of extension of identities (3.1 5.2) and the continuity of the scalar product show that h is an isomorphism of E onto a subspace of F, which, being complete and dense, must be F itself; the inverse of h satisfies the conditions of (6.4.2). Its uniqueness follows from the fact that it is completely determined in G and continuous in E (3.15.2). Under the condition of (6.4.2), the Hilbert space F is often identified with the Hilbert sum of its subspaces F , .
e
Remark (6.4.3)
We can also prove (6.4.2) by establishing first that the sum of the F, n
is direct; indeed, if E x i = 0 with x i €Fi (1 < i < n) we also have i = l.
= 0 for any j
< n, and as (xjIxi) = 0 for i # j , this boils down
126
VI HILBERT SPACES
to llxil12 = 0, hence xi = 0 for 1 < j < n. Then we define the inverse mapping g of h by the condition that it coincides with j,, on each F,,: we at once verify, as above, that g is an isomorphism of H onto G, and then (5.5.4) is applied in the same way. We observe that this argument still applies when F is a prehilbert space and the F, are complete subspaces of F; it proves the existence of an isomorphism of F onto a dense subspace of the Hilbert sum E of the F,,, which coincides withj,, on each F, .
PROBLEM
Let X be a set, E l , Ez two vector subspaces of Cx, possessing reproducing kernels K,, K2 (Section 6.3, Problem 4). Let E = El E2 c Cx,and let F be the Hilbert sum of the Hilbert spaces El , Ez;the kernel of the surjective mapping u : (fl, fz)-fl +fz of F into E is the subspace N of F consisting of the pairs (f,-f) where YE El n EZ. Show that N is closed in F; if H is the orthogonal supplement of N in F, the restriction u of u to H is a bijection of H onto E; transporting by u the Hilbert structure of H, E is defined as a Hilbert space. Show that for that Hilbert space structure, E has a reproducing kernel equal to K1 Kz; the norm llfll in E is equal to inf(llflII: llfzll~)l~zwhen (fl,f2) takes all values in F such thatf=fl +fz(llfl Illand llf~ [I2 being the norms in El and E2 ,respectively).
+
+
+
5. ORTHONORMAL SYSTEMS
If (with the notations of Section 6.4) we take for each E,, a one-dimensional space (identified to the field of scalars with the scalar product (5 I q) = (fj), the Hilbert sum yields an example of an infinite dimensional Hilbert space E, which is usually written l2 (with index R or C to indicate if necessary what the scalars are); the space I: (resp. 1;) is therefore the space of all gi
sequences x = (5,) of real (resp. complex) numbers, such that convergent, with the scalar product ( x I y ) =
c m
l{,,I2 is n= 1
{, ij,
.
n=l
In 12, let en be the sequence having all its terms equal to 0 except the nth term equal to 1; we then have (em1 en)= 0 for m # n, and IJe,,JI= 1 for each n, and we have seen in Section 6.4 that for every x = (t,) in f 2 , we can write x =
c Show that the closure where An > 0, the series
-
A is the set of all the sums of the series
I
C A,, is convergent
129
"= 1
and has a sum equal to I .
"=I
(b) Prove that the diameter of A is equal to d2 but that there is no pair of points A such that /la- b/l = 42 (compare to Section 6.3, Problem 2). Let E be a Hilbert space with a Hilbert basis Let a, = eZn and
a, h of
b, = ezn
1 ++ 1 e z n f lfor every n > 0; letA (resp. B) be the closed vector subspace of E I1
generated by the a. (resp. b"). Show that: (a) A n B = { O } , hence the sum A B is direct (algebraically). (b) The direct sum A B is not a topological direct sum (consider in that subspace the sequence of points h, - a, and apply (5.4.2.)). (c) The subspace A B of E is dense but not closed in E (show that the point m
+
+ +
(b, - an)does not belong to A
+ B).
"=O
Show that the Banach space 9 ( P ; 1 2 ) can be identified with the space of double
sequences U = (a,,,,) such that: (I) the series m
( 2 ) sup n
m
nr= 0
is convergent for every n:
la,,,nlzis finite. The norm is then equal to IlUIj = sup
"
nt=O
method as in Section 5.7, Problem 2(b)). (a) Let u be a continuous linear mapping of show that the series
c m
fl=O
and that their sums are on E and use (6.3.2).)
/cz,,,12 and
0, is closed in the space .H, (E).
2. SPACES O F B O U N D E D C O N T I N U O U S F U N C T I O N S
Let E now be a metric space; we denote by %F(E) the vector space of all continuous mappings of E into the normed space F, by %',"(E) the set of all bounded continuous mappings of E into F. We note that if E is compact, %?F(E) = %F(E) by (3.17.10). I n general we have %F(E) = %?F(E)n BF(E). We will consider %?F(E)as a normed subspace of gF(E), unless the contrary is explicitly stated. If F is finite dimensional, in the decomposition (7.1.2.1) f is continuous if and only if each of the fiis continuous (see (3.20.4) and (5.4.2)). The remarks preceding (7.1.2) then show that in such a case, %'F(E) is a topological direct sum of a finite number of subspaces, each of which
2 SPACES OF BOUNDED CONTINUOUS FUNCTIONS
135
is isometric to Wg(E) (resp %‘,?(E)). I n particular, the real normed space underlying V,‘(E) is the topological direct sum %‘g(E)+ iV:(E).
(7.2.1) The subspace W;(E) is closed in BF(E); in other ~1ords,a uniform limit of bounded continuous functions is continuous. Indeed, let (f,) be a sequence of bounded continuous mappings of E into F, which converges to g in g3,(E); for any E > 0, there is therefore an integer no such that ilf, - 911 < 4 3 for n 2 n o . For any to E E, let V be a neighborhood of to such that ~l.f,,(t) -fflo(to)l\d 4 3 for any t E V. Then, as Ilf,,(t) - g(t)I/ d 4 3 for any t E E, we have /lg(t)- g(t,)ll < E for any t E V, which proves the continuity of g. Well-known examples (e.g. the functions x -+ x” in [0, 13) show that a limit of a simply convergent sequence of continuous functions need not be continuous. On the other hand, examples are easily given of sequences of continuous functions which converge nonuniformly to a continuous function (see Problem 2). However (see also (7.5.6)):
(7.2.2) (Dini’s theorem) Let E be a compact metric space. Ifa n increasing (resp. decreasing) sequence (f f l )of real-aalued continuous functions converges simply to a continuous function g, it conwrges uniformly to g. Suppose the sequence is increasing. For each E > 0 and each t E E , there is an index n(t) such that for rn 2 n ( t ) , g(t) -f,(t) < e/3. As g and f f l ( r ) are continuous, there is a neighborhood V ( t ) oft such that the relation t’ E V ( t ) implies Ig(t) - g(t’)l < 4 3 and 1fnct,(t) - f&,(t’)l d 4 3 ; hence, for any t‘ E V(t) we have g(r’) -Lcr,(t’) d E . Take now a finite number of points r i in E such that the V(ti) cover E, and let no be the largest of the integers n(t,). Then for any t E E, t belongs to one of the V(ti), hence, for n 2 no , d t ) -L(t) -f,,(t) d A t ) -fncr,)(t) 6 8 . Q.E.D.
PROBLEMS
1. Let E be a metric space, F a normed space, (u,,) a sequence of bounded continuous mappings of E into F which converges simply in E to a bounded function u. (a) In order that u be continuous at a point x, E E, it is necessary and sufficient that for any E > 0 and any integer m, there exist a neighborhood V of x , and an index n > m such that Ilu(x) - u.(x)II < E for every x E V. (b) Suppose in addition E is compact. Then, in order that u be continuous in E, i t is necessary and sufficient that for any E > 0 and any integer m,there exist a finite number
136
VII SPACES OF CONTINUOUS FUNCTIONS
of indices n, > m such that, for every x E E, there is at least one index i for which Ilv(.x) - u,,(x>Il < E (use (a) and the Borel-Lebesgue axiom).
2.
For any integer ti > 0, let g. be the continuous function defined in R by the conditions that g,,(t) = 0 for t < O and I 3 2 / n , g,,(I/n)= 1 , and g,(t) has the form orf p (with suitable constants a , p) in each of the intervals [O, l / n ] and [l/n, 2 / n ] .The sequence (gn) converges simply to 0 in R, but the convergence is not uniform in any interval containing t = 0. Let m i r , , be a bijection of N onto the set Q of rational numbers, and let
+
OL.
fn(t) =
C 2-mg.(t - r,,,). The functions f, are continuous (7.2.1 ), and the sequence (A) lfl=O
3.
4.
5. 6.
7.
8.
converges simply to 0 in R, but the convergence is not uniform in any interval of R . Let 1 be a compact interval of R,and ( f n ) a sequence of monotone real functions defined in I, which converge simply in I to a continuous functionf. Show that f i s monotone, and that the sequence (f.)converges uniformly to f i n I. Let E be a metric space, F a Banach space, A a dense subset of E. Let (fJ be a sequence of bounded continuous mappings of E into F such that the restrictions of the functions fn to A form a uniformly convergent sequence; show that ( f n ) is uniformly convergent in E. Let E be a metric space, F a normed space. Show that the mapping (x, N) + u(x) of E x %F(E)into F is continuous. Let E, E' be two metric spaces, F a normed space. For each mappingfof E x E'into F and each y E E', let f, be the mapping x - f ( x , y ) of E into F. (a) Show that if f i s bounded, if each f, is continuous in E and if the mapping y +A, of E' into XF(E) is continuous, thenfis continuous. Prove the converse if in addition E is compact (use Problem 3(a) in Section 3.20). (b) Take E = E' = F = R,and let f ( x , y ) = sin x y , which is continuous and bounded in E x E'; show that the mapping y - f , of E' into '6F(E) is not continuous at any point of E'. (c) Suppose both E and E' are compact, and for any f E KF(Ex E ) , let be the mapping y -f, of E' into KF(E); show that the mapping f-3 is a linear isometry of PF(E x E') onto % ' C ~ ~ ( ~ ) ( E ' ) . Let E be a metric space, F a normed space. For each bounded continuous mappingf of E into F, let G ( f )be the graph of f i n the space E x F. (a) Show that f - G ( f ) is a uniformly continuous injective mapping of the normed space %F(E) into the space a(E x F) of closed sets in E x F, which is made into a metric space by the Hausdorff distance (see Section 3.16, Problem 3). Let r be the image of VF(E) by the mappingf-- G ( f ) . (b) Show that if E is compact, the inverse mapping G - ' of I' onto %,"(E) is continuous (give an indirect proof). (c) Show that if E = 10, 13 and F = R, G - ' is not uniformly continuous. Let E be a metric space with a bounded distance cl. For each x E E let d, be the bounded continuous mapping y d(x, y ) of E into R. Show that x r!, is an isometry of E onto a subspace of the Banach space %g(E). --f
--f
3. T H E STONE-WEIERSTRASS A P P R O X I M A T I O N T H E O R E M
For any metric space E, the vector space %'g(E) (resp.%?E(E)) is an algebra over the real (resp.complex) field; from (7.1.1) it follows that we have in that algebra llfgll < llfll * Ilgll, hence, by (5.5.1), the bilinear mapping (5 g) -fg
3 THE STONE-WEIERSTRASS APPROXIMATION THEOREM
137
is continuous. From that remark, it easily follows that for any subalgebra A of Vg(E) (resp. V,?(E)), the closure m of A in Vg(E) (resp. VF(E)) is again a subalgebra (see the proof of (5.4.1)). We say that a subset A of BR(E) (resp. B,-(E)) separates points of E if for any pair of distinct points x , y in E, there is a function f E A such that f ( x ># f ( Y ) . (7.3.1) (Stone-Weierstrass theorem) Let E be a compact metric space. If a subalgebra A of 'eR(E) contains [he constant functions and separates
points of E, A is dense in the Banach space VR(E).
I n other words, if S is a subset of WR(E) which separates points, for any continuous real-valued function f on E, there is a sequence (g,) of functions converging uniformly t of, such that each g, can be expressed as a polynomial in the functions of S, with real coefficients. The proof is divided in several steps. (7.3.1 .I) There exists a sequence of real polynomials (u,) which in the interval [0, 13 is increasing and concerges uniformly to J t
.
Define u, by induction, taking u1 = 0, and putting (7.3.1.2)
u,+l(t) = u,(t)
+ +(t - u,2(t))
for n 2 1.
We prove by induction that u , + ~2 u, and u,(t) < JTin [0, 11. From (7.3.1.2), we see the first result follows from the second. On the other hand Jt - u , + , ( t ) = J t -
u,(t) - S ( t
= (Jt- u,(l))(l
- U,2(t))
- +(&+
u,(t>>)
and from un(t) < ,/r we deduce f ( J t + u,(t)) < Jt< 1. For each t E [0, 11, the sequence (u,(t)) is thuj increasing and bounded, hence converges to a limit r ( t ) (4.2.1); but (7.3.1.2) yields f - v 2 ( t ) = 0 and as C(f) 2 0, o(t) = Jt. As c is continuous and the sequence (u,) is increasing, Dini's theorem (7.2.2) proves that (u,) converges uniformly to D.
(7.3.1.3)
For any function f
E
A, If 1 belongs to the closure
A of A on VR(E).
Let a = (1 f (1. By (7.3.1.1), the sequence of functions u , ( f 2 / a 2 ) , which belong to A (by definition of an algebra), converges uniformly to(fz/a2)'/2 = If I/a in E.
138
VII SPACES OF CONTINUOUS FUNCTIONS
(7.3.1.4) to A.
For any pair of functions f, g in A, inf(f, g) and sup(f, g) belong
For we can write sup(f, g) = f( f + g + 1f - gl) and inf(f, g) = f ( f + g - If - gl); the result therefore follows from (7.3.1.3) applied to
the algebra
(7.3.1.5) c1,
A.
For any pair of distinct points x, y in E and any pair of real numbers
p, there is a function f E A such that f ( x ) = u, f ( y ) = 8.
By assumption, there is a function g E A such that g(x) # g(y). As A contains the constant functions, take f = a + ( p - a)(g - y)/(S - y), where y = g(x),& = d Y ) . (7.3.1.6) For any function f E %,(E), any point x E E, and any E > 0 , there is a function g E A such that g(x) =f(x) and g(y) 2/(n - I). Similarly, consider the subset Mi of M, consisting of the 2"-' functions of M, which are equal to x - a in the interval [a, a ( h - a)/n],and for each function g E MA, consider the set of all functions f E K such that g(x) - ( 2 / n ) < f ( x ) 6 g ( x ) for every x E I. Use a similar construction when (b - a)/&is not an integer.)
+
6. REGULATED FUNCTIONS
Let 1 be an interval in R, of origin a and extremity b (a or b or both may be infinite), F a Banach space. We say that a mapping f of I into F is a step-function if there is an increasing finite sequence of points of i (closure of 1 in R) such that xo = a, xn = b, and that f is constant in each of the open intervals ] x i ,x i + l [(0 6 i < n - 1). For any mapping f of I into F and any point x E I distinct from b, we lim f ( y ) exists; we then write the say that f has a limit on the right if Y€l,Y>X Y-x
limit f ( x + ) . Similarly we define for each point x E I distinct from a, the limit on the feft off at x, which we write f ( x - ) ; we also say these limits are one-sided limits off. A mapping f of I into F is called a regulated function if it has one-sided limits at every point of I . It is clear that any step-function is regulated.
(7.6.1) In order that a mapping f of a compact interid I = [a, b ] into F he regulated, a necessary and suficient condirion is thatf be the limit of a unformly convergent sequence of step-functions. (a) Necessity. For every integer n, and every x E I, there is an open interval V ( x ) = ] y( x) ,z(x)[ containing x, such that Ilf(s) -f(t)II < I/n if either both s, t are in ]y(x),x[ n I or both in ]x, z ( x ) [n I . Cover I with a finite number of intervals V ( x , ) ,and let ( c ~be the ) strictly ~ ~ increasing ~ ~ sequence consisting of the points a, b, x i , y ( x i ) and z(xi). As each c j is insome V(xi), c j + ]is either in the same V ( x i )or we have c j + , = z(xi), f o r j d m - 1 ; in other words if s, t are both in the same interval ] c j ,c ~ + ~ then [ , IIf(s) - f(t)ll < I/n. Now define g, as the step-function equal to f at the points c j , and at the midpoint of each interval ] c j , c j + ] [ , and constant in each of these intervals. It is clear that Ilf- grill d l/n.
~
VII SPACES OF CONTINUOUS FUNCTIONS
146
(b) Suficiency. Suppose f is the uniform limit of a sequence (f,) of step-functions. For each E > 0 there is an n such that l\f-fnll Q 43; now for each x E I , there is an interval ]c, d[ containing x and such that Ilf,(s) -f,(t)ll < ~ / 3if both s and t are in ]c, x [ or both in ]x, d [ ; hence under the same assumption we have IIf(s) -f(t)II Q E , and this proves the existence of one-sided limits off at x, since F is complete (3.14.6). Another way of formulating (7.6.1) is to say that the set of regulated functions is closed in BF(E), and that the set of step-functions is dense in the set of regulated functions. (7.6.2) Any continuous mapping of an interval I c R into a Banach space is regulated; so is any monotone mapping of I into R.
This follows from the definition, taking into account (3.16.5) and (4.2.1).
PROBLEMS
1. Let f be a regulated mapping of an interval I c R into a Banach space F. Show that for each compact subset H of I, f ( H ) is relatively compact in F; give an example showing that f(H) need not be closed in F. 2. The function f(x) = x sin(l/x) ( f ( 0 )= 0) is continuous, hence regulated in I = [0, I], and the function g(x) = sgn x (g(x) = 1 if x > 0, g(0) = 0,g(x) = - 1 if x > 0) is regulated in R, but the composed function g o f is not regulated in I. 3. Let I = [a, b ] be a compact interval in R. A function of bounded variation in I is a mapping fof I into a Banach space F, having the following property: there is a number V >, 0 such that,for m y strictly increasing finite sequence (fi)ogrgnof pointsof I, the inequality n- 1
C llf(ti+~)-f(ti)ll
i=o
G V holds.
(a) Show that f(1) is relatively compact in F (prove that f(1) is precompact, by an indirect proof). (b) Show thatfis a regulated function in I (use (a) and (3.16.4)). (c) The function g defined in [0, I ] as equal to xz sin(l/xz) for x # 0 and to 0 for x = 0 is not of bounded variation, although it has a derivative at each point of I.
CHAPTER Vlll
DIFFERENTIAL CALCULUS
The subject matter of this chapter is nothing else but the elementary theorems of calculus, which however are presented in a way which will probably be new to most students. That presentation, which throughout adheres strictly to our general “geometric” outlook on analysis, aims at keeping as close as possible to the fundamental idea of calculus, namely the “ local ” approximation of functions by linear functions. In the classical teaching of calculus, this idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse when dealing with functions of several variables : one thus arrives, for instance, at the classical formula (8.9.2) giving the partial derivatives of a composite function, which has lost any trace of intuitive meaning, whereas the natural statement of the theorem is of course that the (total) derivative of a composite function is the composite of their derivatives (8.2.1), a very sensible formulation when one thinks in terms of linear approximations. This “ intrinsic” formulation of calculus, due to its greater “ abstraction,” and in particular to the fact that again and again, one has to leave the initial spaces and to climb higher and higher to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas. But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of calculus on a differentiable manifold, which we shall develop in Chapters XVI to XX. Of course, he will observe that in these applications, all the vector spaces which intervene have 147
148
Vlll
DIFFERENTIAL CALCULUS
finite dimension; if that gives him an additional feeling of security, he may of course add that assumption to all the theorems of this chapter. But he will inevitably realize that this does not make the proofs shorter or simpler by a single line; in other words, the hypothesis of finite dimension is entirely irrelevant to the material developed below ; we have therefore thought it best to dispense with it altogether, although the applications of calculus which deal with the finite dimensional case still by far exceed the others in number and in importance. After the formal rules of calculus have been derived (Sections 8.1 t o 8.4), the other sections of the chapter are various applications of what is probably the most useful theorem in analysis, the mean value theorem, proved in Section 8.5. The reader will observe that the formulation of that theorem, which is of course given for vector valued functions, differs in appearance from the classical mean value theorem (for real valued functions), which one usually writes as an equality J’(h) - f ( a ) =f’(c)(b - a). The trouble with that classical formulation is that: ( I ) there is nothing similar to it as soon as f has vector values or when there are a finite number of points where f ’ is not defined; (2) it completely conceals the fact that nothing is known on the number c, except that it lies between a and b, and for most purposes, all one need know is that f ’ ( c ) is a number which lies between the g.1.b. and 1.u.b. off’ in the interval [a, b] (and not the fact that it actually is a value off’). In other words, the real nature of the mean value theorem is exhibited by writing it as a n inequality, and not as an equality. Finally, the reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the “ Riemann integral.” It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence t o Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “ theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration (see Section 13.9, Problem 7). Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis (at the level of this first volume), but close enough to the continuous functions t o dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions (sometimes called the “ Cauchy integral ”). When one needs a more powerful tool, there is no point in stopping halfway, and the general theory of (“ Lebesgue”) integration (Chapter XIII) is the only sensible answer.
1 DERIVATIVE OF A CONTINUOUS MAPPING
149
1. DERIVATIVE OF A C O N T I N U O U S MAPPING
Let E, F be Banach spaces (both real or both complex) and A a n open subset of E. Let h g be two continuous mappings of A into F; we say that f and g are tangent at a point x, E A if lim lif(x) - g(x)l)/llx- xoII = 0 ; x-xn,x+xo
this implies of course that f ( x o )= g(xo). We note that this definition only depends on the topologies of E and F; for i f h g are tangent for the given norms on E and F, they are still tangent for equivalent norms (Section 5.6). Jff, g are tangent at xo, and g, h tangent at xo, thenf, h are tangent at x, as follows from the inequality 11 f(x) - h(x)(j < lif(x) - g(x)ll + IIg(x) - h(x)/J. Among all functions tangent at x, to a function f , there is at most one mapping of the form x - f ( x , ) + u(x - x,) where u is linear. For if two such functions x +f(xo) + u l ( x - x,), x 4f ( x o )+ uz(x - x,) are tangent at xo, this means, for the linear mapping u = u1 - u 2 ,that lim Ilu(y)ll/llyll = 0. But this implies L' = 0, for if, given E > 0, there is r > 0 such that ljyll < r implies llv(y)ll d E llyll, then this last inequality is still valid for any x # 0, by applying it to y = rx/ilxil; as E is arbitrary, we see that u(x) = 0 for any x. y-+O,yfO
We say that a continuous mapping f of A into F is diflerentiable at the point x, E A if there is a linear mapping u of E into F such that x + f ( x o ) + u(x - x,) is tangent to f at x,. We have just seen that this mapping is then unique; it is called the derivative (or total derivative) o f f at the point x,, and writtenf'(x,) or Df(x,).
(8.1 .I)If the continuous mapping f of A into F is diferentiable at the point xo, the deriuative f '(x,) is a continuous linear mapping of E into F.
Let u = f ' ( x o ) .Given E > 0, there is r such that 0 < r < 1 and that lltll < r implies llf(xo t ) -f(xo)ll < 4 2 and llf(xo + t ) -f(xo) - 4l)ll < & IllIlP; hence IltlI d r implies Ilu(t)ll < E , which proves u is continuous by (5.5.1).
+
The derivative (when it exists) of a continuous mappingfof A into F, at a point x, E A, is thus an element of the Banach space Y(E; F) (see Section 5.7) and not of F. In what follows, for u E 9 ( E ; F) and t E E, we will write u . t instead of u ( t ) ; we recall (Section 5.7) that Ilu .tll < llull . lltll and that
150
Vlll
DIFFERENTIAL CALCULUS
When E has finite dimension n and F has finite dimension m , f ' ( x o ) can thus be identified to a matrix with nz rows and n columns; this matrix will be determined in Section 8.10. Examples
(8.1.2) A constant function is differentiable at every point of A, and its derivative is the element 0 of 9 ( E ; F). (8.1.3) The derivative of a continuous h e a r mapping u of E into F exists at every point x E E and Du(x) = u. For by definition u(xo) + u(x - x o ) = u ( x ) . (8.1.4) Let E, F, G be three Banach spaces, ( x , y ) --* [x . y ] a continuous bilinear mapping of E x F into G. Then that mapping is differentiable at every point (x,y ) E E x F and the derivative is the linear mapping (s, t ) --* [x . tl [s . y l .
+
For we have [(x
+ s) . ( y + t ) ] - [ x . y ] - [ x . t ] - [ s . y ] = [s. t ]
and by assumption, there is a constant c > 0 such that l\[s * t]ll < c . llsll * (It/l (5.5.1). For any E > 0, the relation sup(llsll, Iltll) = II(s, t)l\ Q E / C implies therefore
"I
+ ). . ( y + t>l - [x ul - [x . 11 - 1s - YlII *
,< & IKS, t)11
which proves our assertion. That result is easily generalized to a continuous multilinear mapping. (8.1.5) Suppose F = F, x F, x ... x F, is a product of Banach spaces, and f = (f i , . . . ,f,) a continuous mapping of an open subset A of E into F. In order that f be differentiable at xo , a necessary and suficient condition is that each fi be differentiable at xo , and then f ' ( x o ) = ( f ; ( x o ) ,. . . ,fA(xo)) (when 2 ( E ; F) is identiJied with the product of the spaces Y(E; FJ).
Indeed, any linear mapping u of E into F can be written in a unique way u = (ul, . . . , u,), where u i is a linear mapping of E into F i , and we have by definition \lu(x)II = sup(llul(x)(~, . . . , ~ ~ u , ( x ) ~whence ~), it follows
2 FORMAL RULES OF DERIVATION 151
(by (5.7.1) and (2.3.7)) that llull
= sup(jlu, I/,
identification of 2 ( E ; F) with the product
m
. . . , IIuJ), which allows the
fl Y(E; Fi). From
the defini-
i= I
tion, it follows at once that u is the derivative off at x , if and only if ui is the derivative of f i at x, for 1 < i Q m. Remark. Let E, F be complex Banach spaces, and E,, F, the underlying real Banach spaces. Then if a mapping f of an open subset A of E into F is differentiable at a point x o , it is also differentiable with the same derivative, when considered as a mapping of A it7tO F, (a linear mapping of E into F being also linear as a mapping of Eo into F,). But the converse is not true, as the example of the mapping z + Z (complex conjugate) of C into itself shows at once; as a mapping of R2 into itself, u : z + 5 (which can be written (x,y ) -+ ( x , - y ) ) is differentiable and has at each point a derivative equal to u, by (8.1.3); but u is not a complex linear mapping, hence the result. We return to that question in Chapter IX (9.10.1). When the mapping f of A into F is differentiable at every point of A, we say that f is differentiable in A ; the mapping x + f ’ ( x ) = Df(x) of A into Y(E; F) will be writtenf’ or Dfand called the derivative o f f in A.
2. FORMAL RULES OF D E R I V A T I O N
(8.2.1) Let E, F, G be three Banach spaces, A an open neighborhoodof x, E E, f a continuous mapping of A into F, yo = f ( x , ) , B an open neighborhood of y o in F, g a continuous mapping of B into G. Then iff is differentiable at x, and g differentiable at y o , the mapping h = g o f (which is defined and continuous in a neighborhood of x,) is differentiable at x, , and we have h’(x0) = S’(Y0) o f ‘(xo).
By assumption, given E such that 0 < E < 1, there is an r > 0 such that, for 1)s11 < r and litil Q r, we can write f ( x 0 + s> =f(x,>+f’(xo) . s + 01(s) S(Y0 + t ) = d Y 0 ) + S’(Y0) . t
+ OAt)
with i\ol(s)I\ Q ~ l l s l land l\02(t)II< ~jltll.On the other hand, by (8.1.1) and (5.5.1), there are constants a, b such that, for any s and t, llf’(xo>*
SII
Q a llsll
and
IlS‘(Y0)
. tll
< b lltll
152
Vlll
DIFFERENTIAL CALCULUS
hence Ilf’(x0)
*
+ o,(s)ll G (a + 1) llsll
for /Is/l < r. Therefore, for ~~s~~ < r/(a and
Iloz(f’(x0)
+ I),
. s + Ol(S))lI
II9’(Yo)
we have Q (a f
I)&Ibll
. Ol(S>ll G bE llsll
hence we can write
h(x0 + $1 = d Y 0 +f’(xo) * s + OI(S)) = S(Y0) + S’(Y0) . (f’(xo>. s) + ads) with Ilo3(s)ll Q (a
+ b + 1)E I l ~ l l ,
which proves the theorem.
(8.2.1)has of course innumerable applications, of which we mention only the following one: (8.2.2) L e t x g be twto continuous mappings of the open subset A of E into F. I f f and g are differentiable at x, , so are f + g and af ( a scalar), and we have (f+ g)‘(xo)= f ‘ ( X 0 ) + g’(x0) and (af)’(x,>= af‘(x0). The mapping f + g is composed of (u, v) 4u + v, mapping of F x F into F, and of x + ( f ( x ) , g ( x ) ) , mapping of A into F x F; both are differentiable by (8.1.3)and (8.1.5),and the result follows (for f + g ) from (8.2.1).For af the argument is still simpler, using the fact that the mapping u+au of F into itself is differentiable by (8.1.3).Of course, (8.2.2) could also be proved very simply by direct arguments. Let E, F be two Banach spaces, A an open subset of E, B a n open subset of F. If A and B are homeomorphic, and there exists a differentiable homeomorphism f of A onto B, it does iiot follow that, for each x, E A, f ‘ ( x o ) is a linear homeomorphism of E onto F (consider e.g. the mapping 5 + of R onto itself).
c3
(8.2.3) Let f be a homeomorphism of an open subset A of a Banach space E onto an open subset B of a Banach space F, g the inverse homeomorphism. Suppose f is differentiable at the point x, ,and f ’(x,) is a linear homeomorphism of E onto F; then g is differentiable at y o = f(xo) and g‘(y,) is the inverse mapping t o f ‘ ( x o )(cf. (10.2.5)).
2 FORMAL RULES OF DERIVATION 153
+
By assumption, the mapping s + f ( x , s) - f ( x o ) is a homeomorphism of a neighborhood V of 0 in E onto a neighborhood W of 0 in F, and the inverse homeomorphism if t +g(yo + t ) -g(y,). By assumption, the linear mappingf’(xo) of E onto F has an inverse u which is continuous, hence (5.5.1) there is c > 0 such that Iju(t)/I 9 c Iltll for any t E F. Given any E such that 0 < E < 1/2c, there is an r > 0 such that, if we write f ( x o + S ) - f ( x o ) = f ’ ( x , ) a s + ol(s), therelation llsll < r implies Ilol(s)II 9 E IIsII. Let r‘ now be a number such that the ball lltli 9 r’ is contained in W and that its image by the mapping t -+ g(y, t ) - g(y,) is contained in the ball llsll 9 r. Let z = g(y, + t ) - g(y,); by definition, for lltll < r ’ , this equation implies t = f ( x , z ) - f ( x o ) and as llzll 9 r, we can write t = f ’ ( x , ) . z + ol(z), with liol(z)II 9 E llzll. From that relation we deduce
+
+
u * t = u * (f’(x,) * z)
+u
*
ol(z) = z
+u
*
ol(z)
+
by definition of u, and moreover IIu * ol(z)II 9 c llol(z)IJ < CE llzll < 1 1 ~ 1 1 , hence IIu . t/l 2 llzll - 4 llzll = -t llzll; therefore llzll < 2Ilu. tll < 2cIltl1, and finally IIu . ol(z)II < CE llzll < 2c% Iltll. We have therefore proved that the relation lltll < r‘ implies IIg(y, + t ) - g(y,) - u . t 11 < 2cZc Iltll, and as E is arbitrary, this completes the proof. The result (8.2.3)can also be written (under the same assumptions)
PROBLEMS 1. Let E be a real prehilbert space. Show that in E the mapping x + llxli of E into R is differentiable at every point x # 0 and that its derivative at such a point is the linear mapping s+(sIx)/IIx/l. 2. (a) In the space (co) of Banach (Section 5.3, Problem 5) show that the norm x + J/xlj is differentiable at a point x = (8.) if and only if there is an index no such that > I f . 1 for every n # n o . Compute the derivative. (b) In the space I’ of Banach (Section 5.7, Problem I ) , show that the norm x + llxll is not differentiable at any point (use (8.1.1) and Problem l(c) of Section 5.7).
ltnol
3. Let f be a differentiable real valued function defined in an open subset A of a Banach space E. (a) Show that if at a point xo E A,freaches a relative maximum (Section 3.9, Problem 6), then Df(xo) = 0. (b) Suppose E is finite dimensional, A is relatively compact,fis defined and continuous in A, and equal to 0 in the boundary of A. Show that there exists a point x,, E A where D f ( x o )= 0 (“Rolle’s theorem”; use (a) and (3.17.10)).
154
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DIFFERENTIAL CALCULUS
3. DERIVATIVES IN SPACES
O F C O N T I N U O U S LINEAR F U N C T I O N S
(8.3.1) Let E, F , G be three Banach spaces. Then the mapping (u, v) -+ v u (also written vu) of 9 ( E ; F) x 9 ( F ; C ) into 9 ( E ; G) is diferentiable, and the derivative at the point (uo , vo) is the mapping (s, t ) -+ vo s + t uo . 0
0
0
If we observe that, by (5.7.5), the mapping (u, v) -+ v u is bilinear and continuous, the result is a special case of (8.1.4). 0
(8.3.2) Let E , F be two Banach spaces, such that there exists at least a linear homeomorphism of E onto F. Then the set 2 of these linear homeomorphisms is open in 9 ( E ; F ) ; the mapping u -+ u - l of iff onto the set Y? of linear homeomorphisms of F onto E is continuous and diferentiable, and the derivative of u -+ u-l at the point uo is the linear mapping (of Y ( E ; F ) into Y(F; E ) )
-'
s --f
-24;'
0
s
0
u;'.
1. We consider first the case F = E, and write 1, for the identity mapping of E. Then:
(8.3.2.1) If llwll < 1 in 9 ( E ; E), the Zinear mapping 1, + w is a homeomorphism, its inverse (1, + w)-' is equal to the sum of the absolutely convergent series
m
C (n=O
l)nw", and we have
(8.3.2.2)
11(IE
We have
N n=O
+ w)-'
- l E + wII
IIwII" = (1 - IIw IIN")/(l -
- Ilwll>.
~~K'~\2/(1
Ilwll)
< l/(l - Ilwll),
hence, by
m
(5.7.5), (5.3.1), (5.3.2) and (5.7.3), the series convergent in 9(E; E). Moreover, we have (1E+W)(1E-W+w2
= (1, - w
(-1)"
is absolutely
n=O
+"'+(-l)NWN)
+ w2 +
* "
+ ( - l ) N W N ) ( l E + w) = 1 E - ( - l ) N + ' W N + l .
and as w N + l tends to 0 with 1/N, we have by definition and by (5.7.5),for the W
element v
= n=O
( - 1 ) " ~ ~ of 9 ( E ; E), (1,
+ w)v = v ( l E + w) = l E , which
proves the first two statements; the inequality (8.3.2.2) follows from the relation (1, + w)-' - 1, + w = w2(lE - IY f M!2 and from (5.7.5) and (5.3.2).
+
. . a ) ,
4
DERIVATIVES OF FUNCTIONS OF ONE VARIABLE
155
2. Consider now the general case; suppose s ~ 9 ( EF); is such that ( ( s ( (. Ilui'll < I ; then the element 1, + u0-'s, which belongs to 9 ( E ; E), has an inverse, due to (5.7.5) and (8.3.2.1);as we can write uo s = uo(1, ui's), the same is true for uo + s, the inverse being (1, u;'s)-'u,'; hence we have
+ +
+
(ug
+ s)-'
Applying (8.3.2.2)to Il(u0
+ s)-'
- t!;' = ( ( 1 E
UJ = ui's,
- u;' Il(u0
- 1E)u;'.
we obtain, for llsll < l/llu;'II
+ u;'su;'ll
Therefore, if we take llsll
+ u,'s)-'
d Ilu,'113. lIsll'/(l - lu;'II
*
Ilsll).
< 1 / 2 ~ ~ u ~we' ~ have l,
+ s)-'
- u;'
+ u,'su,'ll
d c lls11'
' ~ this ~ ~ ends , the proof. with c = 2 ~ ~ u ; and
4. DERIVATIVES OF F U N C T I O N S O F O N E VARIABLE
When we specialize E to a one-dimensional vector space (identified to R or C), we know that 9 ( E ; F) is naturally identified to F itself, a vector b E F being identified to the linear mapping 5 -+ b( of E into F (5.7.6). If f is a differentiable mapping of an open set A c E into F, its derivative D f ( t 0 )at a point toE A is thus identified to a vector of F, and the mapping Df to a mapping of A into F. If F itself is one-dimensional (identified to R or C), we obtain the classical case of the derivative (at a point) as a number. The general results obtained above boil down in that last case to the classical formulas of calculus; for instance, (8.3.2),when E and F are one-dimensional, is simply the formula giving the derivative of 1/4 as equal to - 1/(' for 5 # 0. We explicitly formulate the following consequence of (8.2.1):
(8.4.1) Let E, F be two real (resp. complex) Banach spaces, f a differentiable mapping of an open subset A of E into F, g a diferentiable mapping of an open subset I of R (resp. C) into A; then the derivative at 5 E I of the composed mapping h =f .g of I into F is the vector of F equal to Df(g( 0, k > 0. (b) The real functionfequal to x z sin(l/x) for x # 0, to 0 for x 0, is differentiable in R, but ( f ( x ) - f ( y ) ) / ( x - y ) has no limit when ( x , y ) tends to ( 0 , O ) in the set of pairs such that x > 0, y > 0, x # y . (c) In the interval I = [O, I], the sequence of continuous functions f. is defined as follows: fo(t) = t ; for each n > 1, fn has the form at fl in each of the 3" intervals
+
+
:
+
k 3"
-Sf 0, such that lim t, = 0 and
x
4.
t2)
t2 t:.)
+
n-m
that Ilt;'f(tnan)ll = a. tends to co; one can suppose that the sequences ( I , ) and 1,) are decreasing and tend to 0. Define a continuous mapping g of [0, 11 into E such that g(0) = 0, that g'(0) exists and is equal to 0, and that g ( J i t,) = faan).) 5. (a) Let E, F be two Banach spaces,fa continuous mapping of an open subset A of E into F. Show that if is diffcrentiable at x o t A, it is quasi-differentiable at xo and its quasi-derivative is equal to its derivative.
(din
158
Vlll DIFFERENTIAL CALCULUS
(b) Suppose E has finite dimension. Show that iff is quasi-differentiable at xo E A,
f is differentiable at xo . (Use contradiction: let u be the quasi-derivative o f f at xo , and suppose there is a > 0 and a sequence (x,) of points of A, tending to xo , such that llf(x,) -f(xo) - u (x. - xo)II> aIIx, - xo(I. Using the local compactness of E, show that one may suppose that the sequence (((x, - xo 11) is decreasing, and that the sequence of the vectors z. = (x,- xo)/Ilx,- xo 11 tends to a limit in E; then define a continuous mapping g of [0, 11 into E such that g(0) = xo , that g'(0) exists, but that
-
u(g'(0)) is not the derivative of t + f ( g ( t ) ) at t = 0.)
6. Let I = [0, I], and let E be the Banach space WR(I). In order that the mapping x -+ llxll of E into R be quasi-differentiable at a point xo , it is necessary and sufficient that the function 1 Ixo(t)l reaches its maximum in I at a single point to E I ; the quasi-derivative of x + l/xll at xo is then the linear mapping u such that u(z) = z(to) if -+
xo(to)> 0, u(z) = -z(to)
if xo(to)< 0 (compare Section 8.2, Problem 3). (To prove the condition is necessary, suppose lxol reaches its maximum at two distinct points t o , tl at least; let y be a continuous mapping of I into itself, equal to 1 at t o , to 0 at t l ; examine the behavior of ( I/xo hyll - /IxoIl)/has the real number h # 0 tends to 0. To prove the condition is sufficient,let h + z, be a continuous mapping of I into E, having a derivative a f E at h = 0 and such that zo =O; observe that if r, is the largest number in I (or the smallest number in I) where t Ixo(t) z,(t)l reaches its maximum, then t, tends to to when h tends to 0.) Deduce from that result that the mapping x -+ j/x/I of E into R is not differentiable at any point (compare to Section 8.2, Problem 2). 7. Letfbe a continuous mapping of an open subset A of a Banach space E into a Banach space F. Supposefis lipschitziun in A : this means (7.6, Problem 12) that there exists a constant k > 0 such that llf(xl) -f(xZ)ll < k I/xl - xz/I for any pair of points of A. Let xo E A, and suppose there is a linear mapping u of E into F such that, for any vector a # 0 in E, the limit of (f(xo f a t ) --.f(x0))/t when t # 0 tends to 0 in R,exists and is equal to u(a). Show that f i s quasi-differentiable at xo . llu rb /I of 8. (a) Let u , b be two points in a Banach space E. Show that the mapping t R into itself has a derivative on the right and a derivative on the left for every t E R (prove that if 0 < t < s, then ([la bril- Ilall)/t < (ila bsll- Ilall)/s and use (4.2.1)). (b) Let u be a continuous mapping of an interval I E R into E. Show that if at a point t o E I, u has a derivative on the right, then t Ilu(t)ll has at to a derivative on the right and
+
+
-+
-+
+
+
+
--f
(D+ IlulI)(to)
< IID+u(to)ll
(apply (a)). (c) Let U be a continuous mapping of I into Y(E; E). Show that if at a point to E I, U has a derivative on the right and U(t0) is a linear homeomorphism of E onto itself, then the mapping f l\(U(t))-lII= f ( t ) , which is defined in a neighborhood of t o , has a derivative on the right at t o , and that
-
5. T H E M E A N VALUE THEOREM
Let I = [ct, p] be a compact interval in R, f a continuous mapping of I into a Banach space F, cp a continuous mapping of I into R. We suppose that there is a denumerable subset D such that, for each 5 E I - D, f and cp have both a derivative at 5 with respect to I (8.4), and that (1 f'(5)II < cp'(5j. Then I l f (PI - f(4I1 G cp(P) - c p ( 4
(8.5.1)
5 THE MEAN VALUE THEOREM
159
Let n -+ p n be a bijection of N onto D ; for any E > 0, we will prove that left hand side being independent of E , this will complete the proof. Define A as the subset of I consisting of the points 5 such that, for a < 5 < 5 ,
IIf(p) -f(a)II < cp(p) - cp(a) + E(P - a + 2 ) ; the
It is clear that a E A ; if 5 E A and a < v] < 4, then v] E A also, by definition; this shows that if y is the 1.u.b. of A, then A must be either the interval [a, y [ or the interval [a, y ] ; but in fact, from the definition of A it follows at once that A = [ E , y ] . Moreover, from the continuity of f a n d cp it follows that (8.5.1.1)
Ilf(Y) -f(.>II< d Y ) - cp(4 + 4 Y - 4 + E
c
2-"
Pn 0, ex is strictly increasing (by (8.5.3)), and hence e = e' > e0 = 1. The function ex is also written exp(x) or exp x. The function log, x is written log x and it follows from (8.2.3)and (4.2.2) that D(log x) = I/x for x > 0. Furthermore D(a") = log a * a".
PROBLEM
Study the variation of the functions (1
+
!-)"".
(1
+ $)Y
(1 +!)(I
+ ;):
(I
+
y+'
for x > 0 , p being a fixed arbitrary positive number; find their limits when x tends to
+to.
9. PARTIAL DERIVATIVES
Let f be a differentiable mapping of an open subset A of a Banach space E into a Banach space F; Dfis then a mapping of A into 9 ( E ; F). We say that f is continuously diferentiable in A if Dj i s continuous in A. Suppose now E = E, x E, . For each point (al, a z )E A we can consider the partial mappings x1 +f ( x , , a z ) and x, +f(a,, xz) of open subsets of El and E, respectively into F. We say that at (a,, a,), f is d/ferentiab/e with respect to the first (resp. second) variable if the partial mapping x1 +f(xl, a 2 ) (resp. x2 +f ( a , , xz)) is differentiable at a, (resp. a z ) ; the derivative of that mapping, which is an element of -%'(El ;F) (resp. 9 ( E z ; F)) is called the partial derivative off at ( a l , a,) with respect to the first (resp. second) variable, and written D,f(a,, a,) (resp. D,f(a,, a,)).
PARTIAL DERIVATIVES
9
173
(8.9.1) Let f be a continuous mapping of an open subset A of El x E, into F. In order that f be continuously differentiable in A, a necessary and suficient condition is that f be differentiable at each point with respect to the first and the second variable, and that the mappings (x, , x,) -+ D, f ( X I , x,) and (x,, x,) -+ D,f(x,, x2) (of A into 2’(El; F) and 9 ( E 2 ; F) respectioely) be continuous in A. Then, at each point (x,, x,) of A, the derivative off is given by (8.9.1.1)
Df(x1,
~
2
)(11, t 2 )
=Dif(xi,
~ 2 ) ti.
+ Dzf(x1,
~ 2 ) t. 2 .
(a) Necessity The mapping x, -)f(x,, a,) is obtained by composing f and the mapping x1 -+ (x,, a,) of El into El x E2 , the derivative of this second mapping being r, - + ( t , , 0) by (8.1.2), (8.1.3), and (8.1.5). Then by (8.2.1), x1 f(x,, a,) has at (a,, a,) a derivative equal to t , -+ Df(a,, a,) . (tl, 0). If we call i, (resp. i,) the natural injection t , .+ ( t , , 0) (resp. t2 + (0, t , ) ) , which is a constant element of 2’(El; E, x E2) (resp. 2’(E2; E, x E,)), we therefore see that D, f ( a l ,a,) = Df (al, a,) i,, and similarly D2f (al, a,) = Df(a,, a2) i, (all this is valid iff is simply supposed to be differentiable in A). As the mapping ( 0 , u ) -+ v u of 9(El x E,; F) x 9( E l ; El x E,) into 9 ( E l ; F) is continuous ((5.7.5) and (5.5.1)), the continuity of D, f and D, f follows from that of Df; finally, as ( t , , t , ) = il(tl) + i2(t2),we have (8.9.1 .I). (b) Suficiency Write -+
0
0
0
f ( 0 , + 4 > a2 + f 2 ) - f (a19 a,) = (f(a1
Given
E
+ t , , 0 2 + t 2 ) -f(a, + t , , 0 2 ) ) + ( f b l + t , , a,)
-f(a,, a,)).
> 0, there is, by assumption, an r > 0 such that, for IItlll d r IIf(a1
+ t , , a2) -f(a,,
02)
- D,fta,,a2), fill
E
Iltlll.
On the other hand, we have in a ball B of center (al, a,) contained in A, by (8.6.2) lIf(a1
+ t , , a2 + 12) -f(a1 + t l , 0 2 ) - D,f(a1 + t l , 0 2 )
*
t2II
The continuity of the mapping D, f therefore implies that there is r‘ > 0 such that for /lt,Ij < r’ and Ilt,II < r‘, we have IIf(a,
+ t,, a, +
-f(a,
+ t , , 0,) - D2f(al + I , ,
0,)
- t,Il
and on the other hand
I/ D2f (01 + t17a,) - D2f (al, a,)ll
<E
hence, by (5.7.4) lIDzf(a1 + t ~02) ,
t2
- D,f(ai,
02)
. t ~ l ld E llt,ll.
G E IIt211
174
DIFFERENTIAL CALCULUS
Vlll
Finally, for sup((lt,I(, IIt211) < inf(r, r') we have IIf(a1 + t,, a2
+ tz) - f ( a , ,
a,)
- D,f(a,, a,) * tl - Dzf(a1, 6 ) .hIl
G 4 E SUP(IItl I1 I1t z Ill 9
which proves (8.9.1.1); the continuity of Df follows from the fact that (8.9.1.1) can be written Df= D, f 0 prl D, f 0 pr, and from (5.7.5). Theorem (8.9.1) can be immediately generalized to a product of n Banach spaces by induction on n ; if we combine that result with (8.2.1), we obtain
+
(8.9.2)
Let f be a continuously direrentiable mapping of an open subset
A of E =
n n
i=l
Ej into F, and, for each i, let g i be a continuously differenti-
able mapping of an open subset B of a Banach space G into E i , such that (g,(z),. . ,gn(z))E A for each z E B. Then the composed mapping h = f 0 ( g l , . . . , gn) is continuously differentiable in B, and we have
.
=c(Dk f(gl(z), . n
Dh(z)
k= 1
* * 9
gn(z)))
Dgk(Z)*
PROBLEMS
1. Let E, F be two Banach spaces,f a continuous mapping of an open subset A of E into F. Suppose that for each x E A, there is an element u ( x ) E Y(E; F) such that, for every vector y E E, the limit of ( f ( x ty) - f ( x ) ) / f when t # 0 tends to 0 in R,exists and is equal to u(x) . y. Suppose in addition that x + u(x) is a continuous mapping of A into Y(E; F). Show that f is then continuously differentiable in A and that u(x) = D f ( x ) for every x E A. (Apply the mean value theorem to the function f + f ( x f y )for t E [0, 1 I.) 2. Let E be the space (co) of Banach (Section 5.3, Problem 5 ) ; let F be the complex Banach space (co) i(co),consisting of all sequences z = (,O of complex numbers such that lim 5. = 0, with the norm llzll= sup 15.1. We denote by Fo the real Banach space
+
+
+
"
n- m
underlying F (Section 5.1). Let I C R be an open interval containing 0, and for each integer n 3 0, let f. be a continuous mapping of I into C such that the condition lim I, = 0 implies lim f&) = 0; this defines a mappingf: (6")*(fa((")) of E into Fo n-m
n-m
.
(a) Suppose f i s continuous in a neighborhood of 0. In order that f be quasi-differentiable at the point 0 (Section 8.4, Problem 4), it is necessary and sufficient that for each n the derivative f.'(O) exist and that there exist two numbers A > 0 and 6 > 0 such that the relation It1 < 6 implies If.(t) -f.(O)I < Alrl for every n ; this implies that S U P lL(0)l < +a. (b) In order that f be differentiable at 0, it is necessary and sufficient that for every E > 0, there is a 6 > 0 such that the relation It1 G 6 implies Ifn(t)-fn(O) -K(O)tI G E It1 for every n.
10 JACOBIANS 175
(c) In order that the derivativef' exist in a neighborhood of 0 in E and be continuous at 0, a necessary and sufficient condition is that there exist a neighborhood J c I of 0 such that: (1) eachf,' exists in J ; (2) sup Ifk(0)l < + a ;(3) the sequence(f.')isequiconn
tinuous a t the point 0 (Section 7.5). (See Section 8.6, Problem 3.) (d) Let f . ( t ) = e""/n for every n 2 1,Yo(()= I . Show that f is quasi-differentiable at every point x E E; if u(x) is the quasi-derivative o f f at the point x , show that the mapping ( x , y ) u(x) . y of E x E into Fo is continuous, but that f i s not differentiable at any point of E. Let f be a continuous mapping of an open set A of a Banach space E into a Banach space F. Suppose that for any x E A and any y E E, lim ( f ( x t y ) - f ( x ) ) / t = g ( x , y ) --f
+
c-o,r+o
exists in E. If, for y l E E, 1 < i < n, and xo E A, each of the mappings x + g ( x , y l ) is
+ + +
I:
continuous at xo , show that g ( x o , y1 yz . . . y.) = g(xo,y l ) (apply the mean 1=1 value theorem). Let E l , Ez , F be three Banach spaces,fa continuous mapping of an open subset A of El x E2 into F. In order that f be differentiable at ( a l , a 2 )E A, it is necessary and sufficient that: (1) Dlf(al, a 2 ) and D2f(al, a 2 ) exist; (2) for any E > 0, there exists 6 > 0 such that the relations IIfJ < 6, Ilt2i1< 6 imply Ilf(a1
+ t l , a , + t z ) -fh+ t l ,
0 2 ) -f(a1,
a2
+
t 2 ) +f@l,
aJil < E(llflll+ Iltzll).
Show that the secondconditionissatisfied if Dlf(al, az)exists and thereisaneighborhood V of (al, a 2 ) in El x E2 such that D2f exists in V and the mapping ( x I ,x 2 ) - + D 2 f ( x I x, z ) of V into P ( E 2 ; F) is continuous. Let f be the real function defined in R Z by f(x,y ) = (xy/r)sin(l/r) for (x, y ) # (0, O), with r = (x' Y ' ) ' ' ~ , and f ( 0 , O ) = 0. Show that Dlf and D 2 f exist at every point ( x , y ) E R2, and that the four mappings x Dlf(x, b), y Dlf(a, y), x D 2 f ( x , b), y --f D 2 f ( a ,y ) are continuous in R for any (a, b) E R2, but that f i s not differentiable at
+
--f
--f
--f
NO).
Let I be an interval in R , f a mapping of I' into a real Banach space E, such that, for any ( a l , ... , a,) E I,, each of the mappings x , + f ( a ~ ,... , u,.-I, x j . U J + I , . . ,a,) (1 < j < p ) is continuous and differentiable in I, and furthermore, the p functions D j f ( l < j < p ) arc bounded in 1'. Show thatfis continuous in I' (use the mean-value theorem).
.
10. JACOBIANS
We now specialize the general result (8.9.1) to the most important cases. 1. E = R" (resp. E = C"). Iff is a differentiable mapping of an open subset A of E into F, the partial derivative D,f(a,, . . . , a,) is identified to a vector of F (Section 8.4), and the derivative off is the mapping fl
((1,
cn)+
k=l
Dkf(al?
...)'%)lk.
If Dfis continuous, SO is each of the Dk$ Conversely, if each of the mappings DJexists and is continuous in A, then f is continuously differentiable in A. 2. E = R" and F = R" (resp. E = C" and F = C'"). Then we can write
176
Vlll
DIFFERENTIAL CALCULUS
f = (ql,. . . , cp), where the qi are scalar functions defined in E, and by (8.1 -5)f is continuously differentiable if and only if each of the 'pi is continuously differentiable; again, by case I , 'pi is continuously differentiable if and only if each of the partial derivatives D j q i (which is now a scalar function) exists and is continuous. Furthermore, the (total) derivative of f is the linear mapping (Cl,
with
*.*)
Cn)+(Vl,
. . . >V m )
n
Vi
=
C (Djcpi(E1, j=
* * * 3
an>>Cj ;
1
in other words, f',which is a linear mapping of R" into R" (resp. of C" into C"), corresponds to the matrix (Djqi(al, . . . , a,)), which is called the jacobian matrix o f f ( o r of cpl,. . . , cp), at (al, . . . , a,,). When m = n, the determinant of the jacobian (square) matrix off is called thejacobian off(or of cpl, . . ., cp,). Theorem (8.9.2) specializes to Let cpj (1 < j < m ) be m scalar functions, continuously diferentiable in an open subset A of R" (resp. C"); let $i (1 < i < p ) be p scalar functions, continuously diflerentiable in an open subset B of R" (resp. C") containing the image of A by (ql, .. ., cp;), then $ O,(x) = $i(cp,(x), ..., cp,(x)) for x E A and 1 < i < p , we have the relation
(8.10.1)
(Dk Oi) = (Dj II/i)(Dkq j ) between the jacobian matrices; in particular, when m = n = p , we have the relation det(D,Oi) = det(Dj$,) det(Dkcpj) between the jacobians.
a
We mention here the usual notations f;,(tl, . . . , t,,),-f ( t l , . . . , t,,),
at i
for Dif(tl, . . . , tn),which unfortunately lead to hopeless confusion when substitutions are made (what does f,'(y, x) or f J x , x ) mean?); the jacobian det(Djcpi(tl, . . . , t,)) is also written D(cpl, . . ., cp,)/D(t,, . . . , t,) or d(cp1, * * * cp")/a(tl, . . * , t"). 9
11. DERIVATIVE O F A N INTEGRAL DEPENDING O N A PARAMETER
(8.1 1.I)Let I = [cr, /3] c R be a compact interaal, E, F real Banach spaces, f a continuous mapping of I x A into F (A open subset of E). Theti g(z) = ff(t, z ) d( is continuous in A.
11 DERIVATIVE OF A N INTEGRAL DEPENDING ON A PARAMETER
177
Given E > 0 and zo E A, for any 5 E I, there is a neighborhood V(5) of 5 in I and a number r(5) > 0 such that for q E V(5) and ))z- zo))< r(5), 11 f ( q , z) -f(5, zo)II < E . Cover I with a finite number of neighborhoods V(ti), and let r = inf(r(5J). Then IIf ( 5 , z) -f ( 5 , zo)Il < E for llz - zoII < r and any 5 E I; hence, by (8.7.7) for llz - z0II < r. Q.E.D.
It&) - g(z0)II
< E(B - 4
(8.11.2) (Leibniz’s rule) With the same assumptions as in (8.11.1), suppose in addition that the partial derivative D2f with respect to the second variable exists and is continuous in I x A. Then g is continuously diflerentiable in A, and
(observe that both sides of that formula are in Y(E; F)). The same argument as in (8.51.I) applied to D,f, shows that given E > 0 and zo E A, there exists r > 0 such that IID, f ( 5 , z ) - D, f ( 5 , zo)I(< E for ))z- zo)I< r and any 5 E I; hence, by (8.6.2)
+ t ) -At, ZO) - D, f(5, zo) tll < E lltll for any 4 E I and any t such that lltll < r. By (8.7.7)we therefore have IIf(5,
20
*
Butby(8.7.6)and(5.7.4)wehaveJa’(D2f(t;,zo).t)d5 =(fD2f(5,zo)d5) . t for any t , and this ends the proof.
PROBLEMS
1. Let J C R be an open interval, E, F two Banach spaces, A an open subset of E, f a continuous mapping of J x A into F such that D,fexists and is continuous in J x A, 0: and fl two continuously differentiable mappings of A into J. Let
/*(,,”c, O(Z)
g(z) =
4d6.
178
Vlll
DIFFERENTIAL CALCULUS
Show that g is continuously differentiable in A, and that y’(z) is the linear mapping
2.
(apply (8.9.1) and (8.11.2)). Letf,g be two real valued regulated functions in a compact interval [a, b ] ,such thatfis decreasing in [a, b] and 0 < y ( t ) < 1. Show that
s,:
where A
=
s.”
g(1) dt.
1f
( t ) dt
< /abf(t)g(t)dt < JOa+h) dt
When is there equality? (Consider the integrals
Jba’+h”)f(t) dr, where h(y) =
1
g ( t ) df, as
s:
f ( t ) g ( t ) dt and
functions of y , and similarly for the other
inequality.) 3. Let the assumptions be the same as in Problem 1, except that CL and p are merely supposed to be continuous, but not necessarily differentiable, but in addition it is supposed that f(ar(z), z)= 0 and f(P(z), z ) = 0 for any z E A. Show that g(z) is continuously differentiable in A, and that g’(z)
L P(Z)
=
Dzf(E, z ) d[. (Use Bolzano’s theorem (3.19.8)
to prove that if 5 belongs to the interval of extremities p(zo) and p(z), there i3 z’ t A such that llz’ - zo/I< Ilz ZOII and [ = p(z’); if M is the 1.u.b. of llDzfll in a neighborhood of (/3(zo),zo), use the mean value theorem to show that Ilf([, z)ll < M Ilz - zoI/. 4. Let 1 [a, b ] ,A = [c, d ] be two compact intervals in R , f a mapping of I x A into a Banach space E, such that: ( I ) for every y E A, the function x + f ( x , y ) is regulated in I and for every x t 1, the function y -.f(x, y ) is regulated in A; ( 2 ) f is bounded in I x A ; (3) if D is the subset of I i( A consisting of the points (x, y ) where f is not continuous, then, for every xo E I (resp. every y o E A), the set of points y (resp. x) such that (xo,y ) E D (resp. (x,y o ) t D) is finite. ~
Jgb
(a) Show that the function g ( y ) = f ( t , y ) dt is continuous in A. (If
E
> 0 and
y o E A are given, show that there is a neighborhood V of y o in A and a finite number of intervals Jr c I (1 < k < n) such that the sum of the lengths of the Jr is < E and that, if
W =I -
u J, , f i s continuous in W x V; to prove that result, use the Borel-Lebesgue
k = l
theorem (3.17.6).) (b) Deduce from (a) that
/:lrv Jabf(x,Y ) dx = jabdxJcd f ( x , Y ) dy.
s: Jab
(Consider the two functions z + dy
f ( x , y ) dx and z -->
(Cf. Section 13.21). (c) Deduce from (b) that if a = c, b = d , then
(consider the function equal to f ( x , y ) for y
< x , to 0 for y > x).
y ) dy for z E A.)
12 HIGHER DERIVATIVES 5.
179
(a) Let f be a strictly increasing continuous function in an interval [0, a ] , such that
r’
f(0) = 0 ; let g be the inverse mapping, which is continuous and strictly increasing in the
.c,p <
O,
ysR;
for x>O,
y20, p>l,
q>l,
a > 0,
1
1
-+-=1, P Q
b > 0 and (pa)4(qb)p> 1.
12. HIGHER DERIVATIVES
Suppose f is a continuously differentiable mapping of an open subset A of a Banach space E into a Banach space F. Then Dfis a continuous mapping of A into the Banach space 9 ( E ; F). If that mapping is differentiable at a point xo E A (resp. in A), we say thatfis tiisice drfSerentiable at x, (resp. in A), and the derivative of Df at xo is called the second deriuatiue of f a t x,,, and writtenf”(x,) or D2f(xo). This is an element of 2(E; 9 ( E ; F)); but we have seen (5.7.8) that this last space is naturally identified with the space 9 ( E , E; F) (written Y2(E; F)) of continuous bilinear mappings of E x E into F: we recall that this is done by identifying u E 2 ( E ; 9 ( E ; F)) to the bilinear mapping (s, t ) ( u * s) t ; this last element will also be written u . (s, t ) . --f
(8.12.1) Suppose f is tii’ice diferentiuble at x,; then, for any jixed t E E, the derioative of the mapping x-)Df(x) . t of A into F, at the point x o , is s + D2f(x0) . (s, t ) .
If we observe that x -+ Df(x) . t is composed of the linear mapping u + u . t of 9(E; F) into F and of the mapping x Df(x) of E into Y (E; F) the result follows from (8.2.1) and (8.1.3). -)
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180
DIFFERENTIAL CALCULUS
(8.12.2) If f is t,cice diferentiable at xo, then the bilinear mapping ( s , t ) --t D2f(xo) . (s, t ) is symmetric, in other rvords D2f(xO) . (s, t )
= D2f(xO)
Consider the function of the real variable
. ( t , $1.
5 in the interval [0, 11:
g(5) =f ( x o + 5s + 0 - f ( x o + 5s)
where s, t are such that llsll < f r , lltll < f r , the ball of center xo and radius r being contained in A. From (8.6.2) we get
and
II fYx0 + 5s) - f’(x0) - f”(xo> . (5s)/I
E
IIs I1
hence
lls’(5) - ( f ” ( X 0 ) . t >
*
sII G 2E llsll . (llsll
+ lltll)
and therefore
Ildl)- d o )
- (f”(X0) . t ) . sII
+ +
+
< 6.2 llsll (llsll + Iltll).
t ) -f(xo But g(1) - g(0) = f ( x o s t ) - f ( x o in s and 1, hence, by exchanging s and t , we get li(f”(xo)
+ s) +f(xo) is symmetric
< 6~(ilsll+ 11t11)’. < +r‘; but if we replace s and t
. t > . s - (f”(xO>+ s> . tll
Now this inequality holds for ~~s~~ < fr’, lltll by Is and I t , both sides are defined and multiplied by 1/212, hence the result is true for all s and t in E, in particular for llsll = /It11 = 1, which proves (by (5.7.7))that
llf”(xo> . ( 1 , s) -f”(x,> . (s, t>ll < 248 114 . lltll for all s and 1 ; as E is arbitrary, this ends the proof. In particular,
12 HIGHER DERIVATIVES
181
(8.12.3) Let A be an open set in R” (resp. C“); f a mapping f of A into a
Banach space F is tccdce differentiable at xo , then the partial derivatives D if are differentiable at xo, and Di Dj f ( x 0 ) = Dj Di f ( x 0 ) for 1 < i d n , 1 0 such that 11 [x . y]ii d c llxll . lly//in E x F; by definition ofthe norm in Y ( E x F; G) (5.7.1), we have llg(x-3Y)I/ Q C ( l I ~ l l + IlJ~ll)6 2c SUP(ll.~lI> IIJ~II)
hence g is a continuous linear mapping of E x F into 9 ( E x F; G), and therefore (x, y ) + [x . J!] is twice differentiable and its second derivative at (x, y ) is (by (8.1.3) and (8.12.1)) (($1,
tl)?(s2 t2)) - + b l 3
. f 2 1 + i s 2 . tll.
This is a mapping independent of ( x ,y ) , hence the result. (8.12.10) Let E, F, G be three Baiiacli spaces, A an open subset of E, B an open subset of F; iff is a p times continuously diflerentiable inupping of A into B, g a p times continuously clifferetitiable mapping of B into G , tlien 11 = g o f is a p times continuously chflc~rentiahleniappitig of A into G. For p = 1, the result follows from (8.2.1) and from the fact that u is a bilinear continuous mapping of 2 ( E ; F) x 9 ( F ; G) into (u, 11) + I ’ 9 ( E ; C) by (5.7.5). Use now induction on p ; as li’(x) = g ’ ( f ( x ) ).J”(x), and ,f and g’ are p - I times continuously differentiable, the induction hypothesis shows that g’ c , f ’ is p - 1 times continuously differentiable; from (8.12.6) and (8.12.9), it then follows that 11’ is p - 1 times continuously differentiable, hence Ii is p times continuously differentiable by (8.12.5).
184
Vlll
DIFFERENTIAL CALCULUS
Example
Suppose there is a linear homeomorphism of a Banach space E into a Banach space F, and let &? c 9 ( E ; F) be the open set of these homeomorphisms i n Y ( E ; F) (8.3.2). Then the mapping u .+ u-' of &? onto 2- is indejitiitely differentiable. We prove by induction on p that u -+ u-' is p times differentiable, the property being true for p = 1 by (8.3.2). Given v and w in 9(F; E) = M, let f ( v , w) be the linear mapping t - u t IO of L = 9 ( E ; F) into M ; it is clear that f is bilinear (and maps M x M into 9 ( L ; M)) and (5.7.5) proves that Ilf(v, w)II 6 \lull * I\wII, hence f is continuous, and therefore indefinitely differentiable by (8.12.9). Now the first derivative of u -+ u-' is, by (8.3.2), the mapping u - + f ( u - l , u - ' ) ; by (8.12.6) and (8.12.10) it follows that if u-+u-' is p times differentiable, so is u -+f(u-', u - ' ) , and therefore, by (8.12.5), u -+ u-' is p 1 times differentiable. (8.12.11)
-+
+
Remark. When f is a mapping of an interval J c R into a real Banach space F, we have defined earlier (Section 8.4) the notion of derivative off at toE J with respect to J. By induction on p , we define the pth derivative o f f at to,with respect to J, as the derivative at to (with respect to J) of the ( p - 1)th derivative o f f (which is therefore supposed to exist in a neighborhood of to in J); it is an element of F, written again Dpf(to) orf(p)( a n , f ( " ) ( f= ) 0 has at least n - 1 distinct roots in 1- 1 , 1[. (Show by induction on k that there is a strictly increasing sequence xk, < xk. < . . . < x k , of points of 1- I ,I[ such that f(')(xr, i ) f ( k ) ( ~ k , +,) < 0 for I < i < k - 1; use Rolle's theorem.)
,
,
186
4.
Vlll
DIFFERENTIAL CALCULUS
Let E, F be two Banach spaces, A an open subset of E, f a n n times differentiable mapping of A into F. Let xo E A, hi E E ( I < i < n ) be such that xo f o r O < ( , < l , l < i < n . Wedefinebyinductiononk(1 < k < n )
E
A
+ hi) - f ( ~ o )
A'f(xo; h l ) =f(xo
Akf(x,; h i , . . . ,h,)
"
+ tihr
= A"'gr(xo
; h i , ...,h k - i )
with SkW =f ( x
+
-Ax).
hk)
(a) Show that
IlA"f(xo;hi, ..., h " ) l l ~llhill' I l h 2 I l ~ ~ ~ I IlD"f(z)/I lh~Il~~~ Z t P
where P is the set of points x o (b) Deduce from (a) that Ilhlf(xo ; hi,
< llhi I/
"
+ C tih i ,0 < ti< I . (Use induction on n.) i=i
...,h.1-
.
D"f(xo) (hi, . . ,hnN
llhz /I' ' . llhnll SUP IID"f(z)
- D"f(xo)ll.
Z E P
5.
Let f be a continuously differentiable mapping of an open subset A of R2 into a Banach space E. Suppose that in a neighborhood V of (a, h ) E A, the derivative D2(Dlf) exists and is continuous. (a) Let (x,y ) E V; show that for every E > 0, there exists 6 > 0 such that the relations /hi < 6 , lkl < 6 imply lIA2f(x, Y ; h, k ) - Dz Dif(x,
~ W l! P. and the conclusion follows from (5.5.7). 0
I/
0, there exists an indefinitely differentiable mapping f of R P into E such that D"f(0) = ca for every a. (c) Deduce from (a) that if g is an indefinitely differentiable mapping of a closed
14 TAYLOR'S FORMULA
193
interval I c R into E, and J an open interval containing I, there exists an indefinitely differentiable mappingfof R into E which coincides with g in I and with 0 in R - J. 5. Let f b e a mapping of an interval 1 c R into a Banach space E, and supposefis n times differentiable at a point a E I. Show that
(use induction on n and (8.5.1) with v([) = ([ a)"-'). 6 . Let I c R be an interval containing 0, f an n - 1 times differentiable mapping of I into a Banach space E. Write -
which definesf, in I - {O}. (a) Show that iffis n p times differentiable at t = 0, f.can becontinuouslyextended to I and becomes a function which is n p - 1 times differentiable at all points t # 0 in a neighborhood V of 0 in I, and p times differentiable at t = 0; furthermore f:(O) = k! f'"''(0) for 0 < k < p , and lim fAP+'"'(t)tk = 0 f or 1 G k ~ n - l . r-0, t t o . r E v (n k ) ! (Express the derivatives off. with the help of the Taylor developments (Problem 5 ) of the derivatives off, and use Problem 2 of Section 8.6.) (b) Conversely, let g be an n p - 1 times differentiable mapping of I - {0} into E, such that lim g ' p f k ' ( f ) fexists k for 0 < k < n - 1. Show that the function g can
+
+
+
+
r-+O.tto,re~
be extended to a p - I times differentiable mapping of I into E, and that the function g(t)t" is n p - I times differentiable in 1; if furthermore g'p)(0)exists, then g(t)t" is n + p times differentiable at 0. (c) Suppose I = 1- 1, I[, and supposefis euen in I , i.e.f(-t) = f ( t ) . Show, using (a) and (b), that i f f is 2n times differentiable in I, there exists an n times differentiable mapping h of I into E such that f ( t ) = h(r2). 7. (a) Let f b e an indefinitely differentiable mapping of R" into a Banach space E. Show that
+
. .,X") =fa. . . , 0) + xlfI(x1, .. . ,X") + x2 f2(x* ...,x.) + . . . + x. f.(xJ where fk is indefinitely differentiable in R"-k+l (1 < k < n). Write f ( x l , .. . , x,) = (f(xl,.. . , x.) - f ( O , x 2 ,. . . , x.)) +f(O, x 2 ,. . ., x.) and apply (8.14.2) to the first f(x1,.
1
summand, considered as a function of x l ;with a suitable value ofp (depending on k ) , this will prove that ( f ( x l , ..., x,)-f(O, x 2 , . .., x,))/xl is k times differentiable at (0,. . . , O ) ; finally, use induction on n.) (b) Deduce from (a) that for any p > 0,
where all the fa are indefinitely differentiable and f o ( x )= f ( O , . . . , 0). (c) Let f be an indefinitely differentiable mapping from R" into R; suppose that f(0) = 0, Di f ( 0 )= 0 for 1 < i < n, and that the quadratic form ([I,
. . .*t n ) E Dt Dj f(0)ti i.j +
t j
194
Vlll
DIFFERENTIAL CALCULUS
is nondegenerate. Show that, by using a linear transformation in R" one can assume that DlDjf(0) = 0 for i # j and a1 = D:f(O) # 0 for 1 < i < n. Prove that there exists a neighborhood U of 0 and n indefinitely differentiable functions g , defined in U, such that in U
and gi(0) = 1 for 1 < i < n. (Use again (a) applied to each of the functions fl defined in (a), and then apply the usual method of reduction of a quadratic form to a form having a diagonal matrix.) 8. (a) Let S be a metric space, A , B two nonempty subsets of S, M a vector subspace of the space V,(S) of real continuous functions in S, N a vector subspace of M, u +L(u) a linear mapping of M into the space RAof all mappings of A into R.We suppose that: ( I ) there exists a function u0 E N such that L(uo)is the constant 1 on A ; ( 2 )if u E Nand there is a t E B such that u(t) = 0, then there is x E A such that (L(u))(x)= 0. Let u E M such that L(u) = 0; show that for any function u E M such that u - u E N, and any t E B, there exists 6 E A (depending on t) such that u(r) = u ( t ) uo(f)(L(u))(@ (Observe that u o ( t ) # 0, and therefore there is a constant c (depending on t) such that u ( t ) - v ( t ) - cuo(t) = 0). (b) Suppose S is compact, A is connected and dense in S, and all functions u E N vanish on S - B. Suppose that L(u) is continuous in A for every u E M, and that if a function u E N is such that (L(u))(t)> 0 for any t E A, then u has no strict maximum on B. Show that in such a case condition ( 2 ) of (a) is also verified. 9. (a) Let f be an n times differentiable real function defined in an interval 1; let XI < x 2 < ... < xp be points of I, n i (1 < i < p ) integers > O such that nl n2 . . . np = n. Suppose that at each of the points x l , fCX'(x,)= 0 for 0 < k < nl - 1. Show that there is a point 4 in the interval ] x l , x p [ such that f("-')([) = 0 (apply Rolle's theorem iteratively). (b) Let g be an n times differentiable real function defined in I, and let P be the real ) P(k)(x,)for 0 < k < n, - 1 , 1 < i < p. polynomial of degree n - 1 such that g ( k ) ( x i= Show that for any x E I, there exists 6 in the interior of the smallest interval containing x and the xi (1 < i < p), such that
+
+ + +
(Use Problem 8(a), or give a direct proof, using (a) in both cases.) 10. Let g be a real odd function, defined and 5 times differentiable in a symmetric neighborhood I of 0 in R. Show that, for each x E I s(x)
X =3 (g'(x)
+ 2g'(O)) - 180 g y . 9 XS
where [ is a number belonging to the open interval of extremities 0 and x . Deduce from that result that, iffis a real function, defined and 5 times differentiable in [a, b ] , then
with a < 4 < b (" Simpson's formula").
14 TAYLOR'S FORMULA
195
11. Let I = [a, b] be a compact interval, and let Mo be the vector space of real continuous functions defined in I and such that, for any t E ]a,6[, the limit
.
exists in R.All real functions which are twice differentiable in I belong to Mo (a) Let M be the vector subspace of Mo consisting of functions f for which L (f)is continuous in ]a, b[. Show that any function of f~ M is twice differentiable in ]a, b[ and that L(f)=f". (Use Problem 8(a) and 8(b), taking S = I, A = B = ]a,b [ , and for N the subspace of M consisting of functions f for which f ( a ) = f ( b ) = 0.) (b) Show that the function f ( t ) = t cos(l/t) belongs to M o , although it is not differentiable at t = 0. 12. What are the properties of functions with values in a Hilbert space which correspond to the properties of real functions discussed in Problems 9(b), 10, and 1 1 ? (Cf. Section 8.5, Problem 6.) 13. Let f be an indefinitely differentiable mapping of a compact interval I = [a,b] c R (with a 3 0) into a Banach space. (a) Show that, for any two integers p , 4 such that 0 < p < q,
(For p < n < q, express f ( " ) ( a )using Taylor's formula at the point b.) (b) Under the same conditions, show that
(apply Taylor's formula tof'p) and use Problem 4(c) of Section 8.11). (c) Suppose f is indefinitely differentiable in the interval [a, +a[where a 3 0, and that for every integer n 2 0, there is a finite number M. such that Ilf'"'(x)Ilxn/n! < M. for every x 3 a . Show that for 6 > a and n < q, f ( " ) ( b )= -
s,'
(b - x ) l I - n - l f('J)(x)
(4-n-
I)!
dx
and for a < x < b
where the series and the integral are convergent (use Taylor's formula). Conclude that one has
and
(d) Show that, under the assumptions of (c), the function
(whose values are finite and 30,or f a ) is increasing in [a, +m[.
196
Vlll
DIFFERENTIAL CALCULUS
(e) Supposefis indefinitely differentiable in [a, +co[ and that lim f(")(x) = 0 x++m
for every n > 0.
Prove that the sequence of numbers (finite and 2 0 or equal to +a)
J.
=jotIlf(")(x)II
x"- 1
dx
(n > 1)
is increasing. (When these numbers are all finite, write f(")(x) =
-jx+mf("+l)(t) dt
and use Problem 4(c) of Section 8.1 1.) (f) Suppose f is indefinitely differentiable in [a, +to [ and that the integrals
are finite. Show that, for x 3 a,
14. Let f b e a n indefinitely differentiable mapping of the interval [a, +a[, where a 3 0,
into a Banach space. (a) Show that, for every p > 0,
where both sides may be equal to equal to both limits (w)
lim
+ co. Show that if both sides are finite, they are also
C llf(")(x)ll 2 m
x++m n = p
X'
and
lim n-rm
x"jCm llf(")(x)ll -dX (n - l)! 1
(I
(Use Problem 13). (b) I f f satisfies the assumptions of Problem 13(c), then the limits (**) always exist and are equal to both sides of (*).
CHAPTER I X
ANALYTIC FUNCTIONS
In this chapter, we have tried to emphasize the most general facts pertaining to the theory of analytic functions, and in particular to state as many results as possible for analytic functions of any number of variables; until Section 9.13, the theorems which concern only functions of one variable are inserted in a context in which they appear as technical intermediates to the general statements; it is only in Sections 9.14 to 9.17, and in many problems in this chapter and the next one, that we really deal with properties special to the one variable case. Furthermore, we have discussed simultaneously the case of analytic functions of real variables and of analytic functions of complex variables as long as it can be done (i.e. until Section 9.5). Finally, we have kept throughout our general principle of dealing from the start with vector valued functions; as usual, this does not require any change in the proofs, and the reader will see in Chapter XI how useful the consideration of such functions can be. Of course, one can only expect to find here the most elementary part of the very extensive theory of analytic functions. The definition is given by the local existence of power series representing the function, and it is by the technique of power series that the differential properties of analytic functions are obtained (9.3.5) (the usual definition of analytic functions by the existence of continuous derivatives only applies, of course, to functions of complex variables, and therefore that characterization is postponed until Section 9.1 0). The fundamental results about power series are Abel’s lemma (9.1.2) -from which is derived the vital possibility of substituting power series into power series (9.2.2)-and the principle of isolated zeros (9.1.5), whose most important consequence is the principle of analytic continuation (9.4.2), which expresses the “solidarity” between the values of an analytic function at different points of the domain where it is defined. From that point on, we have to assume that the variables are complex; 197
198
IX ANALYTIC FUNCTIONS
with the exception of the principle of maximum (9.5.9), all additional properties of analytic functions of complex variables derive from a single new idea, that of “ complex integration,” and from its fundamental features, Cauchy’s theorem (9.6.3), Cauchy’s formula (9.9.1), and its generalization, the theorem of residues (9.16.1).The form of Cauchy’s theorem which we give here is not the best possible, for it expresses the integral along a circuit as an invariant of the homotopy class of that circuit, whereas in fact it is an invariant of its homology class. In most applications, however, this has no inconvenience whatsoever, and in contrast to the fact that the proof of the weak form of Cauchy’s theorem needs almost no topological preparation, the proof of the complete theorem would have required some developments of algebraic topology, which we feel are above the level of the present volume. The interested reader will find the complete Cauchy theorem, together with all the necessary prerequisites, in Ahlfors [l], Cartan [8], and Springer [17]; we shall come back to that question in Chapter XXLV. Instead of using more results from algebraic topology in order to obtain such refinements, we have thought it might interest some readers to see how, by the very simple device introduced by S. Eilenberg, it is possible to obtain quite deep information on the topology of the real plane (including the Jordan curve theorem), using merely the most elementary facts about complex integration; this is the purpose of the Appendix (which, by the way, is not used anywhere in the later chapters and may therefore be bypassed without any inconvenience). As we have announced in Chapter I , the reader will find no mention in this chapter of the so-called “ multiple-valued ” or multiform ” functions. It is of course a great nuisance that one cannot define in the field C a genuine continuous function Jz which would satisfy the relation (Jz)’ = z ; but the solution to this difficulty is certainly not to be sought in a deliberate perversion of the general concept of mapping, by which one suddenly decrees that there is after all such a “function,” with, however, the uncommon feature that for each z # 0 it has titto distinct “values.” The penalty for this indecent and silly behavior is immediate: it is impossible to perform even the simplest algebraic operations with any reasonable confidence ; for instance, the relation 2Jz = Jz Jz is certainly riot true, for if we follow the “definition” of J z , we are compelled to attribute for z # 0, ticso distinct values to the left-hand side, and three distinct values to the right-hand side! Fortunately, there is a solution to the difficulty, which has nothing to d o with such nonsense; it was discovered more than 100 years ago by Riemann, and consists in restoring the uniqueness of the value of \/. by “doubling,” so to speak, the domain of the variable z, so that the two values of 2 / z corre“
+
1
POWER SERIES
199
spond to tit'o different points instead of a single z ; a stroke of genius if ever there was one, and which is at the origin of the great theory of Riemann surfaces, and of their modern generalizations, the complex manifolds which we shall define in Chapter XVI. The student who wishes to get acquainted with these beautiful and active theories should read H. Weyl's classic [19] and the modern presentation by Springer [17] of Riemann surfaces, and H. Cartan's seminar [7] and the recent book of A. Weil [IS] on complex manifolds.
1. POWER SERIES
In what follows K will denote either the real field R or the complex field C ; its elements will be called scalars. In the vector space K" over K, an open (resp. closed) polydisk is a product of p open (resp. closed) balls; in other words it is a set P defined by conditions of the form Izi- ail < ri (resp. Izi - ail < ri), 1 < i < p , on the point z = (zl, . . . , z,), with ri > 0 for every index; a = ( a l , . . . , a,) is the center or P, r l , . . . , r, its radii (a ball is thus a polydisk having all its radii equal). (9.1.1) Let P, Q be trr'o open polydisks in KP siich that P n Q # @; f o r any tn'o points x, y in P n Q , the segment (Section 8.5) joining x and y is contained in P n Q ; in particular P n Q is connected.
+
Indeed, if Ixi - ail < r i , lyi - ail < r i , then ltxi (1 - t ) y i - ail < tlxi - ail (1 - t)lyi - ail < r for 0 < t < 1 ; the last statement follows from the fact that a segment is connected (by (3.19.1) and (3.19.7)) and from (3.19.3).
+
,
We introduce the following notation: for any element v = ( n l , . . . , J?,) in NP ( n , integers 3 0 ) and any vector z = (zl,. . . , z,) E ISP, we write * + n p . If E is a Banach space zv = z1'z;'. . . z i p and IvI = n, + n2 + (over K), ( c , ) , , ~ ~a family of elements of E having NP as set of indices, we say that the family(c,z"), E N P of elements of E is a power series in p variables z i (1 < i < p ) , icsith coeflcients c, . E K P be such that b , # 0 f o r 1 < i Q p , and that the family ( c v b v ) be bounded in E. Then f o r any system of radii (ri) such that 0 < r < lbil f o r I < i < p , the portxer series (c,zv) is normally summable (7.1) in the closedpolydisk ofcenter 0 and radii ri ("Abel's lemma").
(9.1.2) Let b = ( b L ,. . . , 6,)
200
IX ANALYTIC FUNCTIONS
For if (/c,b’l( < A for any v EN!’, it follows from the definition of the norm in KP that if (zil < ri < [hi( (1 < i < p ) , we have ( ( c v z v< ( JAq’, with q = ( q l , , . . , q,), qi = ri/Jbil< 1. It follows from (5.5.3) that the family ( q v ) v p N of p positive numbers is absolutely summable, hence the result by (5.3.1). (9.1.3) Uiider the assumptioris of (9.1.2), the sum of the p o u w series (c, z‘) is continuous in the opeii polydisk of center 0 and radii Ibil.
As every point of that polydisk is interior to a closed polydisk of radii ri < / b i / ,the result follows from (7.2.1). Let q be any integer such that 1 < q < p ; for any v = ( n l , . . . , n,), write v‘ = (ti1, . . . , n,), V “ = (nq+,,. . . , n,); consider KP as identified to the product K 4 x KP-,, and for z = (zl . . . , z,) E KP, write z’ =.(z,, . . . , z,), Z” = (z,,,, . . . , z,). With these notations: (9.1 -4) Suppose the pouter series (c,z’) is absolutely summable in the polydisk P of radii ri aiid center 0 iii KP. Then, f o r any v” E NP-, the series (c(,.,, , ~ ~ ~ z ’ ’ ’ ) is absohrtely suinmable in the polydisk P’, projection of P on K4;let g J z ’ ) be its sum. Then, f o r any z’ E P’, the power series (g,.,(z‘)z“’”) is absolutely summable in the polydisk P”, projection of P 011 Kp-q, and its sum is equal to the sum of the series (c’z’). As z’ = Z”’Z’””, the fact that each of the series (q,,,,, , ~ ~ ~ Z ‘ “ Z ” ~ ” (v” ) fixed) is absolutely summable, and that gV,,(z‘)z”‘”= cvzv,follows from (5.3.5)
c
c
V”
V
and from the associativity theorem (5.3.6) for absolutely summable families. If we take Z” E P” such that zi # 0 for q 1 < i < p , the absolute summability of (C(,., ‘“)Z”’) follows.
+
(“ Principle of isolated zeros ”) Suppose (cnzn) is a power series (9.1.5) in one variable which converges in an open ball P of radius r, aiid
let f ( z )
=
2 c,zn.
Then, unless all the c, are 0 , there is r‘ < r such
n=O
that f o r 0 < ( z ( < r ’ , f ( z ) # 0.
Suppose h is the smallest integer such that cl1# 0; then we can write f( z ) = zh (c,, + c,,+,z + * * . + e l l + m ~+m* . * ) and the series (ch+,,,zm)converges in P; i f g ( z ) = c , + c , + , z + ~ * ~ + c , , + , , , z m +. . . , g iscontinuous in P by (9.1.3) and as g(0) = c,, # 0, there is r‘ > 0 such rhat g(z) # 0 for J z J< r’; hence the result.
2 SUBSTITUTION OF POWER SERIES IN A POWER SERIES
201
(9.1.6) Suppose two p o s e r series (a,zv) and (b,z") are absolutely summable and have the same sum in a polydisk P; then a, = b, f o r every v E NP. Use induction on p ; for p = 1, the result follows at once from (9.1.5). Taking the difference of the two power series, we can assume b, = 0 for m
I
every v ; applying (9.1.4) with q = p - 1, we have n=O
gn(z')z; = 0, hence
gn(z') = 0 for every n and every z' in the projection P' of P on KP-'; the induction hypothesis applied to each gn yields then a, = 0 for every v.
PROBLEMS
..
1. Let (cy z") be a power series in p variables zl (1 < i < p ) ; let a = (al, . ,a,) E Kp. In order that a real number Y > 0 be such that, for any r E K such that If) < Y, the series (c,(fa,)"' . . . (fa,)",) be absolutely summable, it is necessary and sufficient that
for all but a finite number of indices v = (nl, . . . , n,) (apply (9.1.2).) In particular, for p = I , there is a largest number R 2 0 (the "convergence radius," which may be .c) such that the series (c.z") is convergent for 121 < R, and that number is given by I /R = lim (s~p(IIc,+~/l'/("+~))), which is also written lim . supllc.ll'~"
+
n-m
k30
"-+W
(cf. Section 12.7). When in particular lim l/c,l/'/" exists, it is equal to l/R. "-m
2. Give examples of power series in one complex variable, having a radius of convergence R = 1 (Problem 1) and such that: ( I ) the series is normally convergent for 121 = R ; ( 2 ) the series is convergent for some z such that IzI = R , but not for other points of that circle; (3) the series is not convergent at any point of Iz/ = R. 3. Give an example of a power series in two variables, which is absolutely summable at
two points ( a l , a2), (bl, b2), but not at the point
"1.
bl, 2 +
(Replace z by
in a power series in one variable.) Let (c,,~"), ( d , , ~ "be ) two power series in one variable with scalar coefficients; if their radii of convergence (Problem 1) are R and R', and neither R nor R' is 0, then the radius of convergence R" of the power series (c,d,z") is at least RR' (takenequal to cu if R or R' is co).Give an example in which R" > RR'.
z1z2
4.
+
+
2. SUBSTITUTION O F P O W E R SERIES I N A P O W E R SERIES
Let Q be a polydisk of center 0 in Kq, and suppose the p power series in q variables ( b r ' u p ) with scalar coefficients are absolutely summable in Q (with p = ( m l , . . . , mq), u = (ul,. . . , uq), up = u y l . . . u,"~). We write
202
IX ANALYTIC FUNCTIONS
gk(u)=
bF’u”, Gk(u)=
c IbF)
[ u p .On the other hand, let (a,zv) be
a power
B
P
series in p variables with coefficients in E, which is absolutely summable in a polydisk P of KP, of center 0 and radii rk (1 < k < p ) . If, in a monomial zv = z;’ . . . z i p , we replace “formally” each z k by the power series gk(u), we are led to take the formal “product” of n, + n2 + + nP series, i.e. to pick a term in each of the n, + ... + nP factors, to take their product and then to “ s u m ” all terms thus obtained. We are thus led to consider, for each v = (nl, n 2 , . . . , nP) the set A, of all finite families ( p k j ) = p where p k j E Nq, k ranges from 1 to p , and for each k , .j ranges from 1 to n,; t o such a p we associate the element tP(U)
= a,
nn P
nk
bfpkJ.
k = l j=1
With these notations:
Suppose sl, . . . , sq are q numbers > O satisfying the conditions Gk(sl,. . . , sq) < rk for 1 < k < p . Then, f o r each u in the open polydisk S c Kq of center 0 and radii si (1 < i < q), the family (t,(u)) (where p ranges through the denumerable set of indices A = A,) is absolutely summable,
(9.2.1)
u
and i f f ( z ) =
VENP
1avzv,its sum is equal t o f ( g , ( u ) ,g2(u),. . . ,g,(u)). V
In other words, under the conditions G,(s,, . . . , sq) < rk (1 6 k < p ) , “substitution” of the series gk(u) for zk ( 1 < k < p ) in the series f yields an absolutely summable family, even before all the terms t,(u) having the same degrees in u l , . . . , uq have been gathered together. To prove (9.2.1), we need only prove that the family (t,,(u)) is absolutely summable; that its sum is f ( g , ( u ) , . . . , g,(u)) follows by application of the associativity theorem (5.3.6) to the subsets A, of A, and by using (5.5.3), which shows that t,(u) is equal to a,(gl(u))”* . . . (gP(u))“P.To prove the PEA,
family (t,(u)) ( p E A) is absolutely summable, we apply (5.3.4). For any finite subset B of A, we have, by (5.3.5) and (5.5.3)
and by assumption, the right-hand side of that inequality is the element of index v of an absolutely summable family; hence the result. Write t,(u) p k j = (mkj,,.
= cPul,
with ,A
= (,Al,
. . , mkjq)). From (9.2.1)
. , . , ,Aq), Izi
1 c mkji (if P
=
k=l
nk
j=1
we have
and (5.3.5) it follows (taking all the ui
3 ANALYTIC FUNCTIONS 203
to be ZO,u E S), that for each A, the family of the c p , where p ranges over all elements of A which correspond to the same 2, is absolutely summable in E; if d , is its sum, we see, by the associativity theorem (5.3.6),that
(9.2.1.1)
f(g,(u), ' .
. ?
g,(u)) =
1d i ui i
the series on the right-hand side being absolutely summable in the polydisk S. By definition, that power series is the ponier series obtained by substituting g k ( U ) to z k ,for 1 < k < p , in the power series ( a , 2').
(9.2.2) I f the point (gl(0),. . . ,gp(0))of KP belongs fo P, then there exists in Kq an open polydisk S such that, for u E S, the series gk(U) may be substituted to zk (1 < k < p ) in the poii'er series (a, z"). Observe that by definition, Gk(0)= Igk(0)l for 1 < k 0 (1 < i < q ) such that G,(s,, . . . , sq) < rk for 1 < k < p follows at once from the assumption.
3. A N A L Y T I C F U N C T I O N S
Let D be an open subset of KP. We say that a mapping f of D into a Banach space E over K is analytic if, for every point a E D, there is an open polydisk P c D of center a, such that in P, f ( z ) is equal to the sum of an absolutely summable power series in the p variables zk - a, (1 < k < p ) (that series being necessarily unique by (9.1.6)). Suppose K = C, let b be a point of D, and let B be the inverse image of D by the mapping x + b + x of RPinto Cp. Then it follows at once from the definitions that x +f ( b + x) is analytic in the open subset B of RP.
(9.3.1) Let (a,z") be an absolutely summable powjer series in an open polydisk P c KP. Then f ( z ) = a,zv is analytic in P; more precisely, if ri (1 < i d p ) V
are the radii of P, for any point h = (bi)E P, f ( z ) is equal to the sum of an absolutely summable poiver series in the zk - h k in the open polydisk of center b and of radii ri - lbil ( I < i < p ) . This follows at once from (9.2.1)applied to the case q = p , gk(u)= b, + z i k ; we have then G,.(u)= Ibkl + u k , and the conditions Gk(s,,. . . , sp) < rk (1 < k < p ) boil down to s, < rk - lbkl (1 d k O; suppose lim a,/b, = s. n- m
(a) Suppose the series (b.2") is convergent for / z /< 1, but not for z = 1 (which means k
that if ck = C b,, lim ck = n=o
k-m
+ m). Show that the series (a.z") is absolutely convergent
for Iz/ < I , and that, if I = [0, I[,
(Observe that, for any given k ,
lim 2-1,
IEI
( cb,z")
=
+a).
nak
(b) Suppose the series (b,z") is convergent for every z. Show that the series (ant")is absolutely convergent for every z, and that if J is the interval [0, +a[in R, then lim ( F a n z n ) / ( g $ z n ) = s . z + + m . z ~ J n=O
(Same method.)
c c m
(c) Show that if the series (a,) is convergent and
a,
= s,
then the series (a.z") is
"=O
absolutely convergent for /zI < 1, and that
lim 2-1.
*El
m
a,, z" = s. (Apply (a) with b, = 1
"=a
for every n; this is "Abel's theorem ".) (d) The power series ((- 1)"~") has a radius of convergence 1, and its sum l / ( l z ) tends to a limit when z tends to 1 in I, but the series ((-1)") is not convergent (see Problem 2). 2. Let (a.2") be a power series in one variable having a radius of convergence equal to 1 ; letf(z) be its sum, and suppose thatf(1-) exists. If in addition lim nu. = 0, show that
+
n-m
the series (a,)is convergent and has a sum equal to f(1->. ("Tauber's theorem": observe that if 1na.l < E for n 1 k, then, for any N > k , and 0 < x < 1
and
3. Let (a,z")be a power series in one variable having a radius of convergence r > 0, and I) and 191 < r. let (b,) be a sequence of scalars # O such that q = lim ( b n / b n + exists n-ra
Show that, if C.
= sob.
+
lim (c,/b.) exists and is equal to f ( q ) . n- m
bn-i
+ ... + a.60
4 THE PRINCIPLE OF ANALYTIC CONTINUATION
207
4. Let (pnz"),(q.z") be two power series with complex coefficients, and radius of convergence #O, and let f(z) = C p n z " ,g(z) qnznin a neighborhood U of 0 where both
=c "
0
series are absolutely convergent. Suppose qo = g(0) # 0 ;then there is a power series c,z" which is absolutely convergent in a neighborhood V c U of 0 and has a sum equal to f(z)/g(z) in V (remark that the series (z") is convergent for IzI < 1 , and use (9.2.2)). If all the qn are >O, the sequence (q.+&.) is increasing, the pn are real and such that the sequence (p,/qn) is decreasing (resp. decreasing), show that c, 2 0 (resp. c, < 0) for every n >, 1 . (Write the difference
P"
Pn- 1
4.
4"-1
as an expression in the qk and c k , and use induction on n.) Deduce from that result that all the derivatives of x/log(l - x ) are 0 be such that IzI < r is contained in B and lh(z)l d +lb,l for IzI d r (9.1.3). Write b, = lb,lc with [[I = 1 ; by (9.5.7) there is a real t such that emit= [-';for z = reit, we therefore have 11
+ bmzm+ zmh(z)I= 11 + Ibmlrm+ zmh(z)I 3 1 + tlbmlrm
which contradicts the assumption If(z)l 6 IcoJin B. The result (9.5.9) does not hold if C p is replaced by RP, as the example
of the power series l/(l
+ z 2 )=
m
n=O
(- 1 ) " ~(for ~ " IzI < 1 ) shows.
214
IX ANALYTIC FUNCTIONS
(9.5.10) Let f be a complex valued analytic function dejined in an open subset A c CP, and which is not constant in any connected component of A. For any compact subset H c A, the points z E H where If ( z ) l = sup1f(x)I xeH
(which exist by (3.17.10)) are frontier points of H.
Follows at once from (9.5.9) and the principle of analytic continuation (9.4.1).
PROBLEMS
1. Show that if W(z) < 0, then, for any integer n > 0
2.
(use Taylor's formula (8.14.2) applied to t +P). Prove that, for real x
1
COS x -
(
1-
xz
+ x4 - . . . + (-
and the difference has the sign of (-
I)"+l;
)I 1X1Zn+2 +
1)" -
(X22 4n !
0, 1 - s < e-s.)
+
+
similar to the one obtained in (c), observing that
216 9.
IX ANALYTIC FUNCTIONS
(a) Let f i k (1 < j < m, 1 < k < n) be scalar analytic functions defined in a n open connected subset A of Cp; let tcik be real numbers 30. Show that the continuous n
~fi~(~)~a'k~f~~(~)~'zk~f~~(~)~umk cannot reach a relative maximum at a point ofA,unless each of the products ~fIr(z)lu1"~~ Ifmk(z)lumk(l < k < n) is constant in
function u(z) = C
k=1
A. (Observe that if f(z) is analytic in A and f(zo) # 0, then, for every real number A, there is a function gA(z) which is analytic in a neighborhood of zo and such that IgA(z)j= If(z)lA in that neighborhood; use Problem 8(c) to that effect.) Extend the result to the case in which the tcJk are arbitrary real numbers, provided none of the hkvanishes in A. (b) Generalize to u(z) the result of Problem 3(a). 10. Let f ( z ) be a complex function of one complex variable, analytic in the open set A defined by R1< IzI < Rz (where 0 < R1 < Rz). For any r such that Rl < r < Rz,let M(r) = suplf(z)l. Show that if R1 < rl < rz < r3 < R , , then IT1 = r
log M(rz)
0 such that It - t’( d E , 15 - 5’1 < E imply Icp(t, )+ g i j ( Y j + I ( t i > ) >
= 0.
But yj(ti) and yj+l(ti) both belong to Q i - l , n Q i j for 1 < i < r, hence, by what we have seen above gij(yj(ti)>- gij(Yj+l(ti>> = gi-1, j(yj(ti)) - gi-1, j(Yj+l(ti)) hence the left-hand side of (9.6.3.1) is reduced to gr-1, j(Yj(tr)) - gr-1, jbj+l(tr))- gOjbj(t0)) + 90j(yj+l(to)). But as y j and yj+l are circuits, we have yj(to)= yj(tr)and yj+l(to)= y j + , ( t r ) ; moreover, these two points belong to Q o j n Q r . - l , j , which is connected; the difference g r - l , - goj is thus constant in that set by (8.6.1), and this ends the proof. (9.6.4) Let yl, y 2 be two roads in an open set A c C,having same origin u and same extremity v , and such that there is a homotopy cp of y1 into y2 in A which leaves u and vfixed (i.e., q(a, I;) = u and cp(b, I;) = v for every I; E [a, /?I if cp is defined in [a, b] x [a,PI). Then, for every analytic function f in A,
j-p) dz j n f ( 4dz. =
+
Let yy be the road opposite to yl, and let y 3 ( t ) = yy(t - b a ) for b < t < 26 - a ; y 3 is a road equivalent to y y . By definition, y1 v y 3 and y 2 v y 3 are circuits. Moreover these circuits are homotopic in A, for if we define $ ( t , I;) as equal to cp(t, 5) for a < t < b, to y 3 ( t ) forb < t < 2b - a, $ is a loop homotopy in A. Applying (9.6.3), we get / y 2 f ( z )dz
+ jJ(4
dz. Q.E.D.
Yl
f ( z ) dz
+ JY, f ( z ) dz =
7 PRIMITIVE
OF
A N ANALYTIC FUNCTION
221
1
7. PRIMITIVE OF A N ANALYTIC F U N C T I O N IN A SIMPLY CONNECTED DOMAl N
A simply connected domain A c C is an open connected set such that any loop in A is homotopic in A to a loop reduced to a point; it is clear that any open subset of C homeomorphic to A is a simply connected domain.
Example (9.7.1) A star-shaped domain A c C with respect to a point a E A is an , segment joining a and z is contained open set such that for any ~ E Athe in A. Such a set is clearly connected ((3.19.1) and (3.19.3)); if y is any loop in A, write q ( t , 5 ) = a + (1 - t ) ( y ( t )- a ) for 0 < 5 < 1 ; 40 is a loop homotopy of y into the loop reduced to a. An open ball is a star-shaped domain with respect to any of its points. (9.7.2) If A c C is an open connected set, for any two points u, u of A there is a road of origin u and extremity u. We need only prove that the subset B c A of all extremities of roads in A having origin u is both closed and open in A (Section 3.19). If x E A n B, there is a ball S of center x contained in A, and by assumption S contains the extremity u of a road y of origin u ; the segment of extremities u, x is contained in S, and if y is defined in [a, b], the road y1 equal to y in [a, b], to y l ( t ) = u ( t - b)(x - v) in [b, b + 11is in A and has origin u, extremity x; hence x E B. On the other hand, if y E B, there is a ball S of center y contained in A ; for any u E S, the segment of extremities y , u is contained in S and we define in the same manner a road of origin u, extremity u, which is in A, hence S c B. Q.E.D.
+
(9.7.3) I f A c C is a simpIy connected domain, any function f analytic in A has a primitive Icjhich is analytic in A. Let a, z be two points of A, yl, yz two roads in A of origin a and extremity z; then J y , f ( x )dx = f ( x ) dx. Indeed, we may suppose, by replacing yz In by an equivalent road, that y1 is defined in [b, c] and y z in [c, d ] ; then y = yI v y y is a circuit in A, which is therefore homotopic to a point in A,
222
IX ANALYTIC FUNCTIONS
hence
j7f (x) dx = 0 by
Cauchy's theofem, and this proves our assertion.
We can therefore define g(z) as the value of f ( x ) dx for any road y in A J-7 of origin a and extremity z, and by (9.7.2), g is defined in A. Now for any z o E A, there is an open ball B c A of center zo in which f ( z ) is equal to a convergent power series in z - z,; by (9.3.7) there is therefore a primitive h off in B which is analytic, and such that h(z,) = g(zo); hence we have for ZEB
h ( z ) - h(z0) =
sd
f ( z o + t(z - z ~ ) ) (z zO) dt.
Jg
But the right-hand side is by definition f ( x ) dx, where CJ is the road t + zo t(z - z o ) defined in [0, 11; as that road is in B c A, we have g(z) - g(z,) = Jo f ( x ) dx by definition of g , and therefore g(z) = h(z) in B. Q.E.D.
+
8. INDEX OF A P O I N T WITH RESPECT TO A CIRCUIT
(9.8.1) Any path y defined in an interval I = [a, b] and such that y(1) is contained in the unit circle U = { z E C 1 ( z I = I}, has the form t -+ ei*(*), where $ is a continuous mapping of I into R; if y is a road, $ is a primitive of a regulated function.
As y is uniformly continuous in I, there is an increasing sequence of points t k (0 < k < p ) in I such that to = a, t, = b, and that the oscillation (Section 3.14) of y in each of the intervals 1, = [tk, tk+l] (0 < k < p - 1) be < 1. This implies that ?(I,) # U ; if 8, E R is such that eiek# y ( I k ) (9.5.7), then x + e'@' e k ) is a homeomorphism of the interval 10,274 on the complement of eiek in U (9.5.7). If q k is the inverse homeomorphism, we can therefore write, for t E I,, y ( t ) = eiJlk(*), where $k(t) = ( P k ( Y ( t ) ) +e, is continuous in I k . By (9.5.5), we have $k+l(tk+l)= $k(tk+l) + 2nkn with nk an integer (0 < k < p - 2 ) . Define now $ in 1 in the following way: $(t) = $ , ( t ) for t E 10; by induction on k, We put $(t) = $ k ( t ) $(tk) - $k(tk) for f k < t < tk+,. By induction on k , it is immediately seen that $ ( t k ) - $ k ( t k ) is an integral multiple of 271 for 0 < k < p - 1 ; therefore y ( t ) = ei*(') for t E I, and $ is obviously continuous in I. Moreover, if y(t) = a ( t ) + iP(t), we have a ( t ) = cos $ ( t ) , P ( t ) = sin $ ( t ) , and one of the numbers cos $ ( t ) , sin $ ( t ) is not 0; from (9.5.4), and (8.2.3) applied to one of the functions cos x, sin x at a point where it has a derivative # 0, we deduce that if y has a derivative at a point t , so has $, and i$'(t) = y'(t)/y(t),which ends our proof.
+
8 INDEX OF A POINT WITH RESPECT TO A CIRCUIT
223
(9.8.2) For any point U E C , and any circuit y contained in C - { a } , Jydz/(z- a ) has the form 2ntri, bvhere n is a positive or negative integer. By a translation, we can suppose a
= 0.
Suppose y is defined in I
=
[b, c ] ;
the function q(t, 5) = 5 y ( t ) + ( 1 - = j ( x ; y ) by Cauchy’s theorem. As the set Z of integers is a discrete space, the conclusion follows from (3.19.7).
+
-=
224
IX ANALYTIC FUNCTIONS
Exainple
(9.8.4) Let c, be the circuit t -+ enitdefined in 1 = [0, 2n], M being a positive or negative integer; we have &,(I) = U ; E, is called “the unit circle taken n times”. We observe that the open set C - U has tii’o connected components, namely the ball B: IzI < 1 and the exterior E of B defined by IzI > 1. Indeed, B is connected as a star-shaped domain (9.7.1); and by Section 4.4and (9.5.7) E is the image of ]I, +a[x [0, 2n] by the continuous mapping (x, t ) -+ xe“, hence the result by (3.19.1), (3.20.16), and (3.19.7) (a similar argument also proves the connectedness of B and of B -{O}); finally in C - U, B and E are open and closed since B is open in C and B = (C- U) n and we have B n E = 0. From the definition and (9.5.3) it follows that j ( 0 ; E,) = n, hencej(z; E,) = n f o r any point z of B. Let us show thatj(z; cn) = 0 f o r any point of E; more generally:
s,
(9.8.5) Ifacivcuit yiscontainedinaclosedballD: Iz - a / < r, thenj(z; y ) = 0 f o r any point z exterior to D. Indeed, suppose y is defined in an interval I = [b, c], and that Iy’(t)l in that interval. By definition,
<M
But as I y ( t ) - a1 < r, we have [ y ( t )- zI 3 [z - a1 - r for any t E I, and therefore, by the mean value theorem,
when Iz - a1 is large enough the right-hand side is < 2n, and as j(z; y) is an integer, this implies j(z; y) = 0. But the exterior of D is connected, as seen above, hence the conclusion by (9.8.3). (9.8.6) For any circuit y in C, defined in I, the set of points x E C - y(1) such that j(x; y ) # 0 is relatively compact in C. For by (9.8.5), that set is contained in any closed ball containing y(1). (9.8.7) Let A c C be a simply connected domain, y a circuit in A. For any point x of C - A , j ( x ; y ) = 0.
9 THE CAUCHY FORMULA
225
By assumption, there is in A a loop homotopy rp, defined in I x J, of y into a circuit reduced to a point. As x $ cp(1 x J), Cauchy’s theorem shows that dz/(z - x) = 0. JY
9. T H E C A U C H Y FORMULA
(9.9.1) Let A c C be a simply connected domain (Section 9.7), f an analytic mapping of A into a complex Banach space E. For any circuit y in A, defined in I and any x E A - ?(I), “e have (Cauchy’s formula)
Consider the function g ( z ) defined in A, equal to ( f ( z ) - f ( x ) ) / ( z - x) for z # x, to f ’ ( x ) at the point x; g is analytic in A, for it is obviously analytic in A -{x} by (9.3.2) and (9.5.1); on the other hand, there is a ball B c A of center x such that for z E B, f ( z ) = f ( x ) + ( z - x ) f ’ ( x ) + .. + ( z - x)’f‘”)(x)/n! ... the series being convergent in B; this proves that for any z E B, g ( z ) is equal to the sum of the convergent series
+
f’(x)
+ i ( z - x)f”(x) +
* * *
+ ( z - x)’-’f@)(x)/n! + . . .,
hence analytic at x. From Cauchy’s theorem (9.6.3) we have and writing
s,
g ( z ) dz
= 0,
yields (9.9.1) by definition of the index. Conversely : (9.9.2) Let y be a road in C, defined in an interval I = [b, c] and let g be a continuous mapping of y(1) into a complex Banach space E. Then f (z)=
s,gox x-z
is defined and analytic in the complement of y(1); more precisely, for any point a E C - y(I), if we write
the power series (c,(z - a)“) is convergent in any open ball B of center a contained in C - y(l), and its sum is equal t o f ( z ) in B.
226
IX ANALYTIC FUNCTIONS
Indeed, suppose Iz - a1 x E y(I), we have
1 -x-z
< q . d(a, y(1))
with 0 c q < 1 ; then, for any
=?
( ;I;) 1
(X-a)
with
I
if 6 = d(a, ~(1)).If IIg(x)11 tEI,
,,=O
1 --
( z - a)" ( x - a)"+'
< M in y(I)
1
( z - a) ( x - a)"+' '
1
5 q",
and Iy'(t)l
< m in I, we have, for any
hence the series of general term r'(t)s(r(t))(z
- a)"
( y ( t ) - a)"+
is normally convergent in I. It follows from (8.7.9) that the series (c,(z - a)") is convergent in the ball )z - a1 < q . 6 and has sumf(z) in that ball. (9.9.3) Under the assumptions of (9.9.1), we have, for every x E A - y(I), and every integer k > 0,
s
j ( x ; y)f'k'(x) = -
2ni k!
( zf (z )Xd)zk + l .
This follows at once from Cauchy's formula, the uniqueness of the coefficients of a power series with given sum (9.1.6), the relations (9.3.5) between these coefficients and derivatives, and finally (9.9.2). (9.9.4) Let A c C pbe an open set, f a continuous mapping of A into a complex Banach space E , such that for 1 < k < p , and an arbitrary point (a,) E CP, the mapping zk +f (al, . . . , a, - zk , a, + 1, . . . , up) is analytic in the open set A(a,, . . . , . . . ,up) c C i f that set is not empty (notation of (3.20.12)). Then f is analytic in A. More precisely, let a = (a,) be a point of A, P a closed polydisk of center a and radii r, ( 1 < k < p ) contained in A; for each k , let y k be the circuit t + a, + rkeit in C (0 < t < 2n), and let
',
9 THE CAUCHY FORMULA
227
0
Then the power series (c,(z - a)') is absolutely summable in P and its sum is equal to f ( z ) .
Using Cauchy's formula, and the fact that j ( 0 ; E have, by induction on p - k and the assumption, (9*9*4.1)
~= )
1 (see (9.8.4)), we
f(x1, . . . , x k , z k + l , * . * , z p )
for /xi - ail = rj (1 < j < k ) and Izj - ail < rj ( k + 1 <j < p ) . On the other hand, for lq - akl < rk (1 < k ,< p ) , we can write, for Ixk - akJ= rk 1 (xl - z,) -(xp
- zp)
=c
(XI
- a,)"' * - - ( z , - a,)"" - U , ) " l + l ..-( x , - ap)n,+l
(2,
the power series on the right hand side being normally summable in the set F defined by Ixk - akl = rk (1 < k < p ) , by (5.5.3). By induction on p - k, if we write gn,+ ' ... n , ( X 1 >
* * *
7
Xk)
we have by the mean value theorem (9.9.4.2)
if
Ilf(xl,
. . ., x,)II
<M
on
F. It
follows
that
the
power
series
(gn,+,. . . . p ( ~ l. ,. . , Xk)(Zk+l - a k + l ) n k + l*** (z, - a,)".) in the zj - aj is absolutely summable in P; using induction on p - k , and applying (5.3.5) and (8.7.9), we see that the sum of that series isf(x,, . ., xk, Z k + l , . . . , z,). The conclusion follows by taking k = 0. Moreover (9.9.4.2), for k = 0, proves
.
that, with the same assumptions and notations as in (9.9.4) (9.9.5)
Ilc,,,,, ...npll
< M/ri' . .
*
r;"
i f Ilf(x)ll < M on the product of the circles Ixk - akl = rk (1 (Cauchy's inequalities).
0 such that the relation /A - pi < 6 implies /y;(t) - y;(t)/ < E for t E [a,b] - D, where D is a denumerable subset. Let now f be a continuous mapping of A into a complex Banach space E, such
that its restriction to A is analytic. Show that Cauchy’s theorem j y o f ( z )dz
=IY,
f ( z ) dz
still holds (use (8.7.8)). 2. Let A be an open subset of C, f a continuous mapping of A into a complex Banach space E, such thatfis analytic in A n D, and A n D-, where D, (resp D-) is defined by 4 ( z ) > 0 (resp 4 ( z ) < 0). Show that f i s analytic in A . (Suppose the disk Izl < r is contained in A.Let y + (resp. y - ) be the circuit defined in [- I , + I ] by y + ( t ) = (2r 1)r for -1 < t < 0, y + ( t ) = r e f f i rfor 0 < t < 1 (resp. y - ( t ) = re”” for - 1 < t < 0, y - ( t ) = (1 - 2t)r for 0 < t < I.) Show that if Iz/ < r and 9 ( z ) > 0, then
+
using Problem 1 ; hence if y is the circuit t
--f
1 2nr
f ( z ) = -,
rezir in [- 1 , + I ] ,
jfE, x-z Y
Then use (9.9.2).)
3. Show that the conclusion of (9.9.4) still holds whenfis merely assumed to be bounded
in each bounded polydisk contained in A, but not necessarily continuous. (Use Problem 6 of Section 8.9; actually, a deep theorem of Hartogs shows that even this weakened assumption is not necessary; in other words, a function which is analytic separately with respect to each of t h e p complex variables ziis analytic in A,)
9
4.
m
Let f ( z )
THE CAUCHY FORMULA
a, z" be an analytic complex-valued function in the circle IzI
229
< R. Show
"=O
that, for 0 < r < R
Deduce from that result another proof of Cauchy's inequalities. 5.
Let f ( z ) =
m
a.z" be an analytic function in IzI
< R, and let
"=O
m
M ~ ( r ; f )= Let also M(r;f)
=
C llanllr".
"=O
sup Ilf(z)ll. 111 = I
(a) Show that for 0 < r
0. Show that f is analytic in 1. (Use (a), and Problem 3(b) of Section 8.12).
10. CHARACTERIZATION O F ANALYTIC F U N C T I O N S O F COMPLEX VARIABLES
A continuously differentiable mapping f of an open subset A of C p into a complex Banach space is analytic.
(9.10.1)
Applying (9.9.4), we are immediately reduced to the case p = 1. To prove f is analytic at a point U E A ,we may, by translation and homothetic mapping, suppose that a = 0 and that A contains the unit ball B: IzI < 1. For any Z E B, and any I such that 0 Q I Q 1, note that 1(1 - I ) z + Iei'l < 1 - I + I = 1, and consider the integral (9.10.1 . I )
By (8.1 1.I) and Leibniz's rule (8.1 1.2), g is continuous in [0, 11 and has at each point of 10, 1[ a derivative equal to g ' ( I ) = Joznjr(z + I(eit - z))ei' dt
(see Remarks after (8.4.1)). But i y ' ( z + I(eif - z))e" is the derivative of t +f(z + i(ei' - z)), hence, for I # 0, g ' ( I ) = 0, and therefore (remark following (8.6.1 )), g is constant in [0, 11. But asg(0) = 0, g ( I ) = 0 for 0 Q A Q 1. In particular, it follows, for I = 1, that
for any z E B (by (9.8.4)), and the conclusion follows from (9.9.2).
Let f be a continuously dlfferentiable mapping of an open set A c RZPinto a complex Banach space. In order that the function g dejined in A (considered as a subset of C p ) , by f ( x l , x 2 , .. ., x p , y l , . ., y,) = g (xl + iyl, . . . , x, + iy,) be analytic in A, necessary and suflcient conditions are that (9.10.2)
in A for 1 < k
< p (Cauchy's conditions).
10 CHARACTERIZATION OF ANALYTIC FUNCTIONS
231
We are again at once reduced to the case p = 1 by (8.9.1). Let (x,y )
df (x,y ) , b = af (x,y ) ; expressing that the be a point of A, and put a = ax 8Y limits lim (g(x iy 12) - g(x + iy))/h and lim (g(x + iy + ih) - g(x iy))/ih
+ +
+
h-+O
( h real and ZO) are the same, we obtain a + ib = 0. Conversely, if that condition is satisfied, for any E > 0, there is r > 0 such that if (h2 + k 2 ) 1 / 2< r, 11g(x iy h + ik) - g(x iy) - a(h + ik)ll < e(h2 + k 2 ) 1 / 2by (8.9.1 .I)and this proves that z -g(z) has a derivative equal to a at the point z = x iy. The result then follows from (9.10.1). h-tO
+ +
+
+
PROBLEMS
1. Show that a differentiable mapping o f f an open subset A of C p into a complex Banach space is analytic in A ("Goursat's theorem"; f' is not supposed to be continuous). (Given any h in ]0,1[, prove (with the notations of (9.10.1)) that g'(h) exists and is equal to 0. First show that, given E > 0, there are points t o = 0 < t l < . .. < 1, = 2n, a number p > 0, and in each interval [ t k , t k + 11 a point 8, such that, if ( k = z &elek - z), whenever (k x =z ( A h)(e" - z), then x) -f((k) -f'((k)xl < & / X I Ihl < p and tk < t < t k + l (prove this by contradiction, using a compactness argument and the existence off' at each point). Compare then each integral
+
+ +
+
+
to the expression
2.
for Ihl < p.) Let A be an open simply connected subset of C; i f f is a continuous mapping of into a complex Banach space E such that
s,
f ( z ) dz = 0 for any circuity in A, show that f i s
analytic in A. (" Morera's theorem"; show that f h a s a primitive in A.) 3. Let A be an open subset of Cp, y a road defined in I = [a, b], f a continuous mapping of y(1) x A into a complex Banach space E. Suppose that for each x E y(I), the function ( z l , . .. ,z,) + f ( x , zl, . .. , z,) is analytic in A, and that each of the functions
af -(x,
.
z l , . , ,z,) is continuous in y(1) x A (1 < k
conditions, the functions g(zl,. .. , z,)
=
2 ) , f a n analytic mapping of A into a complex Banach space E. Suppose that there is an open polydisk P C A, of center b= < k < p and radii rL (1 < k < p ) such that for every point (ck) of P, there is a
232
IX ANALYTIC FUNCTIONS
+
number p < inf(rl, r 2 )such that the function x1 ixz +f(xl, xz , c 3 ,.. . , cp)is analytic in the open subset /xl ixz - (cI icz)l < p of C (identified to R2).Show that the same property holds for every point (ck) E A (use (9.10.2) and (9.4.2)). 5. Let S be the "shell" in RD( p 2 3) defined by
+
+
+ XI+ '. + x', < (R +
(R - E)' < X :
E)'
(0 < E
< R).
Suppose f is an analytic mapping of S into a complex Banach space E, and suppose that for any u = (x3, .. ,x p ) , the mapping x1 ixz + f ( x l , x 2 , u) is analytic in a neighborhood (in C) of every point of the cross section S(u) (if S(u) is not empty). (a) For any u = ( x 3 ,. .., x,) such that lluIl2 = x: . . . x', < RZ,let y(u) be the road in C defined by f +(RZ - IIuljZ)l/zeiffor -7r < f < T.Let
+ + +
.
+
+
where y = x1 ixz , and f ( y , u ) =f(xl, xz, u ) ; g is defined for lzlz l/ull2< RZ,and z+g(z,u)isanalyticforlzl < (R2 - jl~ll~))'/~.Ontheotherhand,foranyv = (x;,. . . , x i ) such that Ilu II < R. let
+
Show that h,(z, u ) = g ( z , u ) for llull < l l ~ l< l llull E and /z /cc (R2- l l ~ l l ~ ) (apply '/~ Cauchy's theorem (9.6.3)). On the other hand, show that g(z, u ) = f ( z , u) for R - E < llull < R and /zI < (RZ- I I U I ~ ~ ) ' ~ ~Conclude . that f can be extended to a function f which is analytic in the whole ball B : x i . . . x: < (R E ) ~(apply (9.4.2) and Problem 3). Is the theorem still true for p = 2 ? (b) When E = C, show that f(B) c f ( S ) . (Apply the result of (a) to the function l/(f- c), where c $f(S).) In particular, iffis bounded in S , f i s bounded in B. Extend that last property to the case in which E is a complex Hilbert space (method of Problem 6 of Section 8.5).
+
+ +
11. LIOUVILLE'S THEOREM
(Liouville's theorem) Let f be an entire function in C p , with values in a complex Banach space E. Suppose there exists two numbers a > 0, N > 0 such that ]]j(z)I] < a (l ( ~ u p l z ~ lin) ~C". ) Then f ( z ) is the sum ofajinite with ~ c,, . . . n p E E and number of "monomials" C , , , , , ~ . ....~. :2~ Z ~ (9.1 1 . I )
+
nl
Let f ( z ) =
+ n2 + + np < N. **.
cvzv in C p , the power series being everywhere absolutely
convergent. The Cauchy inequalities (9.9.5) applied to the polydisk lzj]< R (1 < j < p ) yields, for any v = ( n l , . . . , np) Q a * (I R ) ~ R - ( " , +... + ' p ) . lc,,,
+
Letting R tend to
+ co, we see that c,, ...",
= 0,
unless n,
+ + np < N .
12 CONVERGENT SEQUENCES OF ANALYTIC FUNCTIONS
233
(9.11.2) (The "fundamental theorem of algebra"). Any polynomial f ( z ) = a,z" + a,zfl-l + * . . + a,, (a, # 0, n 2 1 ) with complex coeficients has at least one root in C. Otherwise, l / f would be analytic in C (9.3.2), hence an entire function (9.9.6). Let r be a real number such that rk 2 ( n t l ) ~ a k / afor o ~ 1 d k dn; then, for IzI 2 r
In other words, l / f is bounded for lzl 3 r. On the other hand, l / f being continuous in the compact set IzI < r, is also bounded in that set (3.17.10), hence l / f is bounded in C.Liouville's theorem then implies l / f is a constant, hence also f, contrary to assumption since If(z)I 2 la,] . Izl"/(n + 1 ) for IzI 3 r.
PROBLEMS
If p > 2, show that a function which is analytic in the complement of a compact subset of Cp is an entire function; hence if in addition it is bounded in the complement of a compact subset of C', it is a constant (use (9.1 1.I) and Problems 4 and 5 of Section 9.10). Is the result true for p = 1 ? L e t f b e a complex valued entire function in CP. Show that the conclusion of (9.11.1) is still valid if it is supposed that
Wf(z)) < a * (sy41,Iz,l)'") for any z in the exterior of a polydisk of CP (use Problem 6 of Section 9.9). Let f(z)
m
= "=O
a, z" be a nonconstant
M(r) = sup Ilf(z)II, so that p(r) IZI =I
entire function. For any r
< M(r);
> 0, let p(r) = sup JJanJIr", n
by Liouville's theorem, lim p(r)= f a . I-m
Suppose there are two constants a > 0 , a > O such that p ( r ) < a.exp(r"); show that there are positive constants b, c such that M(r) < brap(r) c. (Observe that llanIl< a(ea/n)"".)
+
12. CONVERGENT SEQUENCES O F ANALYTIC F U N C T I O N S
(9.12.1) Let (f,)be a sequence of analytic mappings of an open set A c C p into a complex Banach space E. Suppose that for each z E A, the sequence (f,(z)) tends to a limit g(z), and that the convergence is uniform in every compact
234
IX ANALYTIC FUNCTIONS
subset of A, Then g is analytic in A, and f o r each v = ( nl, . . . , np)E NP, the sequence ( D v L ( z ) )converges to Dvg(z)f o r each z E A, the convergence being uniform in every compact subset of A. As g is continuous in A (7.2.1), to prove g is analytic in A, we need only prove that each mapping zk +g(a,, . . . , zk, . . . , up) is analytic in A(a,, .. . , a k - 1 , a k f l , . . . , up), by (9.9.4); in other words we are reduced to the case p = 1. For each a E A c C, let B be a closed ball of center a and radius r contained in A, and let y be the circuit t + a reir (0 < t d2n); then, for each z E fi and each n, we have by Cauchy’s formula
+
But by assumption the sequence ( f , ( x ) ) converges uniformly to g(x) for Ix - a( = r, and as ) z - x( 2 r - IzI, the sequence (L(x)/(x- z ) ) ( z fixed) also converges uniformly to g(x)/(x- z ) for Ix - a ) = r ; hence, by (8.7.8)
which proves g is analytic in
fi by (9.9.2).
Moreover, as
by (9.9.3), the same argument (and (9.9.3) applied to g) shows that f ’ ( z ) tends to g’(z) for every z E 8; furthermore, we have by the mean-value theorem
Returning to the general case ( p arbitrary), let us now show that the sequence ( D k f , ( z ) )converges uniformly to D k g ( z ) in any compact set M c A. There is a number r > 0 and a compact neighborhood V of M contained in A, and containing all points of A having a distance d r to M (3.18.2). For any E > 0, let no be such that \lg(z) -f,(z)l) d E for every n > no and every z E V. Then, applying (9.12.1 . I ) to the sequence of functions zk -+&(al9. . . , ak- zk, ak+ . . . , up), we obtain, for every point z E M, (IDkg(z)- Dkffl(z))I< E/r as soon as n 2 n o . This ends the proof of the theorem when n, + * + np = 1 ; the general case is then proved by induction on nl np.
,,
+ . - -+
12 CONVERGENT SEQUENCES OF ANALYTIC FUNCTIONS 235
Observe again here that the theorem does not hold for analytic functions of real variables, since a sequence of polynomials can have as a limit an arbitrary (e.g. nondifferentiable) continuous function in a compact set, by the Weierstrass approximation theorem (7.4.1).
PROBLEMS
1. (a)
Let
(ak)l, O . 3. An endless road in an open subset A C C is a continuous mapping y of R into A such that in every compact interval I C R, y is the primitive of a regulated function. Iff is a continuous mapping ofy(R) into a complex Banach space E,fis said to be improperly integrable alongy if the improper integral
j. f ( y ( t ) ) y ' ( t )
c//
-an
exists (i.e., if both limits
lim /aof(y(/))y'(r) dt exist in E); the value of that -m
lim Jobf(y(t))y'(t) dt and b-fm
a+
integral is then callled the integral off dong y and written J y f ( z )dz. Let B be an open subset of C p ,g a continuous mapping ofy(R) x B into E; suppose that for each x E y(R), the function ( Z I , . . , z,) -+g(x,zI,. . . , 2,) is analytic in B and % that each of the functions - (x,z , , . . . , z,) is continuous in y(R) x B. Finally
.
suppose that for each (zl, . .. , z,) alongy,and that
j
'"
g(y(t), z,,
E
.
B, x + g ( x , zl, . . , z),
. . .,z,)y'(t)
c//
is improperly integrable
tends uniformly to
-n
s,
y(x, z I , . . . , z),
ctx
when (zI,. . . , z,) remains in a compact subset of B and n tends to i-co. Under these conditions, show that the function ( z I ,. . . , z,) 4.
+
ij
g(x,zl,
. . . , z,)
dx is analytic in B
(compare to (13.8.6)). Extend the result of Problem 2 of Section 9.9 to functions of p complex variables, D, (resp. D-) being defined by S(z,) > 0 (resp. .P(z,) i 0). (Observe that, by (9.12.1), for each z, such that .P(z,) = 0 and the intersection B of A with the set C p - ' x (z,) is not empty, the function (zI, , , z P - J+f(zI, . . . , z , - I , 2,) is analytic in B.)
..
13 EQUICONTINUOUS SETS OF ANALYTIC FUNCTIONS
5.
237
In the plane C, let Q be the square of center 0, defined by lg(z)l < I , l.Y(z)l < I . Let Q o , Q1, Q 2 , Q 3 be the images of Q by the mappings I + i
z
2
2'
z+---+-
ZJ-
-l+i 2
+-2'z
z+-
-1-i 2
I-i 2
z
f-, z+-+-. 2
+ + +
+
z
2
Let m,, = 0 , and for any h > I , let mh = 4 4' . . . 4h; if n = mh 4k + j , with h > I , 0 < k < 4h - I , 0 < j < 3, define inductively Q, as follows: let nl = mh-1 k, and let zn1be the center of Q,,; let p.,(z) z., -t- z/2* and take Q. = y,,,(Q). (a) Let B be the unit disk /zI C I , U the unit circle IzJ= 1. Show by induction on n the existence of three sequence of numbers (a"),(cJ, (t,) defined for I I 2 4, having the following properties:
+
~
(I)
O y o , suclz that
n= I
f ( z ) = g,(z) + g 2 ( z ) in S ('' Laurent series" off). Moreover the poii'er series g l , g , hacing these properties are uiiiqzie, and, .for every circuit y in we h a w
s,
By (9.9.4) we have
the series being convergent for Iz( < r l . On the other hand, for IzI > y o , 1x1 = y o , we have y- 1 1 -z - x - 2 z" where the right-hand side is normally convergent for 1x1 = ro ( z fixed); by (8.7.9), we get
the series being convergent for IzI > r o . This proves the first part of (9.14.2). Suppose next we have in S f ( z ) = C a n z n+ U,
(9.14.2.1)
n=O
C bflz-" n;
n= 1
both series being convergent in S ; let first y be a circuit in S, defined in I; there are points t , t' in I such that y ( t ) = inf y(s) = r and y ( t ' ) = sup y ( s ) = r' ssl
S€l
(3.17.10), hence ro < r < y(s) < r ' < r1 for any s E I . But, for r < JzI< r ' , both series in (9.14.2.1) are normally convergent (9.1.2), hence by (8.7.9), for any positive or negative integer in
+
As zk"/(k I ) is a primitive of z' for k # - 1, we have / / k dz = 0 for any circuit a ; (9.14.2) then follows from the definition of the index. If now y is in 5, we remark that there is an open ring S , : ( 1 - E)ro < IzI < ( I + c)r, contained in A (3.17.11), and we are back to the preceding case.
15 ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES
241
15. I S O L A T E D S I N G U L A R P O I N T S : POLES; ZEROS; RESIDUES
(9.1 5.1) Let A be an open subset of C, a mi isolatedpoint of C - A (3.1 0.1 0), r a number > O such that all points of the ball Iz - a1 < r except a belong to A. I f f is an analytic mapping of A into a complex Banach space E, then f o r 0 < Iz - a1 < r, we have f(z)=
2 cn(z - a)" + c dn(Z - a>-, m
a:
n=O
n= 1
where both series are convergent f o r 0 < Iz - a1 < r, and (X
tvliere y is the circuit t -+ a
+ re"
(0 < t
- u ) " - ~ ~ ( dx, x)
< 2n).
This follows at once from (9.14.2) applied to the ring p where p is arbitrarily small. Observe that the series u ( x ) =
< ( z - a ( < r,
C d n x " is an entire function
such that
n= I
u(0) = 0; we say that the function u(l/(z - a)) is the singular part o f f in the neighborhood of a (or at a). When u = 0, f coincides in the open
set U : 0 < Iz - a / < r with the function g ( z ) =
a:
c,(z - a)", which is
n=O
analytic for Iz - a1 < r ; conversely, iff is the restriction to U of an analytic function fi defined for Iz - a1 < r, then f i = g by (9.9.4) and (9.15.1), hence u = 0. When u # 0, we say that a is an isolated singular point off. If u is a polynomial of degree n 2 1, we say a is a pole of order n o f f ; if not (i.e. if d,,, # 0 for an infinite number of values of m) we say a is an essential singular point (or essential singularity) of j : In general, we define the order w(a; f ) or w(a) off at the point a as follows: o(a) = - co if a is an essential singularity; w(a) = -n if a is a pole of order n 2 1 ; w(a) = rn iff # 0, u = 0 -"
m
and in the power series
1 c,(z - a)"
equal to f ( z ) for 0 < Iz - a1 < r,
n=O
m is the smallest integer for which c,, # 0; finally w(a; 0) = +a.When w ( a ;f ) = m > 0 , we also say a is a zero of order m off. Observe that if both A g are analytic in the open set U : 0 < Iz - a1 < r, and take their values in the same space, then w ( a ; f + g ) 2 min(w(a; f ) , w(a; g ) ) ; if one of the functions f, g is complex valued, then w(a;f g ) = w(a;f ) w ( a ; g ) when one of the numbers w ( a ;f ) , w ( a ; g ) is finite. Any function f analytic in U and of finite order n (positive or negative) can be written in a unique
+
242
IX ANALYTIC FUNCTIONS
way ( z - a)yl, where fl is analytic in U and of order 0 at the point a. Finally, iffis analytic in U and complex valued, and of finite order n7, then it follows from the principle of isolated zeros and from (9.3.2) that there exists a number r' such that 0 < r' < r and that I/f is analytic in the open set 0 < Iz - a1 < r ' ; we have then w(a; IF) = - w(a;f). (9.15.2) Let f be analytic in the open set U : 0 < Iz - a1 < r . In order that o ( a ;f ) 2 n where n is a positive or negative integer, it is necessary and sufficient that there exist a neighborhood V of a in C such that ( z - a)-"f ( z ) be bounded in V n U. The condition is obviously necessary, since a function having order 3 0 at a is the restriction of a function analytic in a ball Iz - a1 < r. Conversely, by considering the function ( z - a)-"f(z), we can suppose n = 0. Then it follows from (9.15.1) and the mean value theorem that if 11 f(z)II d M in U , we have, for any p such that 0 < p < r, lld,,t,ll d Mp" for any m 3 1 ; as p is arbitrary, this implies d,,, = 0 for each m 2 1. Q.E.D. The coefficient d, in (9.15.1) is called the residue off at the point a.
PROBLEMS
1. Show that there are no isolated singular points for analytic functions of p > 2 complex variables (in other words, if A is an open subset of C p ,a E A and a mappingfof A - { a } into a complex Banach space E is analytic, it is the restriction of an analytic mapping of A into E; use Problem 5 in Section 9.10). 2. Letfbe a complex valued analytic function of one complex variable having an essential singularity at a point a E C; show that for any complex number h, it is impossible that the function l / ( f - h ) should be defined and bounded in an open set of the form V - {a},where V is an open neighborhood (use (9.15.2)). Conclude that for any neighborhood V of a such that f i s analytic in V - {a},f(V - { a ) ) is dense in C (" Weierstrass' theorem"; see Section 10.3, Problem 8). 3. An entire function which is not a polynomial is called a trunscendental entire function. Let f be a complex valued entire transcendenial function of one complex variable. (a) Show that for any integer n > 0, the open subset D(n) of C consisting of the points z E C such that If(z)i > n is not empty and cannot contain the exterior of any ball (apply Problem 2 to the function f(l/z)). (b) Let K(n) be a connected component (3.19.5) of D(n). Show that K(n) is not bounded and that If(z)i is not bounded in K ( n ) (if u $ K(n), consider the function f(l/(z - a ) ) and use Problem 14 of Section 9.5). (c) Show that there is a continuous mapping y of [0, K>[into C, such that in every interval [0, a ] ,y is the primitive of a regulated function, and that lim iy(t)i = +a
+
r-ti
IU
1 5 ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES
and lim If(y(t))l r-+m
=
243
+a.(Consider a sequence of open subsets L, c C such that L, is
a connected component of D(n), and L.+l C L, for every n ; the existence of such a sequence follows from (b). Use then (9.7.2)) (d) Extend the preceding results to complex valued entire transcendental functions of an arbitrary number of complex variables. (If f ( z I , . . . , z), = a n l . . “,zl n l . . . z p , there exists at least an index k such that there are infinitely many monomials with non zero coefficient R , ~ ,. . and arbitrarily large n k . On the other hand, prove that if (9,) is a denumerable family of entire complex valued functions of p complex variables, none of which is identically 0, then there exist points (cl, . . . , c,) for which g,,(cl,.. . , c,) # 0 for every m ; to do this, use induction on p , and the fact that for a function h(z) of one complex variable, analytic in A c C and not identically 0, the set of solutions z of h(z) = 0 is at most denumerable (see (9.1.5)).) 4. Let ~ ( xbe) an arbitrary increasing and positive real function defined in [0, cc [. Let (k.) be a strictly increasing sequence of integers such that k1 = 1 , and (n(n - 1))”“ > ~ ( n1 ) for n > 1. Show that the power series
+
f(z)= 1
+
+ 5 (L)k” n- I n=2
is convergent for all z E C, and that for every real x > 2, f ( x ) > ~ ( x(in ) other words, there are entire functions which tend to infinity “faster” than any given real function). 5. For any real numbers a , p such that p > 0, let L=J be the endless road (Section 9.12, Problem 3) defined as follows: for t < - I , L,,p(t) = a - ip - t - 1 ; for - 1 < t < I , L&) = a ipr; for f > 1, L,,&) = a ip r - 1. Let G.,p = LnSp(R). (a) Show that if 77/2 < p < 3rr/2, and if x $ Ga,pthe function z (exp(exp z))/(z - x) isimproperlyintegrablealongL..B. Furthermore,isP1,P2aresuchthat I-”(x)l < < p2 of /4(x)1> p2 > or a ( x ) < a, the integrals along and Llr.ozare the same; similarly, if 9 ( x ) < a1< a 2 ,or g1 < cc2 < B(x), or [.f(x)( > p the integrals along and La2,Aare the same (use Cauchy’s theorem). (b) Deduce from (a) that if L = Lo,11,
+ +
+
--f
PI,
PI
can be extended to an entire function. (c) Show that
(prove that the integral along La,” of exp(exp z ) is independent of cc and p (provided n-/2 < p < 3n-12)). (d) Show that if x belongs to the open set A defined by W ( x ) < 0 or i-”(x)l > rr,
where F(:) is bounded in A (express F(x) by an integral along Lo,pwith p < rr, using (a) and (c)). (e) Show that if x belongs to the open set B defined by W ( x )> 0 and IY(x)l < rr, then 1
E(x) = exp(exp x ) - x
GW +x2
IX ANALYTIC FUNCTIONS
244
where G(x) is bounded in B (prove first, using Cauchy's formula, that if and /9(x)1< x , then
-1
< 9 ( x )< 0
where L'= L - l , n . Show next that that formula is still valid for x E B using (9.4.2) and express G ( x )by an integral along L - l . f lwith p > x ) . (f) Let H(x) = E(x)e-"("'; show that H is an entire transcendental function such that lim H(rejB)= O for every real # (use (d) and (e); compare with the result of v-+m
Problem 3). 6. Let f be a complex valued entire function of p > 2 complex variables. Show that if f ( a l ,..., a,) = b, then for every r > 0 , there exists z = (zI. ..., z,) such that Izp - ukI2= r 2 and f ( z I , . . . ,z,) = b (use Problem 5(b) of Section 9.10). k
7. Letfbe an analytic mapping of an open subset A c C p into a complex Banach space E. A frontier point zo of A is called a regular point forfif there is an open neighborhood V of zo and an analytic mapping of A u V into E which coincides with f i n A. A frontier point of A is said to be singular forfif it is not regular. (a) Let R < +a, be the radius of convergence (Section 9.1, Problem I ) of a power series f ( z ) =C a, z" of one complex variable. There is at least one point zo such that
lzol = R which is a singular point for f. (Otherwise one could cover the circle Iz/ R with a finite number of open balls Bk whose centers hk are on that circle, and such that in each open set B(0; R) u Bx there is an analytic function 6 coinciding with f in B(0; R). Show that for any two indices h, k for which Bh n Bx # @,fh andfx coincide in Bh n Bk, using (9.4.2), and conclude from (9.9.1) and (9.9.2) that the radius of convergence of CI, z" would be > R.)
"
(b) With the notations of (a), suppose a, 0 for every n. Show that the point z = R is singular for f. (One may suppose R = I . Let el" be a singular point for f; then for 0 < I' < I, the radius of convergence of the power series ~ f ( " ' ( r e ' Q ) z " /is t i !exactly 1 - r (9.9.1). Observe that If(")(re")l < f ( " ) ( r ) ,and use (9.1.2).) (c) With the notations of (a), suppose R = I . Let b, c be two real numbers such that 0 < h < 1, c 1 - h, and let p be an integer > I . In order that the point z = 1 be a singular point for f, it is necessary and sufficient that the Taylor series 1 y'")(O)u"/ir! ~
"
+
for the function y(u) = f(bu, c u p + ' )have a radius of convergence equal to I . (Observe that if I { I ~ < I , Ibu" c u P f 1 I < 1, and that the two sides of the last inequality can only be equal for u = 1. The proof for the necessity of the condition has to use (10.2.5), in order to show that there is in the neighborhood of z = I an analytic function h(z) such that z = g(h(z)) in that neighborhood.) (d) Suppose (with the notations of (a)) that a, = 0 except for a subsequence ( t i x ) of integers such that nk+l > (1 8)nk for every k , where 6 > 0 is a fixed number. Show that euery point zo on the circle IzI = R is a singular point for f(" Hadarnard's gap theorem "; the circle /zI = R is called a mrlmlboundary forf). (One may suppose
+
+
R = I . Use criterion of (c), taking p > l/8, and let g(u) =
m
C dnu" be
the Taylor
"=O
development of g at u = 0. By assumption, for given E > 0 there is a subsequence (mi) of integers such that llun,,~~ 2 (I- E ) ' " ~ (Section 9.1, Problem I). On the other hand the function F(u) (bup cu,+l)"" =C e.u" has I I = 1 as a singular point, by (b),
=c J
+
"
hence there is a subsequence (4,)of integers such that jeqll > (1 - ~ ) " l . l:dq~ll 2 (1 - E ) 2 4 9 .
Prove that
16 THE THEOREM OF RESIDUES
245
16. T H E T H E O R E M OF RESIDUES
We first recall that any subset S c C the points of which are all isolated is at most denumerable, for the subspace S of C is then discrete and separable (by (3.9.2), (3.20.16), and (3.10.9)), hence S is the only dense subset of S (3 .I 0.1 0).
(9.16.1) Let A c C be a simply connected domain, (a,,) a (jinite or injinite) sequence of distinct isolated points of A, S the set of points of that sequence. Let f be an analytic mapping of A - S into a complex Banach space E, and let y be a circuit in A - S. Then I c v have
where R(a,,) is the residue off at the point a,, and there are only aJinite number of terms # O on the right-hand side (" theorem of residues"). We can obviously suppose each a, is a singular point for f, for we can extend f by continuity to all nonsingular points a,, which does not change both sides of the formula (since R(a,) = 0 if a, is not singular). Under that assumption, for any compact set L c A, L n S is jinite, for L n S is closed in L, as A - S is open in C by definition; hence L n S, being compact and discrete, is finite (3.16.3). Let I be the interval in which y is defined, and let P be the set of points x E A such that j ( x ; y) # 0. We know (9.8.6) that the closure P of P in C is compact, and P does not contain any frontier point of A, for such a point cannot be in y(I), nor have index # O with respect to y , by (9.8.7); as the set of points x in C - y(1) where the index ,j(x;y) takes a given value is open (9.8.3), any point in P which does not belong to y(1) is in P, hence P c A. On the other hand, let q(t, 5 ) be a loop homotopy in A of y into a one-point circuit ( t E I, ( E J, where J is a compact interval). Then M = cp(I x J) is a compact subset of A. Let H c N be the finite set of the integers n such that a,€ M u P; for each n E H, let zr,(l/(z - a,,)) be the singular part off at the point a,. Let B be the complement in A of the set of points a, such that n 4 H ; then B is open, for a compact neighborhood of a point of B, contained in A, has a finite intersection with S. By definition of the singular parts, there is a function g, analytic in B, and which is equal to f ( z ) -
at every point z # a,, ( n E H).
246
IX ANALYTIC FUNCTIONS
As M c B by definition, y is homotopic in B to a one-point circuit; hence, by Cauchy’s theorem, g(z) dz = 0, in other words
si
the result then follows from (9.14.2), applied to each of the functions u,, in an open
“
ring ” of center a,, , containing y(1).
17. M E R O M O R P H I C F U N C T I O N S
Let A be an open subset of C, S a subset of A, all points of which are isolated. A mapping f of A - S into a complex Banach space E is said (by abuse of language) to be meromorphic in A if it is analytic in A - S and has order > -co at each point of S. By abuse of language, we will always identifyfto its extension by continuity to all points of S which are not poles off; the argument used in (9.16.1) then shows that we can always suppose that for any compact subset L of A, L n S is jinite. Iff, g are two meromorphic functions in A, taking their values in the same space, and %whosesets of poles in A are respectively S, S‘, then S u S’ has all its points isolated, due to the preceding remark; f + g is defined and analytic in A - (S u S’), and has order > --GO at each point of S u S‘, hence is meromorphic in A (note that some points of S u S’ may fail to be singular for f + g). Similarly if f and g are meromorphic in A, and g is complex valued, f g is meromorphic i n A. Iff is meromorphic in A, S is the set of its poles, T the set of its zeros, then all the points of S u T are isolated; for if a E A and w(a) =h, then f ( z ) = ( z - a)hfl(z) in 2 set 0 < Iz - a1 < r, where A is analytic in ( z - a ( < r and f i ( a ) # 0; the principle of isolated zeros (9.1.5) shows that there is a number r‘ such that 0 < r‘ < r, and f ( z ) # 0 for 0 < Iz - a1 < r’. This proves our assertion, and shows moreover that L n ( S u T) is finite for any compact subset L of A (same argument as in (9.16.1)). In particular, iffis complex valued, l/fis meromorphic in A, S is the set of its zeros and T the set of its poles. Moreover, with the same notation as above, we have f ’ ( z ) = h(z - ~ ) ~ - l f , ( z () z - a)hf;(z) for 0 < Iz - a1 < r, hence f‘/f, which is analytic for 0 < I z - a1 < r‘, has order 0 at the point a if h = 0, order - 1 and residue h at the point a if h # 0.
+
(9.17.1) Let A be a simply connected domain in C , f a complex valued meromorphic function in A , S (resp. T) the set of its poles (resp. zeros), g an analytic function in A . Then, for any circuit y in A - ( S u T ) , we have
17 MEROMORPHIC FUNCTIONS
247
a$nite number of terms only being # 0 on the right-hand side.
This follows at once from the theorem of residues, for the residue of gf ’If at a point a E S u T is the product of g(a) by the residue off’/ at the point a. (9.17.2) W i t h the assumptions of (9.17.1) let t --t y ( t ) ( t E 1) be a circuit in A - ( S u T). Zfr is the circuit t +f ( y ( t ) ) ,then
For it follows at once from (8.7.4) that
hence the result is a particular case of (9.17.1) for g = 1. (9.17.3) (Rouchk’s theorem) Let A c C be a simply connected domain, f , g two analytic complex valued functions in A. Let T be the (at most denumerable) set of zeros o f f , T’ the set of zeros o f f + g in A, y a circuit in A - T, dejned in an interval 1. Then, if Ig(z)l < 1 f ( z ) l in y(I), the function f g has no zeros on y(l), and
+
+
The first point is obvious, since f ( z ) g(z) = 0 implies If ( z ) l = Ig(z)l. The function h = ( f + g)/f is defined in A - T and meromorphic in A ; we have
(-f + s > ’- -f’ h’ f+s f+h
in A -(T u T’).
Using (9.17.2), all we have to prove is that the index of 0 with respect to the circuit r : t+h(y(t)) is 0. As gif is continuous and finite in the compact set y(I), it follows from (3.17.1 0) and the assumption that r = sup Ig(z)/f(z)I < 1. In other words, is in the ball Iz - 1 I < r, and as SY(1)
0 is exterior to the ball, the result follows from (9.8.5).
248
IX ANALYTIC FUNCTIONS
(9.17.4) (Continuity of the roots of an equation as a function of parameters) Let A be an open set in C,F a metric space, f a continuous complex valued function in A x F, such that for each a E F, z +f ( z , a) is analytic in A. Let B be an open subset of A, whose closure B in C is compact and contained in A, arid let E F be such that no zero o f f ( z , ao) is on the frontier of B. Then there exists a neighborhood W of a. in F such that: (I) for any a E W, f ( z , a ) has no zeros on the froritier of B ; (2) for any a E W, the sum of the orders of the zeros o f f ( z , a ) belonging to B is independent of a. The number of distinct zeros of f ( z , ao) in B is finite; let a,, . . . , a, be these points. For each frontier point x of B, there is a compact neighborhood U, of x , contained in A, such that f ( z , ao) has no zero in U, (9.1.5); if we cover the (compact) frontier of B by a finite number of sets U x j , the union U of 8 and the U,, is a compact neighborhood of B, contained in A and such that f ( z , ao) has no zero in U n (A - 8).Let r be the minimum of the numbers la, - ajl (i # j ) , and for each i (1 < i < n), let Di be an open ball Iz - ail < ri of radius r, < r/2, contained in B ; then D i n Dj= @ if i # j . Let H =U Di); this is a compact set; let m be the minimum value of I f ( z , ao)l in H ; we have m > 0 by (3.17.10). Now, for each X E 8, there is a neighborhood V, of x contained in A and a neighborhood W, of a. in F, such that I f ( y , a)- f ( x , a,)l < m/2 for y E V, and a E W,. As 8 is compact it can be covered by a finite number of sets VXk (1 < k G p ) ; let W = WXk;this is a neighborhood of a. in F, and by definition, for any
-(u
k
W and any y E 8, we have If ( y , a ) -f ( y , ao)l < m. As a first consequence, it follows that f ( y , a) # 0 for y E H and a E W ; on the other hand, as I f ( z , a ) - f ( z , a0)l < If ( z , ao)l in H, RouchC’s theorem, applied to each circuit t + a, rieit (0 < t G27r) shows that the sum of the orders of the zeros off (z, a ) in Di is independent of a E W, hence the theorem. CY
E
+
PROBLEMS
1. Let A c C be an open simply connected set,fa nieroniorphic complex valued function in A, such that each pole off is simple and the residue of f a t each of these poles is a positive or negative integer. Show that there is in A a meroinorphic function g such thatf= g’/g. (If zo is not a pole off, show that for any point z1 E A which is not a pole off, and any road y in A, defined in I c R,of origin zo and extremity zl, and such that y(1) does not contain any pole off, the number exp 2.
0)
v)dx onlydependsonzoandzl,
and not on the road y satisfying the preceding conditions (use the theorem of residues).) Let f be an entire function of one complex variable, such that for real x , y , llf(x i-iy)ll < e l y 1 .Show that, for any z distinct from integral multiples nn of n,
17 MEROMORPHIC FUNCTIONS
249
(- 1)"f(nn)
+
(z
-m
-
m)'
where the series on the right-hand side is normally convergent in any compact subset of C which does not contain any of the points nn (n integer). (Consider the integral
's f(4 2ni
Yn
dx -sin x ( x - z)'
+
where yn is the circuit t + (n $ ) r e " for -n < t < n.Observe that for every E > 0, there is a number C ( E ) > 0 such that the relations Iz - n7r 2 E for every integer n E Z imply lsin zI > c(c)el,F(z)l;and use the theorem of residues.) 3. (a) Show that for z # nn (n integer) 1 cotz=-+C z
22 z2-n2nz ~
"=I
where the right-hand side is normally convergent in any compact subset of C which does not contain any of the points nn. (Use Problem 2 and the relation lim (cot z - l/z) = 0.) I-0
(b) Deduce from (a) that
, the convergence being
where the product is defined as the limit of k=l
uniform f(z)
in
=n(1 +
01)
-m
every
4
--
compact
subset
of
C.
(Consider
the
entire
function
e - z / n n (Section 9.12, Problem l), and use (a) to prove that the
function (sin z ) / z f ( z ) is a constant.) (c) Deduce from (b) the identity
4.
(see Section 9.12, Problem 2). Let f be a complex valued function analytic in an open neighborhood A of 0 in C p ; for convenience we write w instead of zp and z instead of ( z l , . . . , z ~ - ~Suppose ). that f ( 0 , O ) = 0 and that the function f ( 0 , w ) , which is analytic in a neighborhood of w = 0 in C , is not identically 0. Then there exist an integer r > 0, r functions h,(z) analytic in a neighborhood of 0 in C P - l ,and a function g(z, w ) analytic in a neighborhood B c A of 0 in C p and # O in that neighborhood, such that f ( z , w ) = (w'
+ h,(z)w'-1 + .. . + h,(z))s(z, w )
in a neighborhood of 0 in Cp (the " Weierstrass preparation theorem"). (If f ( 0 , w) has a zero of order r at w = 0, use (9.17.4) to prove that there is a number E > 0 and a neighborhood V of 0 in C p - l such that for any Z E V, the function w 4 f ( z , w ) has exactly r zeros in the disc lwI < E and no zero on the circle Iwi =- E . Let y be the circuit t + w i t for -n < t < n;using the theorem of residues, show that there are
250
IX ANALYTIC FUNCTIONS
functions h,(z) (1 < j < r) analytic in V and such that the polynomial F(z, w) = wr+ hl(z)w'-' ... h,(z) satisfies the identity
+ +
for Z E V and IwI > E). Let (f.) be a sequence of complex valued analytic functions in a connected open subset A of C.Suppose that for each z E A, the sequence ( f . ( z ) )tends to a limit g(z), and the convergence is uniform in every compact subset of A. Suppose in addition that each mapping z -.f.(z) of A into C is injective. Show that either g is constant in A or g is injective (for any zo E A, consider the sequence (f.(z) -f.(zo)) and apply (9.17.4) and the principle of isolated zeros). Let be a real valued twice differentiable function in the interval 10,11. Suppose ) q#)cos x = 0 in 1-77, n [ . Iq~(0)l< Iv(l)l, and let xo be one of the zeros of ~ ( 0 Show that the entire function F(z) =
IO1
p(r) sin zt df
has a denumerable set of zeros; furthermore it is possible to define a surjective mapping n -.z, of Z onto the set of zeros of zF(z), such that each zero corresponds to a number of indices equal to its order, and that lim ( z ~ ~ - x "2n7r)=
+
n-fm
lim ( z ~ ~ xo+ -~2n7r) = 0. (Integrating twice by parts, show that one can
n-fm
+
zF(z) = ~ ( 0) p(1) cos z G ( z ) , where IG(z)l < u e l f l ( z ) l / l z / ;minorize j ~ ( 0) ~ ( 1 )cos zI outside of circles having centers at the zeros of that function, in
write
the same was as lsinzl was minorized in Problem 2 ; and use RouchC's theorem in a suitable way.) Treat similarly the cases in which I ~ ( 0 )> l Ip(1)l or ly(0)l = /q~(l)j.
APPENDIX TO CHAPTER IX
APPLICATION OF ANALYTIC FUNCTIONS TO PLANE TOPOLOGY (Eilenberg’s Method)
1. I N D E X OF A P O I N T WITH RESPECT TO A L O O P
(Ap.1.1) If t + y(t) (a < t < b) is a path in an open subset A of C , there is in A a homotopy cp of y into a road yl, such that cp is defined in [a, b] x [0, 13 and ~ ( a5), = y(a) and q(b, t) = y(b)for every 5 E [0, I ] .
Let I = [a, b ] ;as y(1) is compact, d(y(I), C - A) = p is > 0 (3.17.11). Asy is uniformly continuous in I (3.16.5), there is a strictly increasing sequence (tk)06ksmof points of I such that to = a, t, = b, and the oscillation (Section 3.14) of y in each of the intervals [tk, t,,+l] (0 f k f m - 1) is < p. Define (0 < k < m - 1); it is clear that y1 is a road, with yl(a) = y(a), y,(b) = y(b), and yl(I) is contained in A, since y l ( [ t k , tk+,])is contained in the open ball of center Y(tk) and radius p. Define then cp(t, t) = ty,(t) (1 - 0. Let A‘, H’ be the images of A, H under the homeomorphism Z - + S , , ~ ( Z ) of C - {b} onto C - { 1) ; H’ is compact and A‘ is a connected open subset of C - H’, which is bounded and contains 0. Moreover, the frontier points of A‘ in C are points of H’ and (possibly) 1 ; hence A’ is compact and so is A’ u H’. In addition, if 1 belongs to the boundary of A’, this means that A is unbounded, hence has points in common with the exterior of a ball containing H ; but as that exterior is connected (9.8.4), it is contained in A by definition of a connected component (Section 3.19). This shows that there is a ball V of center I , such that V - (1) c A’, hence 1 is not a frontier point of C - A‘, which proves that the frontier of C -A’ is always contained in H’. We have to show that the mapping u -+ u/lul of H’ into U is essential (Ap.2.2). Suppose the contrary; then there would exist a continuous mapping f of H’ into R such that u/lul = eif(”) for u E H’. By the Tietze-Urysohn theorem (4.5.1), f can be extended to a continuous mapping g of A’ u H’ into R. Define a mapping h of C into U by taking h(u) = u/lul for u E C -A‘, h(u) = eig(”)for u E A’; it follows at once from the definition of g that h is continuous in C. Let r > 0 be such that A‘ is contained in the ball B : IzI < r ; the restriction of h to B is inessential (Ap.2.6), and so is therefore the restriction of h to S : IzI = r. But the identity mapping [ -+ [ of U onto itself can be written h, gl,where h, is the mapping z -+ z/lzl of S onto U, and g1 the mapping -+ rc of U onto S. However, h, is the restriction of h to S, hence inessential, and therefore h, 0 g1 would be inessential (Ap.2.2), contradicting (Ap.2.9).
c
0
(Janiszewski’s theorem) Let A, B be two compact subsets of C, a, b two distinct points of C - (A u B). If neither A nor B separates a and b, and if A n B is connected, then A u B does not separate a and b.
(Ap.3.2)
From the assumption and (Ap.3.1) it follows that the restrictions of b(z)/Is,, b(Z)I to A and B are inessential; by (Ap.2.7) the restriction of that mapping to A u B is also inessential, hence the conclusion by (Ap.3.1).
z -+ s,,
4. SIMPLE ARCS A N D SIMPLE C L O S E D C U R V E S
An injective path t -+ y(t) in C, defined in I = [a, 81, is also called a simple path; a subset of C is called a simple arc if it is the set of points y(I) of a simple path. A loop y defined in I is called a simple loop if y(s) # y ( t )
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APPENDIX TO CHAPTER IX
for any pair of distinct points (s, t ) of I, one of which is not an extremity of 1. A subset of C is called a simple closed curue if it is the set of points of a simple loop. Equivalent definitions are that a simple arc is a subset homeomorphic to [0, 11, and a simple closed curve a subset homeomorphic to the unit circle U (9.5.7).
The cornplernerit in C of a simple arc is corinected (in other words, a simple arc does not cut the plane).
(Ap.4.1)
Let y be a simple path defined in I, and let f be the continuous mapping of y(1) onto I, inverse to y . Let a, b be two distinct points of C - ?(I). By (Ap.3.1), we have to prove that the restriction cp of z -+ sa, b ( ~ ) / I ~ , , &z)I to y(1) is inessential. But we can write cp = (cp y) o f ; the continuous mapping cp y of I into U is inessential (Ap.2.6), and so is therefore cp by (Ap.2.2). 0
0
(The Jordan curve theorem) Let H be a simple closed curue in C. Then : (a) C - H has exactly two connected components, one of which is bounded and the other unbounded. (b) The frontier of every connected component of C - H is H. (c) I f y is a simple loop dejned it7 I and such tl7at y(1) = H, then j ( x ; y) = 0 i f x is in the unbounded connected cotnponent of C - H, and j ( x ; y ) = f.1 i f x is in the bounded connected component of C - H . (Ap.4.2)
The proof is done in several steps.
(Ap.4.2.1) We first prove (b) izithocrt any assumption on the number of components of C - H. Let A be a connected component of C - H ; as C - H is open, we see as in (Ap.3.1) that the frontier of A is contained in H. Let z E H , and let f be a homeomorphism of U onto H ; let [ = eie E U be such that f ( [ ) = z. Let W be an arbitrary open neighborhood of z in C, V c W a closed ball of center z ; then there is a number w such that 0 < w < n and that f ' ( e " ) E V for 0 - w < t < 0 + w ; let J be the image of that interval by t + f ( e " ) ; then the complement L of J in H is the image by t +f'(eit) of the compact interval [0 + w - 27t, 0 - w ] (9.5.7), and is a simple arc by (9.5.7). It follows from (Ap.4.1) that the open set C - L 3 C - H is connected. Therefore (9.7.2) for any x E A c C - L, there is a path y in C - L, defined in I = [a, b ] , such that y(a) = x,y(b) = z.
4
SIMPLE ARCS A N D SIMPLE CLOSED CURVES
257
The set y(l) n J is compact and contained in V ; let M be its inverse image by y, which is a compact subset of I , such that a 4 M ; let c = inf M > a. Then the image by y of the interval [a, c[ is a connected set P ((3.19.7) and (3.19.1)), which does not meet J nor L, hence is contained in C - (J u L) = C - H ; as P contains x, it is contained in A by definition. But when t < c tends to c, y ( t ) E A tends to y(c) E V, hence y ( t ) E W as soon as c - t is small enough; this shows that z E A. Q.E.D. (Ap.4.2.2) We next prove the theorem under the additional assumption that H contains a segmtvit S with distinct extremities. Applying to C a homeomorphism z +Az I- ,u, we can suppose S in an interval [ - a , a ] of the real line R. Let p = d(0, H - S) d a, and consider an open ball D: IzI < r , with r < p ; then D n (C - H) = D n (C - S), and it is clear that D n (C - S) is the union of the two sets D,: IzI < r , 4 z > 0 and D,: Jz)< r , f z < 0, which have no common points. It is immediately verified that the segment joining two points of D, (resp. D2) is contained in D, (resp. DJ, hence (3.19.3) that D,, D, are connected. On the other hand, we have seen in (Ap.4.2.1) that every connected component of C - H meets D, hence meets D, or D.,; but if two connected components of C - H meet D, (resp. D2), they are necessarily identical, since D, (resp. D,) is connected and contained in C - H (3.19.4). This proves that C - H has at most t ~ v oconnected components. We prove next that C - H is not conriected, hence has exactly fbvo components. Suppose the contrary, and let x E D,, y E D,; as D is connected, C - D does not separate x and y ; on the other hand, if C - H is connected, H does not separate x and y . But H n (C - D) is the complement in H of the open interval 1-r, r [ ; by (9.5.7), this complement is therefore a simple arc, hence connected. By Janiszewski's theorem (Ap.3.2), the union H u (C - D) does not separate x and y ; but this is absurd, since the complement of H LJ (C - D) in C is D, u D,, and D,, D, are open sets without common points, hence D, LJ D, is not connected. As H is compact, it is contained in a ball of center 0, whose complement in C is connected, hence contained in a connected component of C - H ; this shows one of these components A is unbounded, and the other B is bounded. Moreover, it is clear that j(x;y) = 0 when x E A (9.8.5). On the other hand, D, is contained in one of the components of C - H, D, in the other; all we need to prove therefore is that j ( x l ; y) - j ( x 2 ;y) = 5 1 for one point x, E D, and one point x, E D, (9.8.3). Supposing the origin of y to be the point a E S, let J c I be the inverse image by y of H - S, which is a compact interval [s(, p] and let y, be the path t + y ( t ) defined in J, of extremities - a and a. By (Ap.1.1) there is a homotopy 'p, in C - D of
258
APPENDIX TO CHAPTER IX
y, into a road y 2 , such that cp, is defined in J x [0, I ] and cp,(cc, t) = y(cc), q l ( p , t) = y(P) for any 5. Define cp in I x [0, I ] as equal to cpl in J x [0, I ] and to y ( t ) for any ( t , 5 ) ~ ( -1 J) x [0, I ] ; then for any x1 E D , (resp. x2 E D2), cp is a loop homotopy in C - {x,} (resp. C - {x2})of y into a circuit y 3 . We can therefore limit ourselves to proving that j ( q ; y) - j ( x 2 ;y ) = +_ 1 when y is a circuit defined in I, having the following properties: (1) S c ?(I) and if T is the inverse image y - ’ ( S ) , then T is a subinterval of I and the restriction of y to T is a homeomorphism of T onto S; ( 2 ) y(I - T) is contained in C - D (note that perhaps this new y is not a simple loop). Then the inverse image by y of the interval [ - r , r ] is a subinterval [A, p ] of T ; suppose for instance that ?(A) = - r , y ( p ) = r . We can suppose (replacing y by an equivalent circuit) that I = - 7 1 , p = 0 , and moreover that - r is the origin of y, so that I = [ - 7 1 , w ] with w > 0. Take xl = i t , x2 = -it with 0 < 4 < r ; let cr be the road t + y ( t ) , - 7 1 d t < 0, 6, the road t -+ rei‘, --n b t b 0, 6, the road t -+ r e - i t , - n < t < 0. Then, Cauchy’s theorem applied in the halfplane 4 ( z ) < [ (resp. 9 ( z ) > - 4 ) which is a star-shaped domain (9.7.1) yields dz
dz
and
dz z
+ it‘
Hence
Now the left-hand side is independent of 5 , and when t tends to 0, the righthand side tends to 2ni, using the fact that I y ( t ) l > r for 0 ,< t b w, the mean value theorem (to majorize the last integral), and (8.11.1).
(Ap.4.2.3) We now turn to the case in which H contains no segment with distinct extremities. Let a, b be two distinct points of H, S the segment of extremities a, b ; we may again suppose that S is a closed interval in R. By assumption, there is at least one point x E S n (C - H); let J be the connected component of x in S n (C - H), which is an open interval ] y , z [ since S n (C - H) is open in R ((3.19.1) and (3.19.5)); moreover its extremities y , z are in H. Let g be a homeomorphism of H onto the unit circle U, and let g(y) = e”, g(z) = eid, where we may suppose that c < d < c 271 (9.5.7). Let U,, U, be the simple arcs, images of t -+ e i f , c < t b d , and t -+ eit, d d t < c + 27r, and let H,, H, be their images by the homeomorphism f of U onto H, inverse to g. Using (9.5.7), we see immediately that there is a
+
4 SIMPLE ARCS AND SIMPLE CLOSED CURVES
259
homeomorphismf, (resp.f,) of U, (resp. U,) onto the closed interval J = [ y , z ] , such that f,(eic) =f2(ei‘) = y , f i ( e i d )= f 2 ( e i d )= z. Let h, (resp. I?,) be the mapping of U into C, equal to f in U, (resp. in U,), tof, in U, (resp. to& in U,); the definition of J implies that h,, h, are homeomorphisms of U onto two simple closed curves GI = HI u J, G , = H, u J, each of which contains the segment J. Let w E H,, distinct of y and z ; there is an open ball D of center w, which does not meet the compact set G, . From (Ap.4.2.1), each connected component of C - G, has points in D; moreover, if id, wf’ are two points of D in a same connected component of C - GI, iv’ and M”’ are not separated by GI ; they are not separated either by G, , since they belong to D c C - G, which is connected. But GI n G, = J is connected, hence, by Janiszewski’s theorem (Ap.3.2), iv’ and W” are not separated by GI u G , , nor of course by H c G, u G, . In other words, id and w” belong to the same connected component of C - H. But as C - G, has exactly two connected components, and each connected component of C - H has points in D by (Ap.4.2.1), it follows that C - H has at most two connected components. On the other hand, it follows from (Ap.4.2.2) that there are two points NJ’, iv” in D which are separated by G,. We show they are separated by H. Otherwise, as they are not separated by G, , and G, n H = H, is connected, they would not be separated by G, u H 3 GI (Ap.3.2), contrary to assumption. We have thus shown that C - H has exactly two connected components; the same argument as in (Ap.4.2.2) proves that one of them, A, is unbounded and the other, B, is bounded. Finally, we can suppose y is the origin of the loop y, and, if I = [a, PI, that HI = y([a, A]), H, = y ( [ A , /I]). Define the loops y1 and y, as follows: yl(t) = ( t - a + l)(y - z ) + z for a - 1 < t < M, yl(t) = y ( r ) for a < t < A; y 2 ( t ) = y ( t ) for A < t < /I, y 2 ( t ) = y ( t - P)(z - y ) for P < t < P 1. Using (Ap.1 .I) it is immediately verified that for any point x # G, u G , , j ( x ; y ) = ,j(x; yl) + j(x; y,). With the same meaning as above for D, let again M“, w’’ be two points of D separated by GI; then we have j ( d ; y,) = j ( w ” ; y,) since iv’ and KJ“ are not separated by G, (9.8.3), and j ( s ’ ;y , ) - j ( w ” ; 7,) = k 1 by (Ap.4.2.2). From this it follows that j ( w ’ ; y) - ,j(it,”; y ) = k I , which ends the proof.
+
+
(Ap.4.3) Let H be a simple closed curve in C, D the bounded connected component of C - H. Then, for any loop y in D, j ( x ; y ) = 0 f o r any x E H.
Let U be an open ball of center x, having no common points with the set y(1) of points of y. There exists in U a point z E C - (D u H) = C - D (Ap.4.2), and as U is connected, j ( x ; y) = j ( z ; y ) (9.8.3). But j ( z ; y) = j ( y ; y)
260
APPENDIX TO CHAPTER IX
for all points y of the unbounded connected component C - of H (9.8.3), and there are points y E C - r> which are exterior to a closed ball containing y(1); for such points, j ( y ; y) = 0 (9.8.5),hence the result.
PROBLEMS
1. Let A be a connected open subset of C; show that for any two points a , h of A, there is a simple path y contained in A , having a and b as extremities, and whose set of points is a broken line (Section 5.1, Problem 4; this amounts to saying that y is piecewise linear). (Use a similar argument as that in (9.7.2).If a “ square” Q = 1 x 1 c A (I closed interval with nonempty interior in R) is such that a @ Q, and there is a simple path t + y l ( t ) in A, defined in J c R, with origin a and extremity c E Q, consider the E Q, and observe that the segment of extremities smallest value f o E J such that rl(ro) y l ( t o )and any point of Q is contained in Q.) 2. Is Janiszewski’s theorem still true when A and B are only supposed to be closed subsets of C, even if A n B is compact (and connected)? Show that the statement of the theorem remains true in the two following cases: (1) A, B are two closed sets, one of which is compact; (2) A and B are two closed sets without a common point. (If c is a point sufficiently close to a , consider the mapping z+ l/(z - c), and the images of a, b, A and B under that mapping.) 3. For any simple closed curve H in C, denote by P(H) the bounded component of C H. (a) Let A be a connected open subset of C, H a simple closed curve contained in A. Show that A - H has exactly two connected components, which are the intersections of A and of the connected components of C - H (use Problem 2). (b) More generally, if H i (1 < i < r ) are r simple closed curves contained in A, and such that no two of them have common points, the complement of H i in A has ~
u
+
I
exactly r 1 components (use induction on r ) . (c) If H, H’ are two simple closed curves without a common point in C, show that or the closure of one of the sets P(H), P(H’) is contained in either P(H) n P(H’) = 0, the other. (Observe that if H c P(H’), the unbounded component of C - H’ has no common point with P(H), using (3.19.9).) (d) Suppose a connected open subset T of C has a frontier which is the union of r simple closed curves H, ( I < i < r ) , no two of which have common points. Show that there are only two possibilities: (1) T is unbounded and no two of the sets P(H,) have common points, their union being the complement of T; (2) there is one of the H , , say are contained in P(H,) for 1 < i < r - 1, no two of the -H , , such that the /$Hi) (1 < i < r - 1) have common points, and T is the complement of the union of the ,&H,) (for 1 < i < Y - 1) in P(H,). (If y , is a simple loop whose set of points is H i (1 < i < v ) , observe that the indices j ( x ; y , ) are constant for x E T, and that at most one of them may be # 0; otherwise, using (c), show that one at least of the H i would not be contained in the frontier of T.) 4. Let A be a bounded open connected subset of C, such that for any loop y in A and any z E C - A , j ( z ; y ) = 0. (a) Show that for any simple closed curve H c A, the bounded component P(H) is contained in A. (Observe that otherwise it would contain points of C - A, using (3.19.9) and part (b) of the Jordan curve theorem.)
B(H,)
4
SIMPLE ARCS A N D SIMPLE CLOSED CURVES
261
(b) Let ( z 12’) be the euclidean scalar product xx’ 4 yy’ in the plane C = R2 (with z = x -t iy, z’ = x’ -t if). Let Po be the open hexagon defined by the relations
l(zl I ) I < 4,
l(zl eini3)1< 4,
I(zl e-ini3)l< 3.
For any number a > 0, the set of all hexagons aP,. deduced from aPo by all translations of the form me'"'^ I- n), where m and n are arbitrary integers in Z , is called an hexagonalnet of width a ; the sets aP,,,, (resp. aFn,,,)are called the open (resp. closed) meshes of the net; the boundary of aP,,,, is the union of 6 segments (the sides of UP,,), whose extremities are called the vertices of ctP,,,; the nodes of the hexagonal mesh are all the vertices of the meshes. Every node is a vertex of three meshes, and the other extremities of the three sides issuing from that node are called its neighboring nodes. Let B be the union of a finite number of closed meshes of an hexagonal net. Show that if a node belongs to fr(B), there are exactly two neighboring nodes which also belong to fr(B); conclude that fr(B) is the union of a finite number of disjoint simple closed curves, each of which is a union of sides of meshes of the net. (Start with two neighboring nodes a , , a , in fr(B), and show that one may define by induction a finite sequence (a,) of nodes belonging to fr(B), such that a, and a,,, are neighboring nodes for every n.) (c) Let B be the union of a finite number of closed meshes of an hexagonal net of width a , such that fr(B) is a simple closed curve (union of sides of meshes of the net). Let a , h be two neighboring nodes on fr(B); prove that there exists a continuous mapping (z,t ) --z ~ ( zt .) of B x [O, I] into B such that: ( I ) ~ ( z0). z in B; (2) ~ ( zt ,) = z for every t E [O, I ] and every z on the segment S of extremities a and b ; ( 3 ) ~ ( z1), c S for any z E B. (Use induction on the number N of meshes contained in B; consider one side of extremities c, d, contained in fr(B), such that c and d have the same first coordinate, equal to the supremum of pr,(B), and that d has the largest second coordinate among all the nodes in fr(B) having that supremum as first coordinate; show that if N > I , B = B, u P, where P is the unique mesh contained in B and having c and d as vertices, B, is the union of N 1 meshes, and has in common with P one, two or three sides; examine the various possibilities.) Prove that the interior of B is simply connected. (d) Let B be the union of all closed meshes of an hexagonal net of width a , which are contained in A; a is taken small enough for B to be nonempty. Let D be one of the (open) connected components of h ; show that fr(D) is a simple closed curve. (Use (a) and Problem 3(d), to prove that if fr(D) was the union of more than one simple closed curve, there would be simple loops y in A and points z E fr(A) such that j ( z ; y ) = 1 . ) (e) Conclude that A is simply connected, and is the union of an increasing sequence (D,) of open simply connected subsets, each of which is the bounded component of the complement of a simple closed curve (use (c) and (d)). Conversely, such a union is always simply connected. (f) Extend the result of (e) to arbitrary simply connected open subsets of C (for each n, consider the closed hexagons of the hexagonal net of width I / n which are contained in the intersection of A and of the ball B(0; n ) ) . (g) Let A be an open connected subset of C such that the complement C - A has no bounded component; show that A is simply connected (use (9.8.5)). (h) Prove that any connected component of the intersection of a finite number of open simply connected sets in C is simply connected. 5. Show that the following open subsets of C are simply connected but that their frontier is not a simple closed curve: (I) ThesetA, ofpointsx-I i y s u c h t h a t O < x < l , -22. Define inductively two decreasing sequences of numbers, (an)and (p,), tending to 0, such that pl I , a, is the largest number > O such that Iy(t)l < p. for (tl < an,and p n + I= inf
(. -; 1
8 . 3 9
where 6,, is the distance of 0 to the set of points y(r) such that (11 2 c(.. (a) Prove that if z, z’ are two points of P(H) such that IzI < P . + ~ and J z ’ 1)
which is defined in a neighborhood of 0 and such that lim(f(x)-L(x))/llxll = O . X-0
Furthermore, that solution is an entire function in Kp. (Apply Problem 10 in a neighborhood of 0; reduce the problem to the case K = C, and apply (9.4.2) and (9.12.1) to prove that f i s an entire function.) (b) Show that there is no solution of the equationf(x) - hf(x/h)= x (h > 1 ) defined in a neighborhood of 0 in R and such thatf(x)/x is bounded in a neighborhood of 0. 12. Let I = [O, a ] , H = [ - b , b ] , and let f be a real valued continuous function in I x H; put M = sup If(x, y)l, and let J = [0, inf(a, b / M ) ] . (X.Y) E
1X H
(a) For any x E J, let E(x) be the set of values of y E H such that y = xf(x, y). Show that E(x) is a nonempty closed set; if g l ( x ) = inf(E(x)), g2(x) = sup(E(x)), show that gl(0) = gz(0)= 0, and that lim gl(x)/x= lim g2(x)/x =f(O, 0). x-o.x>o
X+O.X>O
If g1= gz = g in J, g is continuous (cf. Section 3.20,Problem 5). (b) Suppose a = b = 1 ; let E be the union of the family of the segments S. : x = 1/2", 1/4"+'< y 6 1/4" (n 2 0), of the segments S. : y = 1/4, 1/2"< x C 1/2"-' (n 3 ])and of the point (0,O). Define f ( x , y ) as follows: f ( 0 , y ) = 0; for 1/2"< x C 1/2"-' and y S 1/4", takef(x, y ) = ( ( y / x ) d((x, y), E))+ ;for 1/2"< x C 1/2"-' and 1/4" < y < xz, take f ( x , y ) = ( y / x ) - d((x, y), E) and finally, for 1/2"< x < 1/2"-' and y C x 2 , takef(x, y ) = x - d((x, xz), E) (n 3 I). Show thatfis continuous, but that there is no function g, continuous in a neighborhood of 0 in I and such that g ( x ) = xf(x, g(x)) in that neighborhood. (c) Let u0 be a continuous mapping of J into H, and define by induction u.(x) = x f ( x , U " - ~ ( X ) ) for n 3 1 ; the functions u, are continuous mappings of J into H. With the notations of (a), suppose that in an interval [0, c ] C J, lim ( U , , + ~ ( X )- u.(x)) =
+
"-m
0 for every x ,
and g l ( x ) =gz(x); show that lim u,(x) = g l ( x ) for 0 < x C c. n-ro
Apply that criterion to the two following cases: (1) there exists k > O such that If(x, zl) - f ( x , zz)[ C k Izl - z21 for x E I, zl, zz in H (compare to (10.1.1)); (2) for 0 < x < y < a and zl, zz in H, ]f(x,ZI)-f(x, Z Z ) ~< I Z I - JZ(/X. (d) Whenfis defined as in (b), the sequence (u,(x)) is convergent for every x E I, to a limit which is not continuous. (e) Take a = b = 1, f ( x , y ) = y / x for O < x C 1,lyl < x 2 , f ( x , y ) = x for 0 < x < 1,y 2 x',f(x,y) = - x for 0 C x < I , y < -xz. Any continuous function g in I such that [g(x)l < xz is a solution of g ( x ) = xf(x, g(x)) although If(x, zl) - f ( x , zz)/ < Izl - zzI/x for 0 < x < 1, zI, zZ in H ; for any choice of u O ,the sequence (u.) converges uniformly to such a solution. (f) Definefas in (e), and let f l ( x , y ) = - f ( x , y). The function 0 is the only solution of g(x) = x f l ( x ,g(x)), but there are continuous functions uo for which the sequence (u,(x)) is not convergent for any x # 0, although Ifl(x, zl) - f i ( x , z2)I < [zI- zzI/x for O < x < l,zl,zzinH. 13. Generalize the results of Problems 12(a) and 12(c) when H is replaced by a disk of center (0,O) in R2 (use the result of Problem 3 of Section 10.2).
270
X
EXISTENCE THEOREMS
2. IMPLICIT FUNCTIONS
(The implicit function theorem) Let E, F, G be three Banach (10.2.1) spaces, f a continuously di'erentiable mapping (Section 8.9) of an open subset A of E x F into G. Let (xo ,yo)be a point qf A such that f (x,,yo) = 0 and that the partial derivative D , f (xo,yo) be a linear homeomorphism of F onto G. Then, there is an open neighborhood Uo of xo in E such that, for every open connected neighborhood U of xo , contained in Uo , there is a unique continuous mapping u of U into F such that u(xo) = y o , (x,u(x)) E A and f (x,u(x)) = 0 for any X E U. Furthermore, u is continuously di'erentiable in U, and its derivative is given by (10.2.1 .I ) u'(x) = - ( D , f (x, 4 x 1 ) ) - W l f (x,W ) ) . Let To be the linear homeomorphism D, f ( x o ,yo) of F onto G, T i 1 the inverse linear homeomorphism; write the relation f ( x , y ) = 0 under the equivalent form (1 0.2.1.2) y =y - Ti1 . f ( x .y ) and write g(x, y ) the right-hand side of (10.2.1.2). We are going to prove that it is possible to apply (10.1.1) to the mapping
(x'9 Y'>
+
S(X0
+ x', Yo + Y ' ) - Y o
of E x F into F, in a sufficiently small neighborhood of (0,O).As T;'. To = 1 by definition, we can write, for ( x , y,) and ( x , y,) in A,
-
(Dzf(x0 Y o ) ( Y , - Y 2 ) - (f(x, Y1) -f(& Y,))). Let E' > 0 be such that E(IZ';'(~ G 3; as f is continuously differentiable in A, it follows from (8.6.2) and (8.9.1) that there is a ball Uo (resp. V,) of center x , (resp. y o ) and radius c1 (resp. p) in E (resp. F) such that, for x E U, ,y1 E V, ,y 2 E Vo , we have g(x9 Y , )
- g(x,
Y2)
I l f ( x ,Yl)
=Ti'
*
3
-m,YZ) - D2f(xo Y o )
- Y2)ll EllYl - Y2II ; whence Ildx, Y1) - g(x, Y2)Il G E I I T i ' II * llYl - Yzll G -511Yl - Y2lI for any x E U, ,y , E Vo ,Y,EV,. On the other hand, g(x, yo) - yo = - Ti1- f ( x , yo); 3
*
(Y1
as f ( x o , yo) = 0 and f is continuous, we can suppose E has been taken small enough to have 11g(x, yo) - yoI) G p/2 for x E Uo . We can then apply (10.1.1 ), which yields the existence and uniqueness of a mapping u of Uo into V, , such that f ( x , u(x)) = 0 for every x E U,; asf(xo ,yo) = 0, this gives in particular u(xo)= y o ; finally u is continuous in U, . Next we prove that if U c Uo is a connected open neighborhood of xo , u is the unique continuous mapping of U into F such that u(xo) = y o , ( x , u(x))E A and f ( x , u(x)) = 0. Let v be a second mapping verifying these
2 IMPLICIT FUNCTIONS
271
conditions, and consider the subset M c U of the points x such that u(x) = u(x). This set contains x, by definition and is closed (3.15.1); we need therefore only prove M is open (Section 3.19). But by assumption, x -+ DZf(x, u(x)) is continuous in U, , hence (replacing if necessary U, by a smaller neighborhood), we can suppose that D2f(x, u(x)) is a linear homeomorphism of F onto G for x E U,, by (8.3.2). Let a E M ; the first part of the proof shows that there exists an open neighborhood U, c U of a and an open neighborhood V, c V of b = u(a) such that, for any x E U,, u(x) is the only solution y of the equation f(x, y ) = 0 such that y E V,, . However, as u is continuous at the point a, and u(a) = u(a), there is a neighborhood W of a, contained in U , and such that ~ ( x E) V, for x E W ; the preceding remark then shows that u(x) = u(x) for x E W, and this proves M is open, hence u = u in U. Finally we show that u is continuously differentiable in U, , provided E has been taken small enough. For x and x + s in U,, let us write t = u(x + s) - u(x); by assumption f(x s, u(x) + t ) = 0, and t tends to 0 when s tends to 0. Hence, for a given X E U , , and for any 6 > 0, there is r > 0 such that the relation llsll < r implies Ilf(x s, u(x) t ) -f(x, u(x)) - S(x) . s - T(x) tll < S(llsll Iltll) where S(x) = D,f(x, u(x)) and T ( x ) = D2f(x, u(x)) (8.9.1). This is equivalent by definition to
+
+
+
+
IIS(x) * s + T(X) * 41 < 6(llsll
+ Iltll)
and as T(x) is a linear homeomorphism of F onto G , we deduce from the preceding relation (10.2.1.3)
ll(T-'(x) 0 S(x)) * s + tll < ~ l l ~ ~ 1 ~ x ~+lIltll). l~~141
Suppose 6 has been taken such that 6llT-'(x)ll 0 such that Ilf'(xo) . s l l 3 cI/sI/ for all s E E (5.5.1)J 2. Letf= (ft,f2) be the mapping of RZ into itself defined byf,(xl, x2) = xl;f2(x1,x2) = x2 - x: for x: < x 2 , f2(xl, x 2 ) = (x: - x:x2)/x: for 0 < x2 < x:, and finally fz(x1. - XI) = -fz(xt, x2) for x2 0. Show that f is differentiable at every point of RZ;at the point (0, 0), Df is the identity mapping of RZ onto itself, but Df is not continuous. Show that in every neighborhood of (O,O), there are pairsof distinct points x', x" such that f ( x ' ) = f ( x " ) (compare to (10.2.5)).
274
X
EXISTENCE THEOREMS
+
3. Let B be the unit disc IzI f 1 in R2 and let z -f(z) = z g(z) be a continuous mapping of B into R2 such that lg(z)l < Izl for every z such that JzI= 1. Show that f(B) is a neighborhood of 0 in R2 (“Brouwer’s theorem” for the plane, cf. Chapter XXIV). (Let y be the loop t - f ( e “ ) defined in [0,27r]; show thatj(x; y ) = 1 for all points x in a neighborhood V of 0 (see proof of (9.8.3)); using the fact that, in B, y is homotopic to 0, deduce that there is no point of V belonging to the complement off(B).) 4. Let E, F be two Banach spaces, B the unit open ball /lx/l< 1 in E; let uo be a continuously differentiable homeomorphism of B onto a neighborhood of 0 in F, such that uo(0)= 0; suppose uo’ is continuously differentiable in a ball Vo : llyll< Y contained in uo(B), and Duo is bounded in B and Duo’ is bounded in V,. Let V be a ball llyll < p, with p < r. (a) Show that for any a < 1, there is a neighborhood H of uo in the space 9 P ) ( B ) (Section 8.12, Problem 8) such that for any u E H, the restriction of u to U: llxll< a is a homeomorphism of U onto an open set of F containing V, such that the restriction of u-’ to V is a continuously differentiable mapping @(u) of V into E. (Use (lO.l.l).) (b) Show that the mapping u @(u) of H into 9(EI)(V)is differentiable at the point uo ,and that its derivative at uo is the linear mappings + -(uk 0 cD(uo))-’ . (s 0 cD(uo)). 5. Let E, F be two Banach spaces, f a continuously differentiable mapping of a neighborhood V of xo E E into F. Suppose there are two numbers /?> 0, A > 0 such that: (I) Ilf(x0)II < /3/2h;(2) in the ball U: IIx - xoIJ< p, the oscillation off’ is < l/2A; (3) for every x E U,f’(x) is a linear homeomorphism of E onto F such that ll(f’(x))-’ I1 < A. Let (2,) be an arbitrary sequence of points of U ; show that there exists a sequence (x.).,~ of points of U such that x.+’ = x. - (J’(zn))-’ .f(x,) for n 2 0. Prove that the sequence (x,) converges to a point y E U, such that y is the only solution of the equationf(x) = 0 in U. (“Newton’s method of approximation.” Use (8.6.2) to prove by induction on n that IIx, - x.-’ I/ < 2-”p and Ilf(xn)II < /3/2“+‘X). 6. Let E, F be two finite dimensionalvector spaces over K, A a connected open subset of E, f a continuously differentiable mapping of A x F into F. Suppose that the set r of pairs (x, y ) E A x F such f(x, y ) = 0 is not empty, and that for any (x, y) F I?, D2f(x, y ) is an invertible linear mapping of F onto itself. (a) Show that for every point (xo ,yo ) E r there is an open neighborhood V of that point in r such that the restriction of the projection prl toVis a homeomorphism of V onto an open ball of center xo contained in A. (Use the fact that there is an open ball U of center xo in A and an open ball W of center yo in F such that for each x E U, the equationf(x, y) = 0 has a unique solution y E W, and apply (10.2.1).) (b) Deduce from (a) that every connected component G of r (Section 3.19) is open in I’ and that pr,(G) is open in A. It is not necessarily true that pr,(I’) = A (as the example A = E = F = R,f(x,y) = xy2 - 1 shows), nor that if prl(r) = A, prl(G) = A for every connected component G of r (as the example A = E = F = R,f ( x , y ) = xy2 - y shows). Prove that if pr2(I’) is bounded in F, then pr,(G) = A for every connected component G of F. (If xo is a cluster point of pr,(G) in A, show that there is a sequence (x,, y.) of points of G such that lim x, = xo and that limy. exists in F; n-1 m n-m apply then (a).) (c) The notations of path. loop, homotopy, and loop homotopy in A are defined as in Section 9.6, replacing C by E. Suppose there is a connected component G of r such that prl(G) = A; if y is a path in A, defined in I = [a, b] c R,show that thereexists a continuous mapping u of I into G such that prl(u(r)) = y ( t ) for each t E I (consider the 1.u.b. c in I of the points 6 such that there exists a continuous mapping uc of [a, 61 into G such that prl(uc(r)) = y(f) for a < t < 5 and use (a)). Is that mapping always unique? (Consider the case E = F = C, A = C - { O } , f ( x ,y ) = y 2 - x . ) --f
2 IMPLICIT FUNCTIONS
275
Show that if two continuous mappings u, u of I into G are such that prl(u(t)) = prl(u(t)) = y(r) for each t E 1,and if they are equal for one value of t E I, then u = u (use a similar method). (d) Under the same assumptions as in (c), let v be a continuous mapping of I x J into A, where J = [c, d ] C R. Let u be a continuous mapping of J into G such that prl(u([)) = ~ ( a5), for 6 E J ; and for each 5 E J, let uc be the unique continuous mapping of I into G such that prl(uc(t)) = ~ ( t , tfor ) t E I and u&) = ~(6).Show that the mapping ( t , 5) + u,(t) is continuous in I x J. (Given 5 E J, there is a number r > 0 such that for any t E I, the intersection V, of I' and of the closed ball in E x F, of center u,(t) and radius r, is contained in G and such that prl is a homeomorphism of V, onto the closed ball in E of center y ( t ) and radius r. If L = u{(I), let M be the supremum of ll(D2f ( x , y))-' 0 ( D , f ( x , y))/I for all points ( x , y) E G at a distance < r of L. Let E > 0 be such that E < 1-14and E M < r/4. Show that if 6 is such that the relation 15 - 51 < 6 implies Ilq(t,5) - q ( t , 1) of complex numbers (which may be empty) such that /n,l < lanfl/,f(n.) = 0 and for every c E C such thatf(c) = 0, the number of indices n for which a, = c is equal to the order w ( c ; f ) ;when the sequence (an) is infinite, lim lanl = +co (9.1.5). Show (with the notations of Section 9.12, n-m
Problem 1 ) that there exists an entire function g such that
fi E (?, n a,
f ( z ) = egc2)
n=1
-
I),
(" Weierstrass decomposition ").
9. Let A and B be two open neighborhoods of 0 in E = C p ,A being connected; let ( x , y) + U(x,y) be an analytic mapping of A x B into Y(E; E) (identified to the space of p x p matrices with complex elements).
276
X
EXISTENCE THEOREMS
(a) Suppose there exists a sequence (u,,) of analytic mappings of A irito B such that rro(x) -= 0 in A and r/,,(.u) U ( s , r/,8-i(.r)). x in A for Ir > I . Suppose in addition that for every compact subset L of A, the restrictions of the I(,, t o L form a relatively compact subset of 'LE(L). Prove that the sequence ( I ! , ) converges uniformly in any compact subset of A to an analytic mapping u of A into B such that u ( x ) U(.u, r(.u)) ' x in A ; furthermore, u is the unique mapping satisfying that equation (use (10.2.1) and ~~
~~~
(9.13.2)).
(b) Suppose that in E, A and B are the open balls of center 0 and radii u and 7)) Let rp be a continuous mapping of [0, ( I [ Y [0, h[ into R such that 7 increasing in [O, / I [ for every [ E [O, ul and suppose that IIU(x, y)Il < ~(ll.rll,IIJII) in A x' B. Suppose in addition that there exists a continuous mapping 0 of [0, N [ into [0, h[ such that =,[(pr &[))[ in [O, u ] . Prove that under these conditions there is a unique analytic mapping u of A into B such that ds) U ( x , dx)) ' sin A, and in A (use (a); prove the existence of the mappings I/,, by induction that Ilo(x)li3 ~(!lsll) on ti). (c) Suppose A and B are defined as in (b): let #(7) be the 1.u.b. of llU(x,.v)Il for l!xIi < a , ilyll -< 7, when 7 : 0, and take 440) $(O i ). Suppose that #(O) 0 and that the function 7 -t 7 / $ ( ~is )increasing in some interval [0, y [ , where y < / I , and Y/I,!J(Y-) < a. Then there is a unique analytic mapping I ) of the open ball P of center 0 and radius y/i,b(y-) into B, such that u(x) U ( x , dx)) . s in 1'. 10. Let f , g be two complex valued analytic functions defined in a neighborhood of the closed polydisk P c Cz of center (0,O) and radii u , h. Let M (resp. N ) be thc 1.u.b. of y)i (resp. lg(x, y ) l ) for Is1 - N and IyI < /I (resp. for 1x1 < N and Iyi --= 1)). Then, there exist two uniquely determined functions u(,s,t ) , o(s, t ) , analytic for 1 . ~ 1 '- tr/M and jtl .::DIN, such that ( ~ ( s , t ) , t f s , t ) )c P for (.Y, t ) in the polydisk Q defined by tlic previous inequalities and that
-v([,
s(()
7
I
: :
~
~ ( s ,t ) - .vf(u(.s,
t ) , c(s,
1))
0
7
and
u ( s , t ) - tg(u(.s, I ) ,
P(S,
I)) -0
in Q. Furthermore, let
and let h(x,y, s, t ) be a n arbitrary analytic function in P
~,
0 ; show that
h(rr(s, t ) ,u(s, I ) , s , I ) = C c,,,,.s"'/" A(u(.s, f ) , P ( S , t ) , . ~t, ) r r t 3 0 . n ~ 0
for
(s,
t ) E Q, where crnn is the value for .r
~
J'
=I
0 of the function
and the series o n the right-hand side is convergent in Q ; note that c,,,. depends o n .s and t if h does. (" Lagrange's inversion formula." First apply RouchC's theorem (9.17.3) to s - sf(x, y), considered as a function of x; this defines an analytic function W(S, y) such that w(s, y ) - sf(w(s,.~),y ) = 0, by (10.2.4); next apply similarly Rouche's theorem to y tg(w(s, y ) . .v) considered as a function of y . Finally, let y , 6 be the . the repeated integral circuits 0 ->acio, 8+hei0 in C (0 < 0 < 2 ~ ) Consider -
3 THE RANK THEOREM
.r,
i,
277
h ( x , y , S , t ) dx (x
- .Sf(S,
y))(y
~
/g(x, y))
.
On one hand, find the value of that integral by repeated application of the theorem of residues (9.16.1); and on the other hand, consider the power series development of (I [)-'(I 7)Vtin which [ is replaced by sf(x, . v ) / x and 7 by / g ( x , y)/y.) Generalize to any number of complex variables. From the inversion formula for one variable, deduce the formula ~
~
h(u(.s)) where
I/(.Y)
~
cf(r/(s))
=
~
h(0)
0 and
Is/
+f "=I
/I!
Dn-l(h'(0)(f(O))")
< n / M , with M
=
sup If(x)l, h being analytic for 1.~16a
1.Vl ~, (1.
3. T H E RANK T H E O R E M
Let E, F be twoJinite tliniensional vector spaces of dimensions n and m , A an open subset of E, f a continuously differentiable mapping of A into F, The rank of the linear mappingj"(x) at a point x E A is the largest number p such that there is at least a minor of order p in the matrix off'(x) with respect to two bases of E and F, which is not 0 (A.7.3). As these minors are continuous functions of x, it follows that if the rank of,f'(x,) is p , there is a neighborhood of xo i n which the rank o f f ' ( x )is at least p ; but it can be > p at every point x # xo of that neighborhood, as the example of the mapping (-Y, J,) (x2 - y 2 , xy) shows at the point (0,O). --f
(10.3.1) (Rank theorem) Let E be an n-dimensional space, F an m-dimensional space, A an open tieigliborlioocl of a point a E E, , f a continuously dtfferentiable mapping (resp. q times contiiiuously differeiitiable mapping, resp. indefinitely differentiable mapping, resp. analj!tic mapping) of A into F, such that in A the rank of f ' ( x ) is a constant number p , Then there exists: ( 1) an open neighborhood U c A ofa, and a homeomorphism u of U onto the unit ball I" : Ix,I < 1 ( 1 i t i ) in K", ttAic1i is continuously differentiable (resp. q times cotitinuously differentiable, resp. indefinitely differentiable, resp. analytic) as well as its inivrse ; (2) an open neighborhooci \I I> j ( U ) of b = f ( a ) and a homeomorphism u of the unit ball 1" : 1y,1 < I ( 1 i m> of K" onto V, which is continuously cliffi~rentiahle( resp. q times continuously differentiable, resp. indefinitely dtfferentiable, resp. analj~tic)as 1t.ell as its itirersc I. -such that f ' = I' f b u, n h w f o is the mapping
<
p . Let P be the image of E (and of N ) by the linear mappingf'(0); it is a p-dimensional subspace of F, having the elements di = f ' ( O ) * ci (1 Q i Q p ) as a basis; we take a basis ( d j ) l s j i , of F, of which the preceding basis of P form the first p elements, and we write y
m
=
$,(y)dj for any y
E
F, the
j=l
tjj being linear forms. We denote by y - + H ( y ) the linear mapping
y+
P
C tjj(y)ejof F onto the subspace KP of K" generated by the ei of index
j=l
i 0 be such that the ball lxil < r (1 < i < n) is contained in g(U,), and let U be the inverse image of that ball by g, which is an open neighborhood of 0 ; our mapping u will be 1
1
the restriction to U of the mapping x -+ - g(x). r
Up to now we have not used the assumption that the rank off'(x) is constant in A ; this implies that the image P, of E byf'(x) has dimension p for any x E A. Now we may suppose U, has been taken small enough so that g'(x) is a linear bijection of E onto K" for x E U, (8.3.2); as we have g'(x) * s = H ( f ' ( x ) . s) for s E N, the restriction of f ' ( x ) to N must be a bijection of that p-dimensional space onto P,, and the restriction of H to P, a bijection of P, onto KP. Denote by L, the bijection of KPonto P,, inverse of the preceding mapping; we can thus writef'(x) = L, H o f ' ( x ) . 0
3 THE RANK THEOREM
279
Now consider K" as the product El x E, with El = KP, E, = K"-P. we are going to prove that the mapping (zl, z 2 )+fl(zl, 2 , ) = f ( u - ' ( z l , z2)) of 1" into F does iiof depend 017 z2 , i.e. that D, fl(zl, z , ) = 0 in I" (8.6.1). By defini-
rf'(x> . t
'
i:
= f, - H(J'(x)),-
tion, we can writef(x)
r
= Dlfl(;
+ D2 f i for any t E
(10.3.11)
E. This yields D, f I
G(,4), hence by (8.9.2)
; ;w).
H ( f ( x ) ) , G(x)) . W f ' ( x ) . t )
(;
H(S(x)),
G(0
;
(i
H ( f ( x ) ) , G(x)) . G ( t ) = S, * H ( f ' ( x ) . t )
' 1
(i
, G(x) is a linear mapping of KP = El where S, = rL, - Dlfi - H ( f ( x ) ) r
into F. We prove that S, = 0 for any x E U,. Indeed, if t E N, we have G ( t )= 0 by definition, hence S, . H ( f ' ( x )* t) = 0 by (10.3.1.1). But t -+ H ( f ' ( x ) . t ) = g'(x) * t is a bijection of N onto El for x E U,, and this proves S, = 0. From (10.3.1.1) we then deduce
D, for any
f E
fl
(' r
I
H(J'(x)),- C(x)) r
. G(t) = 0
E; but G maps E onto E,, hence by definition,
which is a linear mapping of E, into F, is 0 for any x E U,, . The relation D,f,(z,, z 2 ) = 0 in I" then follows from the fact that
is a homeomorphism of U, onto an open set containing I". We can now write fl(zl) instead of f l ( z l ,z , ) and consider fi as a continuously differentiable mapping of El = KP into F; we then have f(x) = fl
(!r H(,f(,x-))) for
proves that y
x E U ; in other words, y
I
-+
=fl
- H ( y ) is a homeomorphism r
z1 -+fl(zl)the inverse homeomorphism.
(1. 1
- H ( y ) for y E ~ ( U )This . *
of f(U) orito Ip c El, and
280
X
EXISTENCE THEOREMS
Consider now K"' as the product El x E, with E, = K"-". Let T be the linear bijection of E, onto the supplement Q of P in F generated by cl,,,, . . . , d,,, , which maps the canonical basis of K'"-" onto dD+,,. . . , d,, . We define P ( Z ~ z,) , =,f,(z,) + T(z,) for z , E Ip, z 3 E it is obviously (8.9.1) n continuously differentiable mapping. By definition, we have H(P(z,,z 3 ) )= H(f,(z,)) = rz, ;hence the relation i ( z I , z,) = r ( z ; ,z ; ) implies z ; = z , , and then boils down to T(z,) = T(z;),which yields z3 = z ; ; therefore 2: is it?jectire. The relation S , = 0 proved above shows that for any z , E Ip,fS;(zl)= rL, where s is any point i n U such that f ( x ) =f,(z,); the derivative of I' at ( z , , 2 , ) is therefore the linear mapping ( t , , t,) + rL, . t , + T ( t 3 )((8.9.1) and (8.1.3)). But as the restriction of H to P, is injective, P, is a supplement of Q in F, hence r ' ( z l ,z,) is a litiear iiomeotiior~~hist~i of K"' onto F. For any point ( z , , z 3 ) E I"', there is therefore an open neighborhood W of that point in I" such that the restriction of to W is a homeomorphism of W onto an open subset r(W) of F, by (10.2.5). As in addition is injective, it is a homeomorphism of I"' onto the open subset V = r( I"'), whose inverse is continuously differentiable in V. The relation f = o f o I I then follows from the definitions. 11
11
11
0
(10.3.2) I f t l i e raiik of j ' ( a ) is equal to ti (resp. to n i ) , thn7 the coiiclusioti of (10.3.1) holds li'itiz p = n (resp. p = n i ) . Indeed, at the beginning of this section we have seen that there exists then a neighborhood of a in which the rank of ['(.Y) is > / I (resp. > t ? i ) , hence equal to ti (resp. to 171) since it is always at most equal to inf(in, t i ) (A.4.18).
PROBLEMS
1. Let E, F be two Banach spaces, A an open neighborhood of a point x o E E, f a continuously differentiable mapping of A into F. (a) Suppose f ' ( x o ) is a linear homeomorphism of E onlo its image in F; show that there exists a neighborhood U c A of .yo such that f is a homeomorphism of U onto f(U) (use Problem 3 of Section 10.1). (b) Supposef'(x,) is surjective; then there exists a number a .r0 having the following property: if Nisthekerneloff'(xo),for every s c E,one has 1 : f ' ( x O.)s I l 3 ( 1 . inf,It 1 .yii IS
N
(12.16.12). Show that thereexists a neighborhood V c A of xo such thatf(V) is a neighborhood o f f ( x o )in F (use Problem 8 of Section 10.1). 2. Let A be an open subset of Cp, and f a n analytic mapping of A into C". Show that if fis inrjecrioe, then the rank of Df(s) is equal t o p for every .Y F A. (Use contradiction, and induction on p ; for p = I, apply Rouche's theorem (9.17.3). Assume Df(o) has a rank < : p for some u t A ; show first that after performing a linear transformation in F, one may assume that, if f ( z ) ( f i ( z ) ,. . . , f p (z)), then D,/,(N)- 0, and if g ( z ) (f2(z),...,f, ( z ) ) , the rank of Dg(o) is exactly p - I; then there is a neighborhood ~
3 THE RANK THEOREM
281
U c A of a such that Dg(x) has rank p - 1 for x E U. Using the rank theorem (10.3.1), reduce the proof to the case in which n = O,X(z) zk for 2 < k < p.) Is the result still true when C is replaced by R ? 3. (a) Let A be a simply connected open subset of C, distinct from C, and let a, b be two distinct points of fr(A). (Appendix to Chapter IX, Problem 6 . ) There exists a complex-valued analytic function h in A such that ( h ( ~ )=) (z ~ - a ) / @- b) (Section 10.2,Problem 7); h is an analytic homeomorphism of A onto a simply connected open subset B of C (Problem 2 and (10.3.1)); furthermore, B n (-B) = 0, hence there are points of C exterior to B. (b) Deduce from (a) that there exists an analytic homeomorphism of A onto a simply connected open subset of C contained in the disc U : Iz/ < 1 , and containing 0. 4. (a) Let A be a simply connected open subset of C contained in the unit disk U : / z / < 1, containing 0, and let H be the set of all complex valued analytic functions g in A, such that g is an injecliue mapping of A into C, Ig(z)J < 1 , g(0) = 0 and g’(0) is a real number >O. For each compact subset L of A, the set H,. of the restrictions to L of the functions of H is relatively compact in Kc(L) (9.13.2). Show that the set of real numbers g’(0) (forg E H) is bounded (cf. proof of (9.13.1)); let be the 1.u.b. of that set. Show that there is a function go e H such that g,,(O) = h (use the result of Section 9.17, Problem 5). (b) Suppose g E H is such that g(A) # U, and let c E U -g(A). Replacing g by g1 defined by g,(z) = e-‘@g(ze‘@), one can assume, for a suitable choice of 8, that c is real and >O. There exists a function h which is analytic in A and such that 7
h W 2 = (c - g(z))/(l - CdZ)) and h(0) = 2 / c > 0 (same argument as in Problem 3(a)); show that the function g2 defined by
h(z) =
(dY-gz(z))/(l -dc92(z))
belongs to H, and that g;(O) > g’(0). (c) Conclude from (a) and (b) that the function go defined in (a) is an analytic homeoniorphism of A onto U; using Problem 3(b), this implies that for any simply connected open subset D of C, distinct from C, there is an analytic homeomorphism of D onto U (“Riemann’s conformal mapping theorem ”). 5. (a) Let f be a complex valued analytic function in the unit disk U: (z[< 1 such that f ( O ) = 1 and If(z)l < M in U; show that for ( z /< I/M, I f @ - I1 < Mlz( (apply Schwarz’s lemma (Section 9.5, Problem 7) to the function g(z)= M(f(z) - 1 )/(M2 -f(z))). (b) Let f be a complex valued analytic function in U such that f ( 0 ) = O,f’(O) = 1, if’(z)l < M in U ; show that for 1z/ < I/M, i f ( z ) - zI S MIz12/2 (apply (a) tof’). (c) Show that under the assumptions of (b), the restriction o f f t o the disk B(0; I/M) is an analytic homeomorphism of that disk onto an open subset containing the disk B(0; 1/2M) (apply Rouchk’s theorem ((9.17.3), using the result of (b)). (d) For any complex number a E U, let [ ( ( z )= ( z - n)/(Zz - I ) ; for any complex , Jg’(z)l(l - lzl’) = valued function f analytic in U, show that, if g(z) = = f ( i t ( z ) )then If’(u(z))l(l ~ U ( Z ) ( for ~ ) any z E U . (e) Show that there is a real number (“Bloch’s constant”) b > 1/32’3 having the following property: for any complex valued function f analytic in U and such that Y(0) = 1, there exists zo E U such that, if xo = f ( z o ) , the open disk B of center xo and radius h is contained in f ( U ) and there is a function g, analytic in B and such ~
282
X EXISTENCE THEOREMS
that g(B) c U and f(g(z)) = z for z E B. (Consider first the case in which f is analytic in a neighborhood of 0 , and take for zo a point where lf’(z)l(l - I z / ’ ) reaches its maximum; use then (d) to reduce the problem to the case in which zo = 0, and apply in that case the result of (c) to a function of the form a +f(Rz), where a and R are suitable complex numbers. In the general case consider the functionf((1 - E)z)/(~- E ) , where E > 0 is arbitrarily small.) 6. (a) Let 9)) be the set of all complex valued functions f analytic in the unit disk U: /zl < I , such that f ( U ) does not contain the points 0 and I . For any function f~ 91), there is a unique analytic function g in U such that exp(2n(q(z)) =f(z) in U and (.P(g(O))I< rr (Section 10.2, Problem 7); g(U) does not contain any positive or negative integer. Furthermore (same reference) there is an analytic function h in U such that g(z)/(g(z) - 1 ) = ((I h(z))/(l - h ( ~ ) ) ) h(U) ~ : does not contain any of the points 0, 1, and c : = (dn v’n(n integer > 1). Finally, there is an c; = (dnf .\/n-analytic function g, in U such that exp(y(z)) = h(z): y(U) does not contain any of the points log c; 2k7ri, log c t 2krri (k positive or negative integer, n > I). Show that no disk of radius > 4 can be contained in g,(U); using Problem 5(e), deduce from that result that Ig,’(x)l < 4 / N I - 1x1)
+
+
+
+
for 1x1 < I (consider the function t cy(x (I - I x / ) t ) , for a suitably chosen constant c). Conclude that there is a function F(if, v), finite and continuous in (C - (0, I } ) x [0, I[, such that for every function f~ 9X,loglf(z)/ < F(f(O), r ) for any IzI < r < 1 . (b) Let f~ 9) be such that either If(O)! < 1/2 or ! f ( O ) - I 1 < 1/2. Given r such that 0 < r < 1, show that either If(z)i < 5/2 for jz/ < Y, or there exists a point x such that 1x1 < Y and If(x)l > 1/2, l f ( x ) - 11 > 1/2 and Ilif(x)l 2 1/2. Applying the result of (a) to the function f((z - x)/(az - I)), conclude that there is a function F,(u, u), continuous and finite in [0, co[ x [0, 1[, such that for any function f~ 911, the relations If(0)I < s and JzJ< r imply If(z)l < Fl(s, r ) (“Schottky’s theorem”). 7. Let A be an open connected subset of C, and ( f , ) a sequence of functions of the set 911 (Problem 6 ) . Show that for any compact subset L of A, there exists a subsequence (fJ such that either that subsequence is uniformly convergent in L, or the sequence (I/f,,.) converges uniformly to 0 in L. (Using Schottky’s theorem, prove that the points x E A such that lim ( l / f , ( x ) )= 0 form an open and closed subset of A, hence equal to --f
+
“-1
m
A or empty: in the second case, show, using the compactness of L, that there is a subsequence of (6) which is bounded in a compact neighborhood of L, and apply (9.13.1); in the first case, use similarly (9.73.7) applied to the sequence (lif.).) 8. (a) Let f be a complex valued function, analytic in the open set V : 0 < / z - a1 < r , and suppose a is an essential singularity off(Section 9.15). Show that C - f ( V ) is empty or reduced to a single point (“Picard’s theorem”. Let W be the open subset of V defined by r / 2 < Iz - a1 < r and consider in W the family of analytic functions L(z) =f(z/2”); if there are at least two distinct points in C -f(V), apply Problem 7 to the sequence (fJ,and derive a contradiction with Problem 2 of Section 9.15, using (9.1 5.2).) (b) Deduce from (a) that if g is an entire function in C, which is not a constant, then C - y(C) is empty or reduced to a single point (consider g(l/z) in C - (0)). 9. (a) Show that there is an entire function f(x,y ) in C2 satisfying the identity
f(4x, 4Y) - 4 m Y ) = - S W X ,
+
- 2 ~ ) ) ~ 2 ~ ( 2 x -, 2 m
and such that the term of degree < I in the Taylor development o f f a t the point (0,O) are x y (Section 10.1, Problem 11).
+
4 DIFFERENTIAL EQUATIONS
283
(b) Let g ( x , y) = f ( 2 x , -2.~4, and let J(x, y ) = a(J g ) / a ( x ,y ) ; show that J(2x, -2y) = J ( x , y ) , and conclude that J ( x , y ) = -4 in C2 (express f ( x , y ) and g ( x , y ) in terms of f ( 2 x , -2y) and g ( 2 x , -2y)). Prove that the analytic mapping II : ( x , y) + ( f ( x , y ) , g ( x , y ) ) of C2 into itself is injective (if it was not, it would not be injective in a neighborhood of (0, 0), owing to the preceding expressions). (c) Show that there is a neighborhood of (1, 1) which is not contained in u(C2). (Observe that there exists E such that 0 < E < 1 and that the relations I f ( 2 x , - 2 y ) - l I < ~ , l g ( 2 x , - 2 ~ ) - - < ~ i m p l y J f ( x , y ) - I 1 < e a n d l g ( x , y ) - 11 < E ; conclude that the relations / f ( x ,y ) - 1 < F and lg(x, y) - I I < E would imply IfCO, 0) - 11 < E and (g(0,O)- 11 C F , a contradiction) (compare to Problem 8(b).)
4. DIFFERENTIAL E Q U A T I O N S
Let E be a Banach space, I an open set in the field K, H an open subset of E, f a continuously differentiable mapping of I x H into E. A differentiable mapping u of an open ball J c I into H is called a solution of the differential equation (10.4.1 )
x’ = f ( t , x )
if, for any t E J , we have (10.4.2) u’(t) = f ( t , u ( t ) ) . It follows at once from (10.4.2) that u is then continuously differentiable in J (hence analytic if K = C, by (9.10.1)). (10.4.3) In order that, in the ball J c I of center t o , the mapping u of J into H be a solution of (10.4.1 ) such that u(to) = xo E H, it is necessary and sufJjcient that u be continuous (resp. analytic) in J if K = R (resp. K = C), and such tllat
(10.4.4)
u ( t ) = xo
+
J:
f(s, u ( s ) ) ds
(where, if K = C, the integral is taken along the linear path 5 + to -t- c(t - to), 0 d 5 < 1). This follows from the definition of a primitive, for (when K = C) iff is continuously differentiable and u is analytic, then s + f ( s , u(s)) is analytic (9.1 0.1 ). (10.4.5) (Cauchy’s existence theorem) / f ’ f is continuously differentiable in I x H, for any to E I and any x o E H there exists an open ball J c I of center to such that there is in J a solution u of (10.4.1) such that u(t,) = x,, .
We first prove a lemma:
284
X
EXISTENCE THEOREMS
(10.4.5.1) Let A be a conipact iiietric space, F a metric space, B a coinpact subset of F, g a conti~iuousniappiiig of A x F into a metric space E. Tlieii there is a neighborhood V of B in F such that g(A x V) is hounded irz E.
For any t E A and any z E B, there is a ball S , , z of center t i n A and a ball UI,zofcenter z in F such thatg(S,,: x U , , z ) is bounded, sincey iscontinuous. For any z E B, cover A by a finite number of balls S , , , z and let V, be the ball Ur,,:of smallest radius. Then g(A x V,) is bounded (3.4.4). Cover now B by finitely many balls Vz,; the union V of the Vz, satisfies the requirements (3.4.4.).
(a) Suppose first K = R. Let J , be a compact ball of center to and radius a, contained in I . By (10.4.5.1) there is an open ball B of center xo and radius b, contained in H, and such that M = sup llf’(t, x l l and k =
sup
(f.xkJo x
( i , x k J c tx B
B
]lDz.f(t,x)l\ are finite. Let J, for r < a be the closed ball of
center to and radius r, and let F, be the space of continuous mappings y of J, into E, which is a Banach space for the norm Ilyll = sup Ily(t)ii ISJ,
Let V, be the open ball in F,, having center xo (identified to the constant mapping f -+ xo) and radius 6. For any y E V,, the mapping (7.2.1)
t
-+
xo
+ Ii: f ( s , y ( s ) )ds is defined
and continuous in J,, since
J(S)
E
B by
definition, for y E V,; let g ( y ) be that mapping; g is thus a mapping of V, into F,. We will prove that for r small enough,g verifies the conditions of (10.1.2); applying that theorem and (10.4.3) will then end the proof, with J = j,. Now, for any two points y l , in V,, we have, by (8.5.4) j q 2
Ilf’(s9 Y l W )
- . f h vz(s))II
1.
- IlYl(4 -
L’2(s)//
d k . IIYI - Yzll
for any s E J, ; therefore, by (8.7.7), for any t E J,,
hence llg(yl) - g(y2)I(d krlIy,
-
J ~ ~ I I . On
the other hand, for any
lljr:
J’ E
V,,
d M for any s E J,, hence f ( s , .As)) dsI1 d M r by (8.7.7) and therefore I/g(x,) - xo/Id Mr. We thus see that in order to be able to apply (10.1.2),we should have k r < I and Mr < b(I - kr), and both inequalities will be satisfied as soon as r < b/(M + kb). Ilf(s, y(s))II
(b) Suppose now K = C; define J , , J,, B, M , and k as above, and let F, be the space of mappings y of J, into E which are continuous it7 J, and analj~tic in j,. This is again a Banach space for the norm ((yll= suplly(t)ll, by (7.2.1) and (9.12.1). For y
E
V,, the mapping t + xo +
i E
J,
f ( s , y(s))cls again belongs
5
COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS
285
to F,, for it is analytic in 9, since s - t f ( s , y(s)) is (9.7.3); and its continuity in g of V, into F,, and the end of the proof is then unchanged.
J, at once follows from (8.11.1). We therefore have defined a mapping
(10.4.6) Remark. The proof of (10.4.5) shows that the result is still valid when K = R and when f satisfies the following weaker hypotheses: (a) for every continuous mapping t + w ( t ) of I into H, t + f ( t , w ( t ) ) is a regulated function in I (Section 7.6); (b) for any point ( t , x) E I x H, there is a ball J of center t in I and a ball B of center x in H such thatfis bounded in J x B, and there exists a constant k 2 0 (depending on J and B) such that IIf(s, Y , ) - f ( s , Y J I I ,< 4 y l - y,II for s E J, yl, y , in B. Such a function f is said to be locally lipschirzian in I x H ; equation (10.4.2) is then to be understood as holding only in the complement of an at most denumeruble subset of J. This last remark also enables one to replace the open intervals I and J by any kind of interval in R.
5. C O M P A R I S O N O F S O L U T I O N S O F D I F F E R E N T I A L E Q U A T I O N S
We say that a differentiable mapping u of an open ball J c I into H is an approximate solution of (10.4.1 ) with approximation E if we have llu’(t) - f ( f ,
for any t
u(t))ll
d&
E J.
(10.5.1) Suppose lID,f(t, x)ll d k it7 I x H. I f u, u are two approximate solutions of (10.4.1 ) in an open ball J of center t o , with approximations E ~ E, ~ then, for any t E J , we h m e
(For k = 0, (eklt-tol- l ) / k is to be replaced by It - tol).We immediately are reduced to the case K = R, to = 0 and t 2 0 by putting t = to a t , la1 = 1, t 2 0 ; then if ul(t) = u(to a t ) , 01(5) = v(t, + a t ) , u1 and v1 are approximate solutions of x’ = af(ro + at, a - l x ) , whence our assertion. From the relation ilu’(s) - f ( s , u(s))/I < in the interval 0 < s 9 t , we deduce by (8.7.7)
+
+
,
286
X
EXISTENCE THEOREMS
and similarly
l
whence
o(r)
- v(0) -
s:
1
f(s, u ( s ) ) d s = E2 t
From the assumption on D,fand from (8.5.4) and (8.7.7) this yields (10.5.1.2.)
w(t) d w(0)
+ + (El
E2)t
+k
J:
w(s) d s
where w ( t ) = llu(t) - v(t)(l. Theorem (10.5.1) is then a consequence of the following lemma:
u,
(10.5.1.3) in an inferual [0,c ] , cp and $ are two (Gronwall's lemma) regulated functions 2 0, thenf o r ar7y regulateLlfiinction MI 0 in [0,c] satisfying the inequality (10.5.1.4)
"(tj
< cp(t) +
we have in [0, c]
(10.5.1.5) Write y ( t ) ) =
s'
0
1:
4 0 G Y(t) + j'cp(s)$(s)
Ic/(s)w(s) ds
e x P ( [ s k ) di) ds.
0
$(s)it(s)
ds; y is continuous, and from (10.5.1.4) it follows
that, in the complement of a denumerable subset of [0, c ] , we have Y ' ( t ) - $ ( f ) Y ( f ) < cp(t)$(t)
(10.5.1.6)
by Section 8.7. Write z ( t ) = y ( t ) exp( - \' $(s) ds); relation (10.5.1.6) is -0
equivalent to
By (8.5.3) and using the fact that z(0) = 0, we get, for t E [0, c ]
whence by definition Y(t>
~ ; c p ( s l wexP(jsZ$(41d t )
and (10.5.1.5) now follows from the relation
ds
w ( t ) G cp(t)
+y(t).
5
COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS
287
(10.5.2) Suppose f is continuously diflerentiable in I x H . If u, c are tn’o solutions of (10.4.1), dejiied in an opeti ball J of center to and such that u(t,) = u(f,), then u = 2’ in J.
It is enough to prove that u and z‘ coincide in every compact ball L of center to contained in J. This follows from (10.5.1) applied to u and u, provided we know that D, f is bounded in some set L x H’, where H’ is an open subset of H containing both u(L) and z(L). But the existence of such a set follows at once from (10.4.5.1 ).
(10.5.3) Suppose E isjiiite dimensional a n d f i s analytic in I x H. Then any solution of (10.4.1) in an open hall J c I is analytic.
This is immediate by definition if K = C. Suppose K = R, and let E = R”; then for any point ( t o ,xo) E I x H there is a ball Lo c C of center to and a ball P c C” of center x, such that L, n R c I and P n R” c H, and an analytic mapping g of Lo x Pinto C” whose restriction to (Lo n R) x (P n R”) coincides with f (9.4.5). There is by (10.4.5) an open ball L c Lo of center to in C such that there exists a solution u of the differential equation z’ = g(t, z), taking the value xo at the point t o , and D is analytic in L. Using the relation u’(t) = g(t, c ( t ) ) , and the definition of g and it is immediately verified by induction on n that all derivatives d”)(t,) belong to R”; hence (9.3.5.1) u ( t ) belongs to R” for t E L n R. This proves that the restriction u of u to L n R is a solution of (10.4.1) (see Section 8.4, Remarks), such that u(to) = xo . But by (10.5.2),any solution M.’ of (10.4.1 ) in a ball M of center t , such that iv(to) = x o coincides with u in L n M , hence is analytic at the point t o . Q.E.D. 21,
(10.5.4) Remark. When K = R, the proof of (10.5.1) shows that the inequality (10.5.1 .I ) is still valid when f i s lipschitziaii in 1 x H for a constant k 3 0, i.e. such that condition (a) of (10.4.6) is satisfied and that ll,f(t, xl) - f ( t , x2)ll d k . llxl - x,I/ for any t E I, x,,x 2 in H ; J can then be taken as an interval of origin (or extremity) t o , containing t o , u and u are primitivesof regulatedfunctions in J, and the relations Ilu’(t) - f ( t , u(t))11 < cl, llu’(t) - f ( t , v(t))II < E, are only supposed to hold in the complement of an at most denumerable subset of J. The uniqueness result (10.5.2) holds likewise (when K = R) under the only assumption thatfis locally lipschitzian (10.4.6) in I x H, when one takes for J an interval having to as origin or extremity, and one only requires that the relations u‘(t) = f ( t , u ( f ) ) ,u ’ ( t ) = g(t, u(t)) hold in the complement of an at most denumerable subset of J.
288
X
EXISTENCE THEOREMS
(10.5.5) Let f be continuously differentiable in I x H i f K = C, locally 11)schitzian in 1 x H i f K = R. Suppose v is a solution of (10.4.1) defined in an open ball J: If - t o / < r, suclz that j c I , that tl(J)c H,and that t + f ( t , 4 t ) ) is bounded in J . Then there exists a ball J‘ : It - toI < r’ contained in I , with r‘ > r, and a solution of (10.4.1) dejined in J’ and coinciding with v in J . (a) K = R. By assumption, we have 11 f ( t , v(t))ll < M for t E J, hence ilu’(t)ll < M in the complement of an at most denumerable subset of J. This implies I ~ D ( S ) - c(t)il < MIS - tI for s, t in J by the mean value theorem (8.5.2). From the Cauchy convergence criterion (3.14.6) we conclude that the limits c((t0 - r ) + ) and u((t, + r ) - ) exist and belong to v(J) c H. By (10.4.6), there exists a solution )ixl (resp. w,) of x’ = f ( t , x ) defined in an open ball U, (resp. U,) of center t , + r (resp. t , - r ) , contained in I , and taking the value v ( ( t o + r ) - ) (resp. u((t, - r ) + ) ) at this point; from (10.5.4)it follows in addition that (resp. coincides with 11 in U, n J (resp. U, n J), and the proof is therefore concluded in that case. (One may observe that it has not been necessary to check the existence of derivatives on the left or on the right for v (extended by continuity) nor for w1 and w,, at the points to - r and to + r.) (b) K = C. For any complex number 4‘ such that l i l = I , put t = t, + is, with s > 0, and P,(s)= ~ ( t+, is). Then the same argument as in (a) proves that u,(r-) exists and is in H ; hence there exists a solution H!, of x’ = f ( t , x ) defined in an open ball V, of center t , + i r , contained in J, such that w,(I, + i r ) = u,(r-). From (10.5.4) it follows that itsi and u coincide in the intersection of J n V, with the segment of extremities t , and t , + ( r ; as these functions are analytic in J n V,, they coincide in J n V, by (9.4.4). Now cover the compact set ( t - t o [= r with finitely many balls Vcc( 1 < i < m ) ; if V,{ n VCj# @, the functions w,, and coincide in VC4n VCj, for both coincide with 11 in the nonempty open set J n V,, n VCj,and we have only to apply (9.4.2) (to show that the preceding intersection is not empty, remark that the assumption implies r l i i - ijl < p i + p i , where p i , pi are the radii of VCi and V,,; hence there is 2 ~ 1 0 I,[ such that rAlii - iil < p i and and r(l - A ) l i i - ijl < p i ; it follows that the point t, + r((l - A)(, Aij) belongs to J n V,, n Vij). There is therefore a solution of x’ = f ( t , x) equal to v in J, to wi, in each of the V l i ,and there is an open ball of center t , and radius Y’ > r contained in the union of these sets (3.17.1 1), which ends the proof. ~
3
~
)
+
(10.5.5.1) It follows from (10.5.5) that if r , is the 1.u.b. of all numbers r such that j c I a n d F ) c H, either ro = co,or, if J, is the open ball It - tol < y o , one of the two relations jo$ I , $ H holds.
+
zi(J,)
5
COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS
289
(10.5.6) Let f, g be two continuously diflerentiable mappings of I x H into E, and suppose that, in I x H , 11 f ( t , x) - g(t, x)II 6 c( and Ij D,g(t, x)I/ 6 k. Let ( t o , xo) be a point qf’ I x H , p, p two numbers > 0, and q($ = ehc - 1 Lieht (a p) -for 2 0. Let u be an approximate solution of x’ = g(t, x),
+ +
c and taking the value y at t o , which contradicts the definition of c.
+
+
+
When K = R,one may, in the statement of (10.5.6), replace J by an open interval ]c, d [ containing the point t o . (10.5.6.1) We again remark that if K = R, we can relax in (10.5.6) the hypotheses on f and g, supposing merely that y is lipschitzian for the constant k, andflocally lipschitzian in I x H. PROBLEMS
1. Let f ( i ,x ) be a real valued continuous function defined in the set I t / < a, 1x1 < b in R2,such that f ( r , x ) < 0 for tx > 0, and f ( r , x ) z0 for t x < 0. Show that x = 0 is the unique solution of the differential equation x’ = f i t , x ) defined in a neighborhood
290
X
EXISTENCE THEOREMS
of 0 and such that x(0) = 0 (use contradiction, and consider, in a compact interval containing 0, the points where a solution reaches its maximum or minimum). 2. Let f ( t , x) be the real valued continuous function defined in R Z by the following conditicns: f ( t , x) = -2, for x > t 2 , f ( t , x) = -2x/t for 1x1 < t 2 , f ( t , x ) = 2 f for x < - t Z . Let (y,) be the sequence of functions defined by y o ( t ) = /’, y.(t)
=
J:
f ( u ,Y,-I(L~))L/u
for n > 1. Show that the sequence (y.(t)) is not convergent for any t # 0, although the differential equation x’ = f ( f , x ) has a unique solution such that x ( 0 ) 0 (Problem I). a, 3. For any pair of real numbers CL > 0, /3 > 0, the function equal to -(t - C L )f o~ r t -: to 0 for a < t < P, to (t - 8)’ for t > /3, is a solution of the differential equation x’ = 2 1 ~ I ”such ~ that x ( 0 ) = 0. Let uo be an arbitrary continuous function defined in a compact interval [a, / I ] , and define by induction
rr.(t) = 2
J’:
~ U - ~ ( . ds S ) for ~ ~ t~ E~ [a, 01.
Show that if y is the
largest number in [a, b] such that u O ( t )= 0 in [a, y ] , the sequence ( i d converges uniformly in [a, b] to the solution of x ’ = 2 1 ~ 1 ’ which /~ is equal to 0 for a < t < y , to ( t y ) 2 for y < r < h. (Consider first the case in which uo(t)= 0 for t < y , u o ( t ) k(t - y)2 for y < t < b. Next remark that, replacing if necessary uo by u l , one may suppose that uo is increasing in [a, b ] ; observe that for any number E > 0, there are two numbers k l > 0, k z > 0 such that in [a, b] -
4.
where vo(t)= 0 if t < 0, vo(t)= t 2 if t > 0.) The notations being those of Section 10.4, suppose K == R,fis continuous and bounded in I x H, and let M = sup llf(r, x)I/. Let x o be a point of H, S an open ball of ( ~ . x )xE HI
center xo and radius r, contained in H . (a) Suppose in additionfis uniformly continuous in I x S (a condition which is automatically satisfied if E is finite dimensional and I is contained in a compact interval I. such that f is continuous in l o x H). Prove that for any E > 0, and any compact interval [ t o ,to h] (resp.[to - h, t o ] ) contained in I and such that h < r/(M E ) , there exists in that interval an approximate solution of x’ =f ( t , x) with approximation E , taking the value xo for t = t o . (Suppose 6 > 0 is such that the relations Itl - f z I < 6, 11x1- xz I1 < 6 imply Ilf(t1, x l ) - ] ( I 2 , x2)l/ < E ; consider a subdivision of the interval [ t o , t o h] in intervals of length at most equal to inf(6, 6/M), and define the approximate solution on each successive subinterval, starting from to.) (b) Suppose E is finite dimensional and I = ]to - a, to a[. Prove that there exists a solution of x’ = f ( r , x ) , defined in the interval [ t o ,to c] (resp.[ro - c, t o ] )with c = inf(a, r/M), taking its values in S, and equal to xo for t = to . (“ Peano’s theorem ”: for each n, let u, be an approximate solution with approximation l/n, defined in J, = [ t o , t o c - ( l / n ) l ,whose existence is given by (a). Observe that for each m , the restrictions of the functions u. (for n 3 m ) to J,, form a relatively compact subset of the normed normed space KE(J,,,) (7.5.7), and use the “diagonal process” as in the proof of (9.13.2); finally apply (10.4.3)and (8.7.8).) 5. Letf’be the mapping of the space ( c o )of Banach (Section 5.3, Problem 5) into itself, such
+
+
+
+
+
+
that, for x
= ( x . ) , f ( x ) = (yJa
with y n =
+ -.n +1 l
Show that f is continuous in
(cd, but that there is no solution of the differential equation x’ - f ( x ) , defined in a
neighborhood of 0 in R, taking its values in (co), and equal to 0 for t = 0. (If there was
5
COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS
291
such a solution u ( t ) = ( u n ( f ) ) ,compute the value of each u,(t) by straightforward integration, and show that the sequence (u.(t)).to does not tend to 0 for t # 0.) (a) The notations being those of Section 10.4, let f be analytic in 1 x H if K = C, locally lipschitzian in 1 x H if K = R. Let lo be an open ball of center to and radius a, contained in I , and S an open ball of center xo and radius Y, contained in H. Let h(s, z ) be a continuous function defined in [0, a [ x [0, r [ c R2, such that h(s, z ) 2 0 and that, for every S E [0, a[, the function z+h(s, z ) is increasing in [0, Y[. Suppose that: ( I ) llf(t, x)l/ < h ( / t- t o / , /lx - xoll) in I. x S ; (2) there exists an interval [0, a ] with dc < a, and a function rp, which is a primitive of a regulated function T’ in [O, dc], and is such that ~ ( 0= ) 0, ~ ( s E) [0, r [ and ~ ’ ( s ) > h(s, ~ ( s ) ) in the interval [0, CL], with the exception of an at most denumerable set of values of s. Show that there is a solution I / of x’ = f ( t , x), defined in the open ball J of center to and radius a , taking its values in S and such that u(to)= xo ; furthermore, in J, llu(t) - xoIl < p(lt tot). (Use (10.5.5) to prove that there is a largest open ball Jo of center t o , contained in l o , and in which there is a solution u of x ’ = f ( t , x ) , taking its values in S and such that lIo(t) - X O / / < y(lt - t o l ) in J o , and furthermore that solution is unique; use then the mean value theorem to prove by contradiction that J c Jo .) (b) Suppose that H = E, and that there is a function h(z) > 0 defined, continuous and ~
increasing in [0, + a [ and , such that
Io+-
- = t m , and that
):
llf(t,
x)ll < h(llxll) in
l o x E. Show that every solution of x ’ = f ( t , x) defined in a neighborhood of t o , is defined in lo (use (a)). (c) If lif(t, x)ll < M in lo s S, then there exists a solution of x ‘ = f ( / , x) in the ball J of center to and radius inf(a, r/M), taking its values in S and such that u to)= x o (take h(s, z ) = M). Suppose K = E = C, and a 2 r / M ; show that, unlessfis a constant, there is an open ball J’ 3 in which u can be extended to a solution of x’ = f ( t , x) taking its values in S . (Observe that, due to the maximum principle ( 9 .5 .9 ) , Iic’(t)l < M for t E J; for any 5 such that 151 = I , consider the function I{&) = u(to 5s); arguing as in (10.5.5), prove that the assumptions of (10.5.5) are satisfied.) I t is not possible to take for the radius of J’ a number depending only on a, r and M , and not on fitself, as x)/2)”” (Section 9.5, Problem 8), with t = xo = 0, a = r = the examplef(t, x) = ((I M = I , shows (n arbitrary integer i I). Letfbe a real valued bounded continuous function in the open polydisk P: It - t o / < a, Ix- xoI < b in R2, and let M = sup I f ( f , x ) l ; let r = inf(a, b/M), and let I =
+
+
+
(1.X) E
P
]to- r , to i.[. Let (D be the set of all solutions I ( of x ’ = f ( t , x). defined in I, taking their values in the open interval ]xo- b, x o b[ and equal to xo for t = to ; the set CI, is not empty (Problem 4(b)). For each t E I , let u ( f , t o , xo) = inf u(t), w(t, t o ,xo) =
+
US@
sup u ( r ) ; show that u and w belong to CD (Section 7.5, Problem 1 I ) ; v(resp. w) iscalled the “ E O
minimal (resp. muximal) solutim of x’ = f ( t , x) in I , corresponding to the point ( t o ,XO). u ( t , t o , xo) in an interval For each T E I , let $. = U ( T , t o , xo). Show that u(t, T , of the form [ T , T h [ , if T 2 t o , of the form IT - h , T ] if T < to (with h > 0). Conclude that there is a largest open interval I t I ,f 2 [ contained in ]to- a , to a[ and containing t o , such that u(t, t o ,x,) can be extended to a continuous function g defined in ]/I, f 2 [ , taking its values in ] x o- b, xo -t /I[, and such that, for every t E PI, t 2 [ , g ( . s )= u(s, t , g ( t ) ) in an interval of the form [t,t h[ if t > t o , of the form It - h , t ] if t < to with h > 0). (Ifg, is another such extension of u ( t , t o , xo) in an interval It;, t i [ ,show that g andg, coincide in the intersection of ] I , , f 2 [ and It;, f ; [by considering the 1.u.b. (resp. g.1.b.) crf the points s in that intersection such that g and gt coincide in [ t o , s[ (resp. in Is, t o ] ) . Furthermore, either t l : to - a (resp. f 2 = to n ) , or g(tl +) = xo Jr b (resp.
6)
+
+
+
y(ti-)
= xo
i b).
+
292
X
EXISTENCE THEOREMS
(a) Generalize Gronwall’s lemma (10.5.1.3) to the inequalities
where v, # 1 , t+h2, 81, O2 are regulated functions 3 0. (Use induction on the number of integrals on the right-hand side.) (b) Let K(t, s) be a continuously differentiable function 3 0 defined in [0, c] x [0, c]. Suppose there are two regulated functions g, h, defined and 3 0 in [0, c ] , such that aK(t, s)/af< g(f)h(s).Show that the inequality w ( t ) < p(t) +lotK(f, s)w(s)ds
for a regulated function w
> 0 implies
w ( t ) G vl(t)
+ el(t)j)(s)w(s)
ds
where v1 and 81 are functions which one can explicitly compute when the functions
v,g and r ( t ) = K ( t , t) are known (consider the function y ( t ) = majorize its derivative). (c) Apply (a) or (b) to the inequality
w ( f )< t
+ h 2 f j : c A S w ( sds) +
fd
1:
K(t, s)w(s)ds and
w(s) ds,
where h > 0. Let w be a real function defined in an open interval I c R, and suppose wis the primitive of a regulated function w’ whose points of discontinuity are isolated in I, and such that in each of these points w’(t+) > w ’ ( t - ) ; suppose in addition that if E is the set of points of discontinuity of w’, the second derivative w” exists in I - E and w”(x) w(x) in I - E. (a) Show that if a, b are two points of I such that w(a) = w(b) = 0, then w(x) < 0 for a < x < b (use contradiction). Conclude that for any three points xl < x < x 2 in I one has w(xl) sinh(x2 - x) w(x2) sinh(x - xl) w(x) G sinh(x2 - xl)
+
(consider the difference w(x)- u(x), where u is a solution of the equation u”(x) - u(x) = 0 taking the same values as w at the points x, and x2).
6. LINEAR DIFFERENTIAL EQUATIONS
The existence theorem (10.4.5) can be improved in special cases: (1 0.6.1) Let I c K be an open ball of center to and radius r. Let j’be continuous in I x E ifK = R,continuously differentiable in I x E i f K = C,and such that
6 LINEAR DIFFERENTIAL EQUATIONS
293
Ilf(t, X I ) - A t , xJll < k(lt - t0I)llxl - x2II for t E 1, XI,x2 in E, where 5 --+ k(5) is a regulated function in [0,r[. Then for every xo E E , there exists a
unique solution u of (1 0.4.1), defined in I, and such that u(to) = xo .
We only have to prove that, if c is the 1.u.b. of the numbers p such that 0 < p < r and that there exists a solution of (10.4.1) defined in It - tot < p and taking the value xo at t o , then c = r (by (10.5.4)). Suppose the contrary; then, by (10.5.4), there is a solution v of (10.4.1) defined in J : It - tol < c and such that v(to) = xo . We are going to show that the conditions of (10.5.5) are satisfied; applying (1 0.5.5) then yields a contradiction and ends the proof. As here H = E, the condition v(J) c H is trivially verified, so we have only to check that t -+f ( t , v ( t ) ) is boundedin J . Now, in the compact interval [0, c], k is bounded and so is the continuous function t -+ 11 f ( t , xo)II in the compact x)ll < set 3; hence there exist two numbers m > 0, h > 0 such that ,?(fI[ mllxll h for t E J and x E E. This implies IIv’(t)ll < rnllv(t)ll h for t E J ; if we write w(5) = IIv(to + A()II with 111 = 1, the mean value theorem shows that
+
+
w(t) < llx0II + hc + rnJ; w(c) dc. We therefore can apply Gronwall’s lemma (10.5.1.3), which shows that Ilv(t)II < aemlr-tol+ b in J (a and b constants), hence u is bounded in J, and so is IIf ( t , v(t))ll < mllv(t)ll + h. (1 0.6.1. I ) Here again, when K = R,the condition of continuity on f can be relaxed to condition (a) of Remark (10.4.6).
A linear differential equation is an equation (10.4.1) of the special form
x’
(1 0.6.2)
= A(t) *
x
+ b(t)
(=f (t,x))
where A is a mapping of I into the Banach space 9 ( E ; E) of continuous linear mappings of E into itself (Section 5.7), and b a mapping of I into E. We have here H = E, and by (5.7.4)
Ilf(r,
x1) -
f(C
X2N
< IIA(t>ll
*
11x1
- x2II
for all t E I , x l , x 2 in E. Applying (10.6.1) and (10.6.1.1) we therefore get: (10.6.3) Let I c K be an open ball of center to . Suppose A and b are regulated in I if K = R, analytic in I if K = C. Then, for every xo E E , there exists a unique solution u of (10.6.2), defined in I and such that u(to) = xo .
Observe that if b = 0, and xo = 0, the solution u of (10.6.2) is equal to 0. From (10.6.3) we easily deduce the apparently more general result:
294
X
EXISTENCE THEOREMS
(10.6.4) The assumptions being the same as in (10.6.3), f o r every s E I and every xo E E, there is a unique solution u of (1 0.6.2) defined in 1 and such that u(s) = xo .
Replacing t by t - t o , we may assume that to = 0. Suppose I is a ball of radius r ; the mapping
t+r2-
t-s St - r2
is an analytic homeomorphism of 1 onto itself, mapping s on 0: indeed, one has t -s t'=r2-=it-1
hence, if It1
(
r_' I---- r2 - Is[') s r2 - St
< r, we have lr2 - it1 < r(r
+ Isl)
whence
our assertion follows from the fact that, conversely,
t=r2-
t' - s
St' - 1
.
Now, if
and
one sees at once that if v is the unique solution of the differential equation X' = A,(t)
defined in I and such that v(0)
.x + bl(t)
= xo, then
is the unique solution of (10.6.2), defined in 1 and such that u(s) = xo.
7 DEPENDENCE OF THE SOLUTION ON PARAMETERS
295
When E = K", A ( t ) = (aij(t))is an n x n matrix, b ( t ) = (b,(t))a vector, the a i j ( t ) and b,(t) being regulated in 1 if K = R , analytic if K = C; if
x = (xi)' 1 (10.6.6)
D"x - a,(t)D"-'x - . . . - a,-,(t)Dx - a,(t)x = b(t)
are equivalent to special systems of type (10.6.5); one has only to write < p < n, and (10.6.6) is equivalent to
x1 = x, x p = D P - ' x for 2
X ~ = X ~ +for ~
(10.6.7)
x;
= al(t)x,,
1 0, let us show next that there exists a neighborhood V c U of zo in P such that IIA(t, z ) - A(t, zo)[I< E and Ilb(t, z ) - b(t, zo)(I < E for t E J and z E V. We only have to remark that for any s E J there is a neighborhood W, of s in J and a neighborhood V, c U of zo in P such that the preceding inequalities holdin W, x V,; then wecoverJ by afinite number of neighborhoods WSi,and take for V the intersection of the V S i We . can now write d ( t , Z ) - u'(t, zo) = A(t, Z )
*
(u(t, Z) - u(t, ~
hence, for t E J and z
+ (A(t,Z ) - A(t, ~ 0 ) )U(t, zo) + b(t, Z ) - b(t, zo)
0 ) )
*
EV
< k . Ilu(t, Z ) - u(t, zo)II + E(M+ 1). Put t = to + 25 with 121 = 1,0 < 5 < r, and 4 5 ) = llu(t0 + At, z>- u(t0 + At, z0)Il; then by the mean value theorem, we have w(5) < E(M + 1)r + k w ( ( ) d( for 0 < 5 < r, and using (10.5.1.3), we obtain w(5) < E(M + ])rek' for 0 < 5 6 r, in other words, we have Ilu(t, z) - u(r, zo)II < E(M + l)rek' for IlU'(t, Z ) - U'(t, z0)II
1;
t E J and z E V; as continuous in J).
E
is arbitrary, this ends the proof (since t -+ u(t, zo) is
(1 0.7.3) In addition to the assumptions of (1 0.7.1), suppose that P is an open subset of a Banach space G, and that (1) ifK = R,f is continuously differentiable in I x H x P; (2) ifK = C, f is twice continuously differentiable in I x H x P (when E and G are finite dimensional, this is equivalent to saying that f is analytic in I x H x P (9.1 0)).Let J, c I be an open ball of center toand T, c P an open ball of center zo such that,for every z E TI, there is a solution t -+ u(t, z ) of x' =f ( t , x, z ) (necessarily unique by (1 0.5.2)) dejned in J, and such that u(to, z ) = xo . Then, for any open ball J of center t o , such that c J,, there
7 DEPENDENCE OF THE SOLUTION ON PARAMETERS
297
exists an open ball T c TI of center zo such that ( t , z) -+ u(t, z ) is continuously differentiable in J x T . Furthermore, for any z E T , t D2 u(t, z ) is equal in J to the solution U ( t ,z) of the linear differential equation -+
U'
(10.7.3.1)
= A(t, Z)
0
U
+ B(t,
such that U ( t o ,z) = 0, where A(t, z) D 3 f ( t , u(t, z),z).
Z)
= D,f(r,
u(t, z), z) and B(t, z) =
Let J be an open ball of center to and radius r, such that S c J,. By (10.4.5.1), there is an open ball S c H of center xo and an open ball T c T, of center zo such that D2f and D 3 f are bounded in J x S x T, let IIDzf(t,x , z)II < a and IID3f(t,x,z)ll < b. Then, by (8.5.2) and (8.9.1), we
have,
for t E J, x,,x2 in S , z,, z, in T. Taking (10.7.3.2) into account, we see that, by (10.5.1), we have, for t E J and zl, z, in T (10.7.3.3)
IIu(t, Zl) - u(t, zz)ll
< C l l Z l - ZZII
with c = b(e" - l)/a. We next prove that, given a point z E T and E > 0, there exists p > 0 such that, for any w E P such that z w E T and llwll < p , and any t E 5, we have
+
Ilf(t, u(t9 z
+ 4,z +
- f ( t , u(t, 4, 4
- A(t, z) * (u(t,z + w) - u(r, z))
(10.7.3.4)
- B(t,z ) *
wII
G E IIwII. Indeed, using (8.6.2), (8.9.1), the continuity of D , f and D3 fi n I x H x P, and relation (10.7.3.3), for any s E J, there is a neighborhood W, of s in J, and a number p(s) > 0 such that relation (10.7.3.4) holds for t E W, and llwll < p(s); covering 5 by finitely many W,:, we need only take for p the smallest of the p(si) to have (10.7.3.4). Due to the definition of u(t, z), (10.7.3.4) can also be written (10.7.3.5)
IID,u(t, z
+ W ) - Dlu(t,
G EIIWII.
Z)
- .4(t, Z)
*
(u(t,z
+ W ) - u(t, z)) - B(t, Z ) . wII
298
X
EXISTENCE THEOREMS
Now the existence of U ( t , z ) in J x T is guaranteed by (10.6.3), since D 2 f and D3f are continuously differentiable when K = C. Put v(t, z , M’)= u(t, z HI)- u(t, z ) - U(t,z ) . it’; this function has a derivative with respect to t equal to
+
D,o(t,
Z,
IV) = D,u(r, z
+ w) - D,u(t, Z ) - A(?,
Z)
. ( U ( t ,Z )
W)
-
B(t, Z) . W ,
by (10.7.3.1 ). Relation (10.7.3.5) therefore can be written IID,u(t, Z ,
IV)
- A(t, Z)
. ~ ’ ( tZ,,
n1)Ild
E lllt’ll,
for any t E J and any $11 such that z + I(’ E T and llwxll < p . In other words, v(t, z , it’) is an approximate solution, with approximation E 11 ~ 1 1 , of the linear differential equation y’ = A ( t , z ) . y.
(1 0.7.3.6)
Furthermore, we have u ( t o , z , cv) = 0 by definition; as jlA(t, z)II < a in J x T, we conclude from (10.5.1) (since 0 is a solution of (10.7.3.6)) that
II4t,
7-7
M’III
< COE I I 4
where co = (ear- l)/a, this inequality being valid for any t E J and any cv such that z 11:E T and llct~ll d p . As E is arbitrary, the definition of the derivative of a function shows that u is differentiable with respect to z at any point ( t , z ) E J x T and that D , u(t, z ) = U(t,z). Finally, from the assumptions and (10.7.2), it follows that U is continuous i n J x T ; on the other hand, D,u(t, z ) =.f’(t,u(t, z ) , z ) is continuous in J x T by (10.7.1). Therefore, by (8.9.1), u is continuously differentiable in J x T, and this ends the proof of (10.7.3).
+
(10.7.4) Suppose K = R, and f is p times continuously diferentiable (resp. E and G are finite dimensional a n d f is indefinitely diferentiable) in I x H x P. Then,for each open interual J of center to suclz that J c J,, it is possible to take T such that u is p times continuously diferentiable (resp. indefinitely diferentiable) in J x T. If p = 1, this is (10.7.3). Using induction on p , suppose we have proved the result for ( p - 1) times continuously differentiable mappings. Then, in the right-hand side of (10.7.3.1), A and B are p - 1 times continuously differentiable mappings in J x T (by (8.12.10)); therefore, by (10.7.3) (applied to U(t, z ) ) , D , u(t, z ) is p - 1 times continuously differentiable in J x T (when T has been conveniently chosen). On the other hand D,u(t, z ) = f ( t , u(t, z ) , z ) is also p - 1 times continuously differentiable in J x T by the induction hypothesis and (8.12.10); therefore Du(t, z) is p - I times continuously differentiable in J x T by (8.9.1), (8.12.9), and (8.12.10); but this
7 DEPENDENCE OF THE SOLUTION ON PARAMETERS
299
implies that u isp times continuously differentiable in J x T, by (8.1 2.5). When E and G are finite dimensional a n d f i s indefinitely differentiable, it follows from (3.1 7.1 0) that in the preceding argument one may choose T independently o f p , since all derivatives o f f are continuous; therefore u is indefinitely differentiable in J x T. Observe that in (10.7.4), one may replace J, by any open interval containing the point t o , and J by any open interval containing to and such that J c J,.
(10.7.5) Suppose that the Banach spaces E and G areJinite dimensional and that f is analytic in I x H x P. Then,for any open ball J of center t o , such that 5 c J,, it is possible to take T such that u is analytic in J x T. If K = C,this follows immediately from (10.7.1), (10.7.3), (9.10.1), and (9.9.4). If K = R, we apply an argument exactly similar to that of (10.5.3), which we accordingly suppress.
(10.7.6) Remarks. There are several improvements and variants of the preceding theorems. For instance, in (10.7.3), when K = R, the existence of D,fis not required to insure that D, u(t, z ) exists: we need only the continuity of D2,f and D3J’ as functions of (x,z), and their boundedness in J x S x T, as well as the fact that t + f ( t , h(t),z ) is regulated in I for any function 12 continuous in I and similarly for D,fand D,f:
PROBLEMS
1. The notations being those of Section 10.4, let I be an open ball in K of center to and radius a , S an open ball in E of center xo and radius Y, G the normed space Vz(I x S) (Section 7.2). For each M > 0, let G , be the ball l l f l l < M in G . Let L be the subset of G consisting of all continuous lipschifzian mappings of I x S into E (10.5.4); for each M > 0, let J, be the open ball of center to and radius inf(a, r/M); for each function f~ L n G , , there is a unique solution LI = U(f) of x’ = f ( t , x ) taking its values in S, defined in J, and such that u(to)= x o (Section 10.5, Problem 6(c)). (a) Let (A,) be a sequence of functions belonging to L n G , , and supposef, converges uniformly in 1 x S to a functionf; show that in the space %Z(J,), every cluster value of the sequence of functions u, = U(f.) is a solution of x’ = f ( f , x ) , taking its values in S, and equal to xo for 1 - to (use (10.4.3) and (8.7.8)).Give an example in which the sequence (/In) has 110 cluster value in VZ(JM)(see Section 10.5, Problem 5).
300
X
EXISTENCE THEOREMS
(b) Suppose in addition that E is finite dimensional; using the result of (a), give a new proof of Peano’s theorem (Section 10.5, Problem 4(b); use Ascoli’s theorem (7.5.7) and the Weierstrass approximation theorem (7.4.1 )). 2. (a) In the polydisk P: It - tol < a , Ix - xol < b in RZ,let g, h be two real valued continuous functions such that g ( f , x ) < h(t, x ) in P. Let u (resp. u ) be a solution of x’ = g ( t , x ) (resp. x’ = h(t, x ) ) defined in an interval [ t o , to c [ , taking its values in ]xo - b, xo b[ and such that u(to) = xo (resp. u(to) = x o ) ; show that u(t) < u ( t ) for to < t < to c (consider the 1.u.b. of the points s in [ t o ,to c[ such that u ( t ) < u ( t ) for to < t < s). (b) Let g be continuous and real valued in P, and let u be the maximal solution of x’ = g(t x ) corresponding to ( t o ,x o ) (Section 10.5, Problem 7); suppose u is defined (at least,) in an interval [ t o , to c[ and takes its values in ] x o - b, xo b[. Show that in every compact interval [ t o , to d ] contained in [ t o , to c [ , the maximal and minimal solutions of x’ = g(t, x ) E are defined and take their values in ]xo - b, xo b[ as soon as E > 0 is small enough, and converge uniformly to u when E tends to 0. (Given E~ > 0, there exists an s > to such that the maximal and minimal solutions of all theequations x’ = g(t, x ) E for 0 < E < eo, corresponding to ( t o , xo), are defined and take their values in ] x o - b, xo b[ fort in [to , s ] ; observe that all these functions form an equicontinuous set in [ t o ,s], and prove the uniform convergence to u in [ t o ,s ] by applying the result of (a), Ascoli’s theorem (7.5.7), (10.4.3), and (8.7.8). Finally, show that the 1.u.b. of the numbers d having the stated property is necessarily equal to c, using in particular the last statement of Section 10.5, Problem 7.) (c) In the polydisk P, let g and h be two continuous real valued functions such that g ( t , x ) < h(t, x ) in P. Let [ t o ,to c] be an interval in which a solution u of x’ = g(t, x ) such that u(to)= xo , and the maximal solution u of x‘ = h(t, x ) corresponding to ( t o ,xo) are defined and take their values in ]xo - b, xo b[. Show that u ( t ) < u ( t ) for to < I < to c (apply (a) and (b)). 3. (a) Show that the conclusions of Problem 6(a) of Section 10.5 are still valid when E is finite dimensional, and the assumptions are modified as follows: (1) f is supposed to be continuous in I x H (when K = R), but not necessarily locally lipschitzian; (2) 9 is the maximal solution (Section 10.5, Problem 7) of the equation z’ = h(s, z ) in [0, a ] , corresponding to the point (0,O). (Use the results of Problems l(a) and 2(b), and apply the diagonal process as in Problem 4(b) of Section 10.5.) (b) Suppose in addition that there exists a sequence (Yn)n30 of real valued functions, continuous in [0, a ] , taking their values in [ O , r ] , such that for n 3 1, Y , ( s ) =
+
+ +
+
+ + +
+
+
+
+
+
+
+
+
/:h([, Yn-l([))d[ for 0 < s < a. Let yo be continuous in J if K = R, analytic in J
if K = C, with values in S, and such that Ilyo(t)- x o I / < Yo(lt- tol) in J. Show that there exists a sequence (yn)nbl of mappings of J into S, which are continuous if
K = R, analytic if K = C, and such that y.(t)= x 0 +Joke, yn-l(0)) do, and that IIy,,(t) - xoII < Y,(lt - t o / )in J for every n 3 1. When K = C, conclude that the sequence (y,) converges in J (uniformly in every compact subset of J) to the unique solution u of x’ = f ( t , x). (Use (9.13.2) and the proof of (10.4.5).) Is this last statement still true when K = R andf is not supposed to be locally lipschitzian (cf. Section 10.5, Problem 2)? 4. (a) Let I = [ t o , to + c [ c R, and let w be a real valued continuous function 3 0, defined in I x R. Let S be an open ball of center x o in E, and let f be a continuous mapping of I x S into E such that for t E I, x 1 E S and x 2 E S, llf(t,xl) - f ( t , x 2 ) I I < w ( t , llxl - x211).Letu,ube twosolutionsofx’=f(t,x),definedin
7 DEPENDENCE OF THE SOLUTION ON PARAMETERS
301
I, taking their values in S, and such that
u(to) = x l , v ( t , x o ) = xz; let w be the maximal solution (Section 10.5, Problem 7) of z’ = w ( t , z ) corresponding to ( t o , llxl - xz I!), and suppose w is defined in I; show that in I, Ilu(t) - u ( t ) I/ < w(t). (For small E > 0, consider the maximal solution w(t, E ) of z’ = w ( t , z ) E corresponding to ( l o , 11x1- xz II), which is defined in [ t o , to d] if d < c, as soon as E is small enough (Problem 2); show that for to < t < to d, Ilu(t) - v(t)ll < w(t, E), using contradiction: consider the g.1.b. tlof the points t such that Ily(t)II > w(t, E), where y ( t ) = u(r) - v ( t ) , and observe that for t > t l
+
Ilu(t)ll- IlY(tl)ll
0; define f ( t , x) = f ( - t , x ) for t < 0. Take a : 0; let u ( t ) = E t for It/ < a , and take for u a solution of x' = f ( t ,x) E for the other values oft.) 7. The notations being those of Section 10.4, suppose E is finite dimensional and f is continuous in I x H; let ( t o ,x o ) be a point of I x H, J an open ball of center to contained in I, S an open ball of center xo such that S c H. Supposefis bounded in J x S, and the following conditions are verified: ( I ) There is af most one solution of x' = f(t, x) defined in an open interval contained in J and containing t o , and taking the value xo for t = t o . (2) There exists a sequence ( u , ) , ~of~continuous mappings of J into S such that
+
u.(t)
= xo
+Itr
f(s, r i n - l ( . s ) ) d.s
for n
>1
and
t E J.
0
+
(3) For every t E J , u " + ~ ( / )- u,(t) converges to 0 when n tends to m. Show that in every compact interval J' c J containing to the sequence(u,)converges uniformly to a solution of x' = f(/, x) equal to xo for t = t o . (Observe that the sequence (u.) is equicontinuous; use Ascoli's theorem (7.5.7), as well as (3.16.4) and (8.7.8).) 8. Suppose E is finite dimensional, w and f verify the conditions of Problem 5(a), and in addition, for every t E 10, a [ , the function z+ w(t, z ) is increasing in [0, +a[. There is then at most one solution of x' = f(t, x) defined in an interval [0, a [ c [0, a[ and taking the value xo for t = 0 (Problem 5(a)). Suppose in addition that there exists, in an interval J = [0, a ] C [0, a[, a sequence (u,Jnb0 of continuous mappings of J into S such that I&)
= xo
+s,'f(s,
U " - ~ ( S ) ) ds
(a) For every t E J, let y,(t)
=
for n > 1 and
l l ~ , , + ~( t u,,(t)/I, ) z.(t)
=
t E J.
supy,+,(t), and w ( t ) = k 2 0
inf z.(t). Show that the functions z, and w are continuous in J (use Problem 1 1 of XbO
Section 7.5). (b) Let t , t - h be two points of J (h > 0); show that, for every 6 > 0, there is an N such that, for n > N, lyn(t)- y.(t
- h)/
z,(t - h)). Hence
(by (8.7.8)).
t-h
(d) Conclude that w(t) = 0 in J (same argument as in Problems 4(b) and 5(a)), and using Problem 7, prove that the sequence (u,) converges uniformly in J to a solution of x = f ( t , x ) taking the value xo for t = 0. 9. The notations being those of Section 10.4, suppose E is finite dimensional, and f is continuous and bounded in I x H. Suppose in addition there is at most one solution of x' = f ( t , x) defined in any open interval J C I containing t o , and equal to xo E H for t = t o . Suppose that, for any integer n > 0, there exists an approximate solution it. of x' = f ( t , x), with approximation I/n, defined in 1 and taking its values in H, and such that u,(to) = xo . Show that in any compact interval contained inI, thesequence ( i t n ) is uniformly convergent to a solution u of x' = f ( t , x), taking its values in H and such that u(to)= x o . (Use the same argument as in Problem 7.)
8. DEPENDENCE OF THE SOLUTION O N INITIAL CONDITIONS
(10.8.1) Let f b e locally lipschitzian (10.5.4) in I x H if K I x H if K = C.Then,for any point (a, b) E I x H :
= R, analytic
in
(a) There is an open ball J cl of center a and an open ball V c H of center b such that,,for every point ( t o ,x,) E J x V, there exists a unique solution t u(t, t o , x,) of (10.4.1) defined in J, taking its values in H and such that d t , to xo) = xo * (b) The mapping ( t , t o , x,) + u(t, t o , x,) is uniformly continuous in JxJxV. (c) There is an open ball W c V of center b such that, for any point ( t , t o , x,) E J x J x W, the equation xo = u(tO,t , x ) has a unique solution x = u(t, t o , x,) in V. --f
9
9
(a) By assumption, there is a ball J, c I of center a and a ball B, c H of center b and radius r such that in J, x B, , 11 f ( t , x)ll Q M, and 11 f ( t , xl) f ( t , x2)l/ < k * I/xl - x211 for tE J,, x l , x 2 in B,, By (10.4.5) and (10.5.2) there is an open ball J, c J, of center I,, and a unique solution v of (10.4.1) defined in J,, taking its values in H and such that v(a) = b. We are going to see that the open ball V of center b and radius 4 2 , and the open ball J of center a and radius p , answer our specifications as soon as p is small enough. Apply
304
X
EXISTENCE THEOREMS
(10.5.6) to the case c1 = = 0; this shows that there exists a solution of (10.4.1) defined in J, with values in Bo , taking' the value xo E V at the point t o E J,
provided we have IIu(t) - 611
(1 0.8.1 .I)
for
every
IIv(t)
t~ J.
But
+ IIu(to) - xollekl'-'ol < r
by
the
mean
value
theorem,
we
have
< M It - a1 < M p forevery t E J ; as by assumption llxo - bll < r / 2 ,
- bll
the inequality (10.8.1 . I ) will be satisfied if p is such that (10.8.1.12)
(M p + -I)e 2 k p < r
Mp+
which certainly will be satisfied for small values of p > 0, since the left-hand side of (1 0.8.1.2) tends to 1-12when p tends to 0. (b) From the mean value theorem, we have (10.8.1.3)
for
to,t,,
(10.8.1.4)
- u(t2
IIu(t1, t o 9 xo)
t o 9 x0)II
9
dM
It2
- 1,
I
t , in J, xo in V. By (10.5.1), we have IIu(t, t o 9 XI)
- u(t, t o
9
x2)ll
< e2kp1x2 - x1 I
for t , t o in J, x , , x2 in V. Finally, (10.8.1.3) for Ilu(t1,
t2
9
xo) - xoll
<M
to = t2
It2
yields by definition
- tl I
and as t + u(t, t , , xo) is the unique solution of (10.4.1) in J which is equal to u(t,, t 2 , xo) at the point t,, we have, by (1 0.5.1) (10.8.1.5)
IIu(r, t , , xo) - u(t9 t 2 , x0)II
< M e 2 k pI f 2 - [,I
for t , t , , t , in J, xo E V . The three inequalities (10.8.1.3), (10.8.1.4), and (10.8.1.5) prove that u is uniformly continuous in J x J x V. (c) By (10.8.1.3), we have IIu(t, t o , xo) - xoII < Mlt - tol < 2 M p in J x J x V. Suppose p satisfies (10.8.1.2) and in addition the inequality 2 M p < r/4 ; then, if W is the open ball of center b and radius 1-14,we have u(t, t o , xo) E V for t , t o in J and xo E W. Let x = u(t, t o , xo)for such values of t , t o , x,; then s -,u(s, t , x ) is defined in J and is the unique solution of (1 0.4.1 ) with values in H which takes the value x at the point t ; but as s + u(s, t o ,xo) has these properties, we have u(s, t , x ) = u(s, t o ,xo) for s E J; in particular xo = u(to, t o , xo) = u(to, I, x ) . Suppose nowy E V is such that u ( t o , t, y ) = x,; then s + u(s, t , y ) is a solution of (10.4.1 ) defined in J and taking the value xo for s = t o ; therefore u(s, t , y) = u(s, t o , xo) for any s E J, and in particular, for s = t , y = u(t, t o , xo) = x , which ends the proof.
8 DEPENDENCE OF THE SOLUTION ON INITIAL CONDITIONS
305
(10.8.2) With the notations of (10.8.1), suppose f is p times continuously differentiable (resp. E is jinite dimensional and f is indejnitely differentiable, resp. E is finite dimensional and f is analytic) in I x H. Then it is possible to choose J and V such that the function ( t , t o , x,) u(t, t o ,x,) is p times continuously differentiable (resp. indejnitely differentiable, resp. analytic) in -+
J X J X V .
+ s, t o , x,) - x,,
Indeed, if we write u(s, t o , x,) = u(t, s -+ u(s, t o , x,) is a solution of the equation
z' = f ( t ,
we see that
+ s, x, + z )
which takes the value 0 at the point s = 0; the result then follows from (10.7.4) and (10.7.5).
For linear differential equations, there are much more precise results. The equation
x'
(10.8.3)
= A(t)*
x
is called the homogeneous linear differential equation associated to (1 0.6.2) ; the difference of any two solutions of (10.6.2) in I is a solution of (10.8.3) in I, and the solutions of (10.8.3) in I constitute a vector subspace 2 of the space qE(1) of all continuous mappings of 1 into E.
(10.8.4) For each (s, x,), let t --+ u(t, s, x,) be the unique solution of (10.8.3) dejned in I and such that u(s, s, x,) = x, . (1) For each t E I, the mapping x, -+ u(t, s, x,) is a linear homeomorphism C(t, s) E 9 ( E ) of E onto itself. (2) The mapping t -+ C(t,s) of I into the Banach space 9 ( E ) is equal to the solution of the linear homogeneous differential equation (10.8.4.1)
U'
= A(t)0
U
which is equal to I, (identity nzapping of E) for t = s. (3) For any three points r, s, t in 1 (10.8.4.2)
C(r, t ) = C(r,s) C(s, t ) 0
and
+ +
C(s, t ) = (C(t,s))-'
It is clear that u(t, s, xl) u(t, s, x2) (resp. Au(t, s, x,)) is a solution of (10.8.3) which is equal to x1 xz (resp. Ax,) for t = s; hence (10.6.4) it is equal to u(t, s, x1 xz) (resp. u(r, s, Ax,)) in I, which proves that the mapping x, -+ u(t, s, x,) is linear; let us write it C(r,s) (we have not yet
+
proved that this mapping is continuous in E).
306
X
EXISTENCE THEOREMS
Now the bilinear mapping ( X , Y ) + X o Y of 2 ( E ) x 9 ( E ) into 2 ( E ) is continuously differentiable (8.1.4); denote by R ( t ) the continuous linear mapping U - i A ( t ) U of 2 ( E ) into itself. From (5.7.5) it follows at once that 0
IIWt)
- W)ll G
I I 4 t ) - 4411
hence, if K = R, t -+ R ( t ) is regulated if t + A ( ? ) is regulated. On the other hand, if t -+ A ( t ) is differentiable, so is t R ( t ) , and its derivative at the point t (identified (Section 8.4) to an element of 2 ( E ) ) is the mapping U-+ A ’ ( t ) U ((8.1.3) and (8.2.1)); hence if t -+ A ’ ( t ) is continuous, so is t -+ R’(t). We can therefore conclude that if K = C and if t + A ( r ) is analytic in I , so is t -+ R(r) (9.10.1). In any case, we may apply (10.6.4) to the equation (10.8.4.1); let V ( t )be the solution of that equation equal to IE for t = s. We have, for any t E I ((8.1.3) and (8.2.1)) 0
D(V(t) * x0) = V’(t) . x0 = A ( t ) . ( V ( t )* xO) and furthermore, for t = s, V ( s ). xo = ZE . xo = x,; it therefore follows from (10.6.4) applied to (10.8.3) that C(t,s) . x,, = V ( t ) . xo for any x,, E E, hence C(t,s) = V(r) for t E I . This proves that C(r,s) E 2 ( E ) and that t 3 C(t,s) is the solution of (10.8.4.1) which is equal to IEfor t = s. Finally, the function r --t C(t,v ) . xo is the solution of (10.8.3) equal to C(s, r ) * xo for t = s; hence, by definition
C(t, v ) . xo = C(t,s) . (C(s, v ) * xo) = (C(t,s) C(s, r ) ) * x, 0
for any xo E E, which proves the first relation (10.8.4.2); as C ( t , t ) = Z,, that relation yields C(t,s) 0 C(s, t ) = ZE. This shows that C(s, t ) is a hijectire linear mapping of E, whose inverse mapping is C(t,s) (hence also belongs to 2(E)). With this we reach the end of the proof of (1 0.8.4). The operator C(t,s) is called the resolvent of (10.8.3) (or of (10.6.2)) in I. (103.5) The niupping (s, t ) -+ C(s, t ) of 1 x I into 2 ( E ) is continuous. We may indeed write C(s, t ) = C(s, to) (C(r, to))-’, and the result then follows from (10.8.4), (5.7.5), and (8.3.2). 0
The knowledge of C(s, t ) enables one to give the explicit solution of (10.6.2) taking the value xo for t = t o : (1 0.8.6)
The function U(t) =
C(?, to) . xo
+
9 THE THEOREM OF FROBENIUS
307
is the solution of (10.6.2) in I which is eyuul to xofor t = to (if K = C, the integral is to be taken along the segment of origin r, and extremity t ) .
Indeed, one may write, by (10.8.4.2)
using (8.7.6); therefore, we have u ( t ) = C(t,to) . z ( t ) , where z ( t ) = xo
Hence ((8.1.4) and (8.2.1))
+
lo
( C ( t o ,s) . b(s))ds.
d ( t ) = C ’ ( t , to) * z ( t )
+ C(t,1,) - z’(t).
But by (10.8.4.1), C‘(t,to) = A ( / ) C(t, to), and on the other hand, z’(t) = C(to,t ) . b(t) by definition; hence 0
-
u’(t) = A ( / ) u ( t )
and as u(t,)
= xo , this
+ b(t)
ends our proof.
When E = K“, and the equation (10.6.2) is then written as a system of scalar linear differential equations (10.6.5), the resolvent C(s, t ) is an invertible n x n matrix (cij(s,t ) )whose elements are continuous in I x I, and t -+ c i j ( t , s) is a primitive of a regulated function in I if K = R, an analytic function in I if K = C.
PROBLEM
(a) Suppose, in the linear differential equation (10.6.2), that A and b are analytic functions in the simply connected open subset H c C. Show that, for any toE H and any x,, E E, there is a unique solution u of (10.6.2), defined in H and such that u(fo) = x o . (Use the same kind of argument as in (9.6.3): (10.6.3) allows one to define a solution of (10.6.2) along a broken line (Section 5.1, Problem 4) in H, and the argument of (9.6.3), along with local uniqueness, yields the result.) (b) Show that the result of (a) is not valid for the scalar differential equation x ’ = f / x : given any simply connected open subset H c C , and any fo E H, there exists an xo E E, such that xo # 0 and that there is 170 solution of the equation defined in H and equal to xo for t = lo,
9. T H E T H E O R E M OF F R O B E N I U S
Let E, F be two Banach spaces over K , A (resp. B) an open subset of E (resp. F), U a mapping of A x B into the Banach space 2 ( E ; F) (Section 5.7).
308
X
EXISTENCE THEOREMS
A differentiable mapping u of A into B is a solution of the total differential equation
(10.9.1 )
Y’ = U ( x ,Y )
if, for any x E A , we have
(10.9.2)
u’(x) = U(x,U(.Y)).
When E = K, 9 ( E ; F) is identified to F (5.7.6), and a total differential equation is thus an ordinary differential equation (10.4.1). When E = K“ is finite dimensional, a linear mapping U of E into F is defined by its value at each of the n basis vectors of E, and, by definition, (10.9.2) is thus equivalent to the system of ii “partial differential equations” (10.9.3)
D i y = f i ( x l ,. . . , x, , y )
(1
< i < n).
In general, such a system will have no solution when 11 > 1 , even if the right-hand sides ,f; are continuously differentiable functions. We say that an equation (10.9.1) is conipletely integrable in A x B if, for every point (xo,yo) E A x B, there is an open neighborhood S o f x o in A such that there is a unique solution u of (10.9.1), dejned in S, with values in B, and such that 4 x 0 ) = Yo. We will suppose in what follows that U is continuously chfferentiuble in A x B; for each (x,y ) E A x B, D, U ( x ,y ) (resp. D, U ( x ,y ) ) is an element of 9 ( E ; 9 ( E ; F)) (resp. 9 ( F ; 9 ( E : F))), which can be identified to the continuous bilinear mapping (sl, s,) -+ (D, U(x,y ) . sl) . s, of E x E into F, written (sl, s2) -+ D, U(x,y ) . (sl, s,) (resp. the continuous bilinear mapping ( t , s) -+ (D,U(x, y ) . t ) . s of F x E into F, written (1, s) -+ D, U ( x ,y ) . ( t , s)) (5.7.8); furthermore, the linear mapping s1 + ( D 1U ( x ,y ) . sl) s, of E into F, for each s, E E, is the derivative at the point (x, y ) of the mapping x + U(x,y ) . s2 of E into F, by (8.2.1) and (8.1.3); similarly, the linear mapping t -+ (D, U(x,y ) t ) * s of F into F, for each s E E, is the derivative at the point (x, y ) of the mapping y -+ U(x,y ) . s of F into F. 1
(10.9.4) (Frobenius’s theorem) Suppose U is continuously differentiable in A x B if K = R, t,tIice continuously riifferentiuble if K = C. In orrler that (10.9.1) be conipletely integrable in A x B, it is necessary and sufljcient that, for each (x,y ) E A x B, the relation (10.9.4.1) Di u(X,Y )
(Si, $2)
(u(x,J’) ’ S1, S2) + D, U(x, Y>. (U(X,Y ) . s 2 , $1)
-t D, U(x,Y ) *
= D, U.u,Y> . ( s 2
9
holds,for any pair (sir s,) in E x E.
s1)
9
THE THEOREM OF FROBENIUS
309
(a) Necessity. Suppose u is a solution of (10.9.1) in an open ball S c A of center xo such that zi(xo)= yo; then, from (10.9.2) and the assumption it follows that u‘(x) is differentiable in S ; moreover, for any s,EE, the derivative at the point xo of the mapping x -+ u’(x) . s, is sI -+ u”(xo) . (s,, s2)by (8.12.1). But by (10.9.2), that derivative is also (using (8.2.1), (8.1.3), and (8.9.1)) s1 -+
(D, U(x0 Y o ) * 3
Sl)
. s2
+ (D, U(x0 Yo) . (u’(x0)
*
7
s1))
*
s2.
Using the relation (10.9.2) again, and expressing that the second derivative of u at the point xo is a symmetric bilinear mapping (8.12.2), we obtain (10.9.4.1) at the point (xo, yo). But by assumption that point may be taken arbitrarily in A x B, hence the result. (b) Suficiency. Let So c A be an open ball of center xo a n d radius ct To c B an open ball of center yo and radius /3, such that U is bounded in So x To, let 11 U ( x ,y)II < M. We consider for a vector z E E the (ordinary) differential equation (where 5 E K) (10.9.4.2)
M” =
U(xo+ 52, w) * z
= f(5,
1.0,
z)
and observe that if u satisfies (10.9.2) in a neighborhood IIx - xoII < p of llzll < p is a solution of (10.9.4.2) in the ball < 1, in K , taking the value yo for 5 = 0 (which already proves uniqueness of u by (10.5.2)). Now the right-hand side of (10.9.4.2) is continuously differentiable for 151 < 2, IIM, - yo/l < /3 and llzll < a/2, and we have ,(fI[ M’, z)ll < Mllzll for such values. Applying (10.5.6) to f and to g = 0, we conclude that for any z E E such that ilzll < /3/2M, there is a unique solution 5 .+ v(5, z ) of (10.9.4.2) defined for 141 < 2, taking its values in H and such that a(0, z ) = y o . We are going to prove that thefimction u ( x ) = z*( I , x - xo) is a solution of (10.9.1) in the ball IIx - xoII < /3/2M. Now, for lizll < /3/2M and 151 < 2, we know from (10.7.3) that u is continuously differentiable, and that t -+ D, ~ ( 5z,) is, for 151 < 2, the solution of the linear differential equation
xo,5 -,u(xo + 5 2 ) for
v‘ = D, f ( 5 , u ( 5 , z), Z ) v + D, f(5, 4 5 , z), Z ) taking the value 0 for 5 = 0. For any sI E E, write g(5) = D, v(5, z) s,; we , z ) . g ( 5 ) + D, f(5, 4 5 , z), z ) . s1 and from the have g ’ ( 0 = D,,f(5, ~ ( 5z), definition 0f.L this can be written
d(5)= NO . M5L 4 + B(5). s1 + W(5) (s1,4 *
O 5z, ~ ( 5z>), , 4 5 ) = D2 U(xo + 52, 4 5 , z)), B(t) = ~ ( X + C(5) = D, U(xo+ 42, v(5, z)). We want to prove that g(5) = 5U(xo + 52, u(5, z)) * s1
with
310
X
EXISTENCE THEOREMS
and we therefore consider thedifferenceh(t)
-
g(5) - tB(5) sl.We have
= g ( ( ) - (U(x0
+ ~ z , v ( (z)), 's1=
h ' ( 0 = A(5) . Mt), z>f B(5) . s1 + W(5). (S1r
4 (K5) z , Sl)
- B(5) . s1 - t C ( 0 . ( z , $1 - IIuil, by (8.3.2.1). Furthermore, for
l i l '111411,
where the series is absolutely convergent, and
a s soon as l i l 3 211~11,we have therefore ll(u - i. l,)-' /I d llull-'. Now, if we had R, = C, ( u - ( . IE)-' would be a n entire function (9.9.6), bounded in C since it is bounded in the compact set Ill d 211ull and bounded in its complement; by Liouville's theorem (9.11 . I ) ( u - [ . 1 , ) - I would be a constant, hence also its inverse LI - i . I , which is absurd. The first part of the proof shows in addition that ( u - i* I,)-' exists and is analytic for l i l > /lull, therefore the spectrum of II, which is closed in C, is compact and contained in the ball
,
lil d I I ~ I I .
It is possible to give examples of operators for which the spectrum is an arbitrary compact subset of C (see Problem 3).
1
SPECTRUM OF A CONTINUOUS OPERATOR
315
PROBLEMS
1. Let E be a complex Banach space, u an element of P ( E ) , sp(u) its spectrum. (a) Show that if a complex number 5 is such that, for a n integer p > I , 15Ip> lluPll, then 5 is regular for u. (Use (11.1.3), and from the convergence of the series m
"=O
5-"pu"p.conclude that the series
x ]a Hilbert basis of E. Let S be an arbitrary infinite compact subset of C, and let (p,,) be a denumerable set of points of S , which is dense in S (3.10.9). Show that there is a unique element L I E 9 ( E ) such that u(e,) = pne, for every n > 1 ; prove that the spectrum of u is equal to S , whereas the eigenvalues of u are the p,, . If 5 E S, 5 is not equal to any of the pn, and vc = u - 5 . l E, show that v : ( E ) is dense in E but not equal to E(use (6.5.3) to prove the first statement). Show that the Tpectrum of the operator u defined in (11.1.1) is the disk 151 S 1 in C; I I has no eigenvalue. If v c = u - 5 . I,, show that for 151 < 1, v,(E) is not dense in E, but for 151 1, vc(E) is dense in E and distinct from E (cf. (6.5.3)). Let E be a complex Banach space, Eo a dense subspace of E. Show that for any element I I f 9 ( E 0 ) , the spectrum of u contains the spectrum of its unique continuous extension ri to E (5.5.4). Give an example in which these spectra are distinct and an example of an operator u E Y(EO)and of a spectral value 5 of u such that, if u 5 = u - 5 . l E , v c is a bijective mapping of Eo onto itself (in Problem 3, consider the subspace Eo of E consisting of the (finite) linear combinations of the vectors en). Note that this is impossible if Eo is a Bunuch space, for every continuous bijective linear mapping of Eo onto itself is then a homeomorphism (12.16.8). Let E be a complex separable Hilbert space, (en),,>] a Hilbert basis of E; let u be a continuous operator in E such that, for every pair of indices h, k , one has (&,)I e x )> 0. (a) Show that the number p(u) (Problem I(b)) belongs to sp(u). (Note that for 151 > &I), if v c (u - 5 . I E ) - I , one has
+
:
5.
6.
:
m
(O;(eh)
I 4 = -"C (un(eh)I e.d5-"-' =O
and use Problem 7(b) of Section 9.15.) (b) Suppose in addition that for some integer n > I , there exist an integer k > 1 such that (u"(ek)I e b )= d > 0. Prove then that p(u) > d''" (observe that for every integer m > I , (unm(ek) I e r )> d"). (c) Suppose p(u) > 0 and that the point p(u) is a pole of the function 5 + u c . Prove
x 5" m
that there exists then an eigenvector x =
5" > 0 for every n. (Let N
en of u corresponding to p(u), such that
" = I
be the order of the pole p(u) of v c , and let
316
XI ELEMENTARY SPECTRAL THEORY
Show that (wN(eh)I ek)< 0 for every pair h, k , and w N # 0 by assumption, and use the fact that U W N = p(u)wN .) (d) Suppose that (u(el) I eA)> 0 for every pair of indices h, k (this, by (b), implies p(u) > 0), and in addition suppose that p(u) is a pole of v c . Show that p ( u ) is then a simple pole of v z . (Observe that ( d / d ( )- v c = v t and prove that, if one had N > I , then one would have wA = 0; this would imply (wN(ek)I eA)= 0 for every k ; using the relatidn uwN = p ( u ) w N , observe that if ( w N ( e k ) 1 eh) = 0 for one index h , then wN(eA) 0.) Prove that there exists an eigenvector z = = x(.en of N corresponding to p ( u ) and such that 1
=c~ “ e is, an eigenvector of the adj,oint u* corresponding to p ( u ) (cf. Section one has 7. > 0 for every n, or 7“ 0 for every (otherwise, one would get the contradictory inequality In I 7. I <x(.I 7.1, using a ” > 0 for every n. Next show that if y
n
=Z
11.5),
II
n
majoration of each 7” derived from p(u)T. =C qr(u(ek)1 en)).Conclude finally that all k
eigenvectors of u corresponding to p(u) are scalar multiples of z (exchange u and (“Theorem of Frobenius-Perron.”)
id*).
2. COMPACT OPERATORS
Let E, F be two normed (real or complex) spaces; we say that a linear mapping u of E into F is conTpact if, for any bounded subset B of E, u(B) is relatively compact in F. An equivalent condition is that for any bounded sequence (x,) in E, there is a subsequence (x,J such that the sequence ( ~ ( x , ~ ~ ) ) converges in F. As a relatively compact set is bounded in F (3.17.1), it follows from (5.5.1) that a compact mapping is continuous.
Examples (11.2.1) If E or F is finite dimensional, every continuous linear mapping of E into F is compact (by (5.5.1), (3.17.6), (3.20.16), and (3.17.9)).
(1 1.2.2) If E is an infinite dimensional normed space, the identity operator in E is not compact, by F. Riesz’s theorem (5.9.4). (11.2.3) Let I = [a, b] be a compact interval in R, E = gC(l) the Banach space ofcontinuouscomplex-valued functions in I (Section 7.2), (s, t ) -+ K(s, t ) a continuous complex-valued function in I x I . For any function f ’ E,~ the mapping r --t K(s, t ) f ( s )ds is continuous in I by (8.11.1); denote this
Jab
2 COMPACT OPERATORS
317
function by U f . Then the mapping f + Uf of E into itself is linear; we prove that it is compact. Indeed, if g = U j ; we can write, for to E I , t E I,
s(r>- d t o ) =
(1 1.2.3.1)
(K(s,
r ) - K(s, t o ) ) f ( s )d s .
.lab
As K is uniformly continuous in I x I (3.16.5),for any E > 0 there is a 6 > 0 such that the relation It - tol < 6 implies IK(s, t ) - K(s, t o ) [< E for any s E I ; hence, for a n y f in E
(11.2.3.2)
Ig(t) - g(t0)l
d E(b - a>llfll
by the mean-value theorem. This shows that the image U(B) of any bounded set B i n E is equicontinuozrs at every point to of I (Section 7.5); on the other hand, for any t E I, we have similarly Ig(t)l d kllf I/ if IK(s, t ) l d k / ( b - a ) in I x I . By Ascoli’s theorem (7.5.7),U(B) is relatively compact in E.
(11.2.4) With the same notations and assumptions o n K as in (11.2.3),let now F be the space of complex-valued regulated functions in I (Section 7.6), which is again a Banach space, when considered as a subspace of the space g,-(l);Uf’ is then defined as in (11.2.3)for any f E F, and the inequality (1 1.2.3.2)still holds. The argument in (11.2.3)then proves that U is a coinpact mapping of F into E.
(11.2.5) If’u,
L’
are two compact mappings of E into F, u + u is compact.
Let (x,) be a bounded sequence in E ; by assumption, there is a subsequence (x;) of (s,)such that (u(x;)) converges in F. As the sequence (x:) is bounded in E, there is a subsequence ( x i ) of (x;) such that (c(xi)) converges in F. Then by (3.13.10)and (5.1.5), the sequence (u(x1)+ xi)) converges in F. Q.E.D.
(1 1.2.6) Let E, F, El, F, be normed spaces, f’ a continuous linear mapping into E, g a continuous linear mapping of F into F,. Then,f o r any compact mapping u of E into F, u1 = g u o f is a compact niapping of El into F,. of El
For if B, is bounded in E 1 , f ( B l ) is bounded in E by (5.5.1), u ( f ( B , ) ) is relatively compact in F by assumption, and g(u(,f(B,))) is relatively compact in F, by (3.17.9).
318
XI
ELEMENTARY SPECTRAL THEORY
(11.2.7) I f u is a compact mapping of E into F, the restriction of u to any rector subspace El of E is a compact mapping of El into u(E,).
For by (11.2.6), that restriction is a compact mapping of El into F. If B is a bounded subset of El,-) is then a compact subset of F, and as __ - u(B) c u(E,), u(B) is relatively compact in u(E,). Example (11.2.8) With the same notations and assumptions on the function K as in (11.2.3), let now G be the prehilbert space defined by the scalar product
( f ) g ) = ~ a b f ( t ) g ( r ) don t the set gC(I)(6.5.1); we write the norm f)”’ = ~ ~ , to f ~ distinguish ~ a it from the norm l l f l i = supIf(t)l, and we still
(fI
f E l
with the norm llfll; the identity mapping f + f o f denote by E the space gC(l) E into G is continuous, since i l f i l z < ( b - a)’/’ * llfll by the mean value theorem; but it is not bicontinuous, nor is G a Banach space. The CauchySchwarz inequality (6.2.1) is written here
With the same notations as in (1 1.2.3), we therefore deduce from (11.2.3.1) and (11.2.8.1) that It, - t,l < 6 implies
Is(t1) - g(t2)I < E(b .llflla and similarly Ig(t)l < k(b . 1l.f 11’ for any t E 1. Hence, by the same
(1 1.2.8.2)
9
argument as in (11.2.3), f - Uf is a compact mapping of G into E; and as the identity mapping of E into G is continuous, ,f -+ U ’ is also a compact mapping of G into G by (11.2.6). (11.2.9.) Let E, F be two Banach spaces, E, (resp. F,) a dense subspace of E (resp. F), u a compact mapping of E, into F, , ii its unique continuous extension as a mapping of E into F (5.5.4). Then u“(E) c F, , and u” is a coiiipact mapping ofE into F, .
I t is immediate that any ball llxil < r in E is contained in the closure of any ball of center 0 and radius > Y i n E, (3.13.13) hence any bounded set in E is contained in the closure of a bounded set B in E, . But ii( f3) is contained in the closure in F of the set C(B) = u(B) by (3.11.4); now, u(B)
2 COMPACT OPERATORS
319
is relatively compact in F,, i.e. its closure in F, is compact, hence closed in F, and therefore equal to its closure in F. This shows that G(B) is contained in F, and relatively compact in that space. Q.E.D. (11.2.10) Let E be a normed space, F a Banach space, (u,) a sequence of nzuppings in Y ( E ; F) (Section 5.7) w~hichconverges to u in Y(E; F). Then, if every u, is compact, u is compact. Let B be any bounded set in E ; as F is complete, all we have to do is to prove that u(B) is precompact (3.17.5). Now B is contained in a ball /IxJId a ; for any E > 0, there is no such that n 2 no implies /Iu - u,II 6 6 / 2 4 and therefore (by (5.7.4)) IIu(x)- u,(x)II 6 ~ / for 2 any x E B. But as u,,(B) is precompact, it can be covered by finitely many balls of centers y j (1 6 j < m) and radius ~ / 2For . any x E B, there is therefore a j such that IIu,,(x) - yjll Q 4 2 , hence IIu(x) - yj/l 6 E , and the balls of centers y j and radius E cover u(B). Q.E.D. In particular, any limit in Y(E; F) of a sequence of mappings of finite rank is compact by (11.2.1) and (11.2.10). Whether conversely any compact mapping is equal to such a limit is still an open problem (see Problem 4).
PROBLEMS
Let E be a Banach space, A a bounded open subset of E, F a finite dimensional vector space. Show that for any p 3 1, the identity mapping f-f of the Banach space 9LP'(A) (Section 8.12, Problem 8) into 9Pp-l)(A)(the latter being replaced by %$' (A) for p I ) is a compact operator. (Use the mean value theorem and Ascoli's theorem.) Let u be a compact mapping of an infinite dimensional Banach space E into a normed space F. Show that there is in E a sequence (x,) such that IIxJ = 1 for every n, and lim u(x,) = 0. (Observe that there is a number a > 0 and a sequence (y") in E such that "+a,
l / y l l = 1 for every n, and lly,, - y, I/ > a for m # n (Section 5.9, Problem 3, and (3.16.1)), and consider the sequence (~(y,,)).) Conclude that if the image by u of the sphere S : /lx/l= 1 is closed in F, it contains 0.
Let E be a separable Hilbert space, (e,) a Hilbert basis of E. If u is a compact mapping of E into a normed space F, show that the sequence (u(e.)) tends to 0. (Use contradiction, and show that it is impossible that the sequence (u(e,)) should have a limit 0 # 0 in F.) If, conversely, F is a Banach space and the series of general term /lu(e,)lj2 is convergent, show that u is compact (use the Cauchy-Schwarz inequality to prove that the image of the ball l/xl/< 1 by u is precompact). Let F be a normed space having the following property: there exists a constant c > 0 such that, for anyfinite subset ( a i ) 1 6 i bof n F, and any E > 0, there exists a decomposition E = M N of E into a direct sum of two closed subspaces, such that M is finite dimensional, d(u,, M) < E for I < i < I ? , and if for any x E F, x = p ( x ) q ( x ) , where
+
+
320
XI
ELEMENTARY SPECTRAL THEORY
< c . d(x, M). Show that, under that assumption, any compact linear mapping of a normed space E into F is a limit in Y(E; F) of a sequence of linear mappings of finite rank (use the definition of precompact spaces). Show that any Hilbert space satisfies the preceding condition, as well as the spaces ( c d (Section 5.3, Problem 5 ) and I' (Section 5.7, Problem 1). 5. Let I = [ a ,b ] be a compact interval in R, K(s, t ) a complex valued function defined in I x 1, and satisfying the assumptions of Section 8.11, Problem 4. Show that if U is defined as in (11.2.3), U is still a compact mapping of E = %',(I) into itself. p ( x ) E M and q(x) E N, then llq(x)ll
3. T H E T H E O R Y OF F. RlESZ
We will need repeatedly the following lemma: (11.3.1) Let u be a continuous operator in a normed space E, v = 1, - 11, L, M two closed vector subspaces of E such that M c L, M # L, and v(L) c M. Then there is a point a E L n M such that llall < 1 and that, for any x E M, lMa>- u(x>Il 2 t .
c
By assumption, there is b E L such that b 4 M, hence d(b, M) = CY > 0. Let M be such that \lb - yll < 2a, and take a = ( b - y)/lib - y / l ; we have llall = 1, and, for any z E M, a - z = ( b - y - Ilb - yllz)/llb - yll; but as y + 116 - yllz E M , we have 116 - y - Ilb - yllzll 2 a, hence ]la - zil 2 for any z E M. But, for x E M, we have u(a) - u(x) = a - (x v(a) - ~ ( x ) )and , by assumption, x u(a) - v(x) E M ; hence our conclusion. y
E
+
+
+
(11.3.2) Let u be a compact operator in a normed space E, and let v = I, - u. Then: (a) the kernel v-'(O) isfinite dimensional; (b) the image v(E) is closed in E; (c) ti(E) hasjinite codimension in E; (d) if ~ ~ ' ( = 0 {) 0 } , then v is a linear honieomorphism of E onto v(E) (cf. (11.3.4)).
(a) For any x E N = v-'(O), we have u(x) = x , hence the image of the ball B: llxll < 1 in N by u is B itself; by assumption u(B) is relatively compact in E, hence in N since N is closed in E. But this implies that N is finite dimensional by Riesz's theorem (5.9.4). there is then a sequence (x,) in E, such that (b) Suppose y = lim ~(x,,) (3.13.13). Suppose first that the sequence (d(x,,, N)) is n-+ m
unbounded; then, by extracting a subsequence, we may suppose that
3 THE THEORY OF F. RlESZ
lirn d(x,, N)
=
+
00.
Let z,
= x,/d(x,
321
, N); it is immediate that d(z, , N) = 1,
fl+W
and therefore there is t, E N such that I/z, - t,l/ < 2. Let s, = z, - t,, and observe that by definition we have u(s,) = u(z,) = o(x,)/d(x,, N), and d(s,, N) = 1. From the assumptions we deduce at once that lirn u(s,) = 0. n+ w
But the sequence (s,) is bounded in E ; as u is compact, there is a subsequence (s,J such that (u(snk)) converges to a point a E E. As lirn (s, - u(s,)) = 0, we also have lirn s,, n-m
v(a) = lim u(s,,)
= a, = 0,
n+
w
hence, as x -+ d ( x , N ) is continuous, d(a, N)
=
1. But
and this contradicts the definition of N.
k-+ w
We therefore can suppose that the sequence (d(x,, N)) is bounded by a number M - I ; there is then a sequence x; such that x, - x; E N and Ilx;l\ 6 M ; as c ( x 3 = u(x,), we may suppose that IIx,II 6 M. Then as u is compact there is a subsequence (xnk) such that (u(xnk))converges to a point b E E ; as xnk- u(x,,) = ~(x,,) tends to y , (x,,,)tends to b + y and by continuity we have u(b + y ) = y , which proves that y E v(E), hence u(E) is closed. (c) T o say that u(E) has an infinite codimension in E means that there exists an infinite sequence (a,) of points of E such that a, does not belong to the subspace Vn-l generated by o(E) and by a,, . . . , a,-, for every n. Now each V, is closed since u(E) is closed (using (5.9.2)). By (11.3.1)we can define by induction a sequence (b,) such that 6, E V,, 6, $ V n - l , llb,/l 6 1, and IIu(b,) - u(bj)I/2 3 for any j 6 n - I . This implies that the sequence (u(b,)) has no cluster point, contradicting the assumption that u is compact. (d) In order to prove that 21 is a homeomorphism of E onto v(E) when V ' ( 0 ) = {0}, it is only necessary to show that for any closed set A c E, u(A) is closed in E (hence in o(E)) (3.11.4).But this is proved by exactly the same argument as in (b), replacing throughout E by A (and N by (0)).
(11.3.3) Under the same assumptions as in (11.3.2),define inductively N, = v - ' ( o ) , N , = U - l ( N k - l )for k > 1, F, = @), F, = V ( F k - 1 ) f o r k > 1. Then : (a) The N , form an increasing sequence of finite dimensional subspaces, the F, a decreasing sequence of finite codiniensional closed subspaces. (b) There is a smallest integer n such that N k + l = N , for k 2 n ; then Fk+l= F, for k Z n, E is the topological direct sum (Section 5.4)of F,and N,, and the restriction of 1: to F, is a linear homeomorphism of F, onto itself. (a) Define by induction
zl1
= 11, ti, = i I k - l
0
t'; I
claim that
ilk
= 1, - u,,
322
XI
ELEMENTARY SPECTRAL THEORY
induction on k , for U , and the result follows at once from the inductive hypothesis and from (11.2.6) and (11.2.5). Then by definition Nk = a ~ ' ( 0 and ) F, = zik(E), and our assertion follows from (11.3.2).
where Zlk =
(IE
uk
is
- Uk-1)
compact:
O
this
is
shown
by
( 1 -~ U ) = 1, - uk-1 - U -k uk-1
O
(b) Suppose Nk # Nk+1 for every k . We have U(Nk+,) C Nk for k 3 1 ; by (11.3.1), there would exist an infinite sequence (xk) of points of E such that xk E Nk, xk $ Nk-1, llxkll < 1 for k > 1 and IIu(xk) - U(Xj)II > f for any j < k . This implies that the sequence (u(xk))has no cluster point, contradicting the assumption that u is compact. Similarly, suppose Fk+l# Fk for every k . We have v(Fk) c F,+, for k > 1 ; by (11.3.1), there would exist an infinite sequence (x,) of points of E such that xk E Fk, xk $ Fk+l, llxkll < 1 for k 2 1, and IIu(xk) - U(Xj)II > for a n y j > k . This again implies contradiction, hence there exists a smallest integer m such that Fk+,= Fk for k 2 m . Next we prove that N, n F, = (0): if y E F, n N , , then there is x E E such that y = v,(x), and on the other hand v,(y) = 0; but this implies that tiZn(x)= 0, hence x E N,, = N, , and y = v,(x) = 0. By definition, we have F, c F, and v(F,) = F,; let us prove that F, = F,. Otherwise, we would have nz > n ; let z be such that z E F,-, c F,, and z $ F,; as U ( Z ) E F, = v(F,), there is a t E F, such that u(z) = v ( t ) , i.e. z - t E N, c N,; but as z - t E F,, we conclude that z = t , and our initial assumption has led to a contradiction. For each x E E we have v,(x) E F, = F,, and as u,(F,) = F, by definition of nz, there is y E F, such that v,(x) = u,(y), hence x - y E N, , and therefore E = F, + N , . This last sum is direct since F, n N, = ( 0 ) ; F, is closed and N, is finite dimensional, therefore (5.9.3) E is the topological direct sum of F, and N,. Finally, the restriction of I' to F, is surjective and its kernel is F, n N, c F, n N, = {0}, hence it is also injective. By (11.3.2(d)) that restriction is a linear homeomorphism of F, onto itself, and this ends the proof.
(11.3.4) Under the same assumptions as in (11.3.2), 11 is injectiile (i.e. v-'(O) = {0)), then v is surjective, hence a linear homeomorphism of E onto itself. For the assumptions imply that Nk = (0) for every k , hence n = 1 and N, is reduced to 0, therefore F, = E by (11.3.3) and the result follows from (11.3.3).
4
SPECTRUM OF A COMPACT OPERATOR
323
PROBLEMS
1. Let E, F be two Banach spaces, f a continuous linear mapping of E into F such that f(E) = F; then, there exists a number m > 0 such that for any y E F, there is an x E E for whichf(x) = y and Ilxll < mllyll (12.16.12). (a) If (y.) is a sequence of points of F which converges to a point b, show that there exists a subsequence ( y n k ) and , a sequence (x,) of points of E, which converges to a point a and is such that f(xk) = y,,, for every k . (Take (y,J such that the series of general term IIy.,+ - ynk/lis convergent.) (b) Let u be a compact mapping of E into F, and let u =f- u. Show that u(E) is closed in F and has finite codimension in F. (Follow the same pattern as in the proof of (11.3.2), using (a).) (c) Define inductively F1 = u(E), Fk+l = u(f-'(Fk)) for k > 1 ; show that there is an integer n such that = F, for k > n (same method). (d) Take E = F to be a separable Hilbert space, and let (en)nS1 be a Hilbert basis of E. Definefand u such that f(e,) = e n - 3for n > 4,f(e,) = 0 for n < 3, u(e,) = e.-z/n for n > 6, u(el)= u(e3)= 0, u(e2)= - e 2 , u(e4)= e l , u(e5) = e2 (e3/5). Define inductively NI = u-'(O), N,+,= u-'(f(Nk)) f o r k > I ; show that the N kare all distinct and finite dimensional. 2. Let E, F be two normed spaces,fa linear homeomorphism of E onto a closed subspace f(E) of F, u a compact mapping of E into F, and let u =f- u. (a) Show that u-'(O) is finite dimensional and u(E) is closed in F; furthermore, if u-'(O) = {O}, u is a linear homeomorphism of E onto u(E). (Follow the same method as in (11.3.2).) (b) Define inductively NI= u-I(O), N,+, = u-'(f(N,)) for k > 1 ; show that there is an integer n such that N k + = N,for k 2 n. (c) Give an example in which, when F1 = u(E), and F,,, = u(f-'(F,)) for k > 1 , the F, are all distinct (take for E = F a separable Hilbert space, and for f a n d u the adjoints (Section 11.5) of the mappings notedfand u in Problem l(d)). 3. Let E be a Banach space, g a continuous linear mapping of E onto itself such that Ilg11 ( 1 - 211g11)/2.) 4. In the space E = I' (Section 5.7, Problem 1; we keep the notations of that problem), ) e 2 k f 2 (k > O ) , f ( e , )= eo, f(e2n+l)= let f be the automorphism of E such that f ( e Z k = e2k-1 for k I , and let u be the compact mapping such that u(e,) = 0 for n # 1 , and u(el)= e o . If u =f- u, and the Fkand N,are defined as in (11.3.3), show that Na+l # Nkand F, + # F, for every k .
+
4. SPECTRUM OF A C O M P A C T OPERATOR
(11.4.1 ) Let u be a compact operator in a complex normed space E. Then: (a) The spectrum S of u is an at most denumerable compact subset of C , each point of which, with the possible exception of 0, is isolated; 0 belongs to S if E i s injinite dimensional.
324
XI
ELEMENTARY SPECTRAL THEORY
(b) Each number A # 0 in the spectrum is an eigenvalue of u. (c) For each A # 0 in S, there is a unique decomposition of E into a topological direct sum of two subspaces F(A), N(A) (also written F(A; u), N(A; u)) such that: (i) F(A) is closed, N(A) i s j n i t e dimensional; (ii) u(F(2)) c F(A), and the restriction of u - A . 1, to F(A) is a linear homeomorphism of that space onto itself; (iii) u(N(A)) c N(A) and there is a smallest integer k = k(A), called the order oj'A (also written k(A; u)),such rhat the restriction to N(A) of(u - A . I , ) k is 0. (d) The eigenspace E(A) of u corresponding to the eigenvalue A # 0 is contained in N(A) (hence finite dimensional). (e) I f A, p are two difeerent points of S , distinct from 0 , then N(p) c F(A). (f) I f E is a Banach space, the function (-+ ( u - ( I,)-', which is dejned and analytic in C - S , has a pole of order k(A) at each point A # 0 of S. Let A # 0 be any complex number; as A-'u is compact, we can apply the Riesz theory (Section 11.3). By (11.3.4), if A is not an eigenvalue of u, I - A- ' u is a linear homeomorphism of E onto itself, and the same is true of course of u - A . 1, = -A(l, - A - l u ) , i.e. A is regular for u, which proves b). Suppose on the contrary A is an eigenvalue of u ; then the existence of the decomposition F(A) + N(A) of E with properties (i), (ii), (iii), follows from (11.3.3), as well as (d) (E(A) is the kernel noted N, in (11.3.3)). To end the proof of (c), we need only show the uniqueness of F(A) and N(A). Suppose there is a second decomposition E = F' N' having the same properties, and write v = u - A . 1., Then, any x E N' can be written x = y + z where y E F(A), z E N(A); by assumption there is h > 0 such that vh(x)= 0, hence vh(y) = 0; as the restriction t o F(A) of vh is a homeomorphism by assumption, y = 0 and x E N(1). This proves that N' c N(A), and a similar argument proves N(1) c N'. Next, if x = y + z E F' with y E F(n), z E N(A), we have vk(x) = vk(y), hence vk(F')c F(A); but as v(F') = F', this implies F' c F(A), Denote by u l , u2 the restrictions of u t o F(A) and N(A), respectively. From the relation (u2 - A . l N ( n ) ) k = 0, it follows by linear algebra (A.6.10 and A.6.12) that there is a basis of N(1) such that the matrix of u2 - I . IN(,, with respect to that basis is triangular with diagonal 0; if d = dim (N(A)), the determinant of u2 - ( lN(l)is therefore equal to ( A - ( ) d and this proves that u2 - ( . lN(n) is invertible if 5 # A. Let us prove on the other hand that u, - ( . lF(A)is invertible for ( - A small enough: we can write u1 - [ . 1 F(i) = v1 + (A - 5) . IF(,, with u1 = u1 - A . We know by (c) that V , is invertible; by (5.7.4), we therefore have I ~ V ; ~ ( X ) I I < IIv;' II * /Ixll in F(A), which can also be written IIu,(x)I/ 2 c . l/xil with c = llv;'/l-l. Now if i # 0 and
+
4 SPECTRUM OF A COMPACT OPERATOR
325
u1 - ( . IF(,) is not invertible, this implies, by (b) (applied to F(A) and u l , using (11.27)) that there would exist an x ;f; 0 in F ( I ) such that u l ( x ) = ('x, hence li- A1 . llxll = ilol(x)j/ 3 c . IIxll, which is impossible if - A1 < c. This shows that for [ # 0, i# I , and - I )< c, u - i .I, is invertible (since its restrictions to F(A) and N(A) are), i.e. ( is not in S; therefore all points A # 0 in S are isolated, and S is at most denumerable. By (b), for each A # 0 in S, there is x # 0 in E such that u(x) = Ax, hence 121 . ~~x~~ < 1 ~ . ~~x~~ ~ ~ by1 (5.7.4), and 111 < IIull, which proves S is compact. To end the proof of (a), suppose E is infinite dimensional; if u were a homeomorphism of E onto itself, the image u(B) of the ball B: ~1~~~ < 1 would be a neighborhood of 0 in E, and as it is relatively compact in E, this violates Riesz's theorem (5.9.4). If p is a point of S distinct from 0 and A, and x E N(p), we can write x =y z with y E F ( I ) , z E N(I). We have seen above that the restriction of w = u - /i . I, to N(A) is a homeomorphism; as wh(x)= 0 for h large enough, and wh(y)E F(I), wh(z)E N(A), we must have wh(y)= wh(z) = 0, which proves statement (e). in C - S follows If E is a Banach space, the analyticity of ( u - i. I,)-' from (11.1.2). With the same notations as above, A is not in the spectrum of u l , hence (by (11.2.7)) (ul - . IF(,,)-' is analytic in a neighborhood of A ; in particular, there are numbers p > 0 and M > 0 such that
+
for x E F(A) and l i - AI < p. On the other hand, we can write u2 -5 . I,(,, = (A - i). I,,,)+ 1i2 with u2 = u2 - A . I,(,, , and we know that for 5 # A, u2 -( . I,(,, is invertible; moreover, we can write
since 11; = 0. From this it follows that there is a number M' > 0 such that l i - 21k . l l ( u ~- i . IN(~.J-'(x)Il< M'llxll for l i - A1 < p, I # A and for any x E N(1). Now any x E E can be written x = y z with y E F(A), z E N(A), and there is a constant a > 0 such that llyll < a/lxl/and llzll < allxll (5.9.3); therefore we see that, for li - I1 < p, i# A, and any ~ E E we , have
+
l i - I l k ll(u - i. 1E)-'(X)Il
< 4 M p k + M')IIxII. < a ( M p k + M') for if A
In other words, l i - Alk . ll(u - i .l,)-'Il and - A1 6 p ; by (9.1.5.2), this implies that A is a pole of order < k for ( u - i* I,)-'. But by definition there is an x E N ( I ) such that z~:-'(x) # 0, hence (i-A)"-'((u - i I,)-'(X)) is not bounded when i# I tends to A, and this proves that A is a pole of order k , and ends the proof of (1 1.4.1). 9
326
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ELEMENTARY SPECTRAL THEORY
We say that the dimension of N(1) is the algebraic multiplicity of the eigenvalue A. of u, the dimension of the eigenspace E(1) its geometric n d tiplicity; they are equal if and only if k(1) = 1 ; when E is a Banach space, this is equivalent to saying that 1 is a simpre pole of ( u - 5 . l&'. (11.4.2) Let E be a Banach space, E, a dense subspace of E, u a compact operator in E,, ii its unique continuous extension to E. Then /he spectra of u and ii are the same, and for each eigenoalue A # 0 of u, N(A, u) = N(1, u"), E(1, u) =: E(1, ii) and k(1, u) = k(1, ii).
We know that u" is compact and maps E into E, , by (11.2.9); if A # 0 is an eigenvalue of u", any eigenvector x corresponding to 1 is such that x = 1-'C(x) E E,, hence A is an eigenvalue of u, and E(1, u") c E(1, u ) ; the converse being obvious, we have sp(ii) =sp(u) and E(A, u) = E(A, u") c E, for each eigenvalue A # 0. Considering similarly the kernels of ( u - A . IEJk and of its extension (ii - A . I E)k we see that they are equal, hence k(1, u) = k(A, u") and N(A, u) = N(1, u") c E, .
PROBLEMS
1. Let E be a coniplex Banacli space, u a compact operator in E; we keep the notations of (11.4.1), and in addition, we write pi (or p i , J and qA=- l E - p i the projections of E onto N(h) and F(h) in the decomposition of E as direct sum F(h) i N(h). (a) Show that -pi is the residue of the meromorphic function (u - 5 . I E ) - l at the pole h, for every A E sp(u) such that h .?; 0. (b) If h1, . .. , h, are distinct points of the spectrum sp(u), show that the projections p i , ( I < j < r ) commute, and that p i , I ' . . -+-pi, is the projection of E onto N(h,) -I . . . -t N(hJ in the decomposition of E as direct sum of that subspace and of F(hl) n F&) n . . . n F(h,). 2. Let E be an infinite dimensional complex Banach space, u a compact operator in E, (uJngl a sequence of compact operators in E, which converges to I I in the Banach space Y(E). (a) Prove that for any bounded subset B of E, the union u.(B) is relatively compact
u"
in E. (Show that it is precompact.) (b) If A E C does not belong to sp(u), show that there is an open disk D of center h and an integer no such that, for n > n o , the intersection sp(u,,) n D = 0(use (8.3.2.1)) and ( [ i n- 5 . IE)-' converges uniformly to (u - 5 . I E ) - I for 5 E D. (c) Let (p.) be a sequence of complex numbers such that p.~sp(u,) for every n ; such a sequence is always bounded. If h is a cluster point of ( p J , show that h E SP(N). (One can assume that h = lim pn 0; there is then x. E E such that llxn/l= 1 and u.(xn) = h.x,; use then (a).)
"'m
+
4
SPECTRUM OF A COMPACT OPERATOR
327
(d) Conversely, let h # 0 be in sp(u). Show that for each n there is (at least) a number E sp(u,) such that h = lim p.. (Otherwise, one can assume that there is an open
p.
"+ m
disk D of center h and radius Y, such that D n sp(u) = {A} and D n sp(u.) = @ (extract from (u,) a suitable subsequence). Let then ^J be the road t + A Yeirdefined in [0,277];consider the integral
+
sy(lln 5 -
* IE)-l(5
-
d[
=0
for k
>0
and use (b) to obtain a contradiction.) (e) Let h # 0 be in sp(u), and let D be an open disk of center h and radius r such that D n sp(u) = {A}; there exists no such that, for n > no the intersection of sp(un) and of the circle 15 hl = r is empty (use (c)). Let pI,... , p, be the points of D n sp(u,),
c k(pj;
~
and write k , =
uJ. Show that there exists nl such that, for n
j=l
> n l , k, > k(h;u).
(Use the same method as in (d), multiplying (u, - 5 . by a suitable polynomial in 5 of degree k , .) Give an example in which k, > k(h; u) for every n. (f) With the notations of (e), let p =pi.", pn=
c pPJ,""; show that
j =
1
limp, = p
n- m
in the Banach space 9 ( E ) (use (b), and Problem I). Deduce from that result that there exists n2 such that, for n 2 n 2 , N, = N(p, ; u,) . . . N(p, ; u.) is a supplement to F(h; u) in E. (Suppose n is such that IIp --p.li < 1/2; if there was a point x, E F(h) n N. such that /lx, /I = I , then the relations p(x.) = 0, p,,(x,) = x, would contradict the preceding inequality. Prove similarly that the intersection of N(h; u ) and of the subspace F(pI ; u,) n . . . n F(p,;u,) is reduced to 0.) 3. Let u be a compact operator in an infinite dimensional complex Banach space E, and let P(5) be a polynomial without constant term; put v = P(u). Show that the spectrum sp(u) is identical to the set of numbers P(h), where h E sp(u); furthermore, for every p E sp(v), N(p; u ) is the (direct) sum of the subspaces N(h,; u ) such that P(&) = p, and F(p; u ) the intersection of the corresponding subspaces F(&; u). (Let V be any closed subspace of E, such that u(V) C V, and let uv be the restriction of u to V. Show that there is a constant M independent of V and n, such that ll(P(uv))'ll < M" lIuqI1. Apply that remark and Problem 1 of Section 11 .I, taking for V a suitable intersection of a finite number of subspaces of the form F(h; u).) 4. Let E be a separable Hilbert space, (en)n3oa Hilbert basis of E. Show that the operator u defined by u(e,) = e.+l/(n 1 ) for n > 0 is compact and that sp(u) is reduced to 0 (more precisely, u has no eigenualue). 5. Let u be a continuous operator in a complex Banach space E. A Riesz point for u is a point h in the spectrum sp(u) such that: (I) h is isolated in sp(u); (2) E is the direct sum of a closed subspace F(h) and of a finite dimensional subspace N ( h ) such that u(F(h)) c F(h), u(N(h))c N(h), the restriction of u - h . 1, to F(h) is a linear homeomorphism and the restriction of u - h . 1, to N ( h ) is nilpotent. (a) If h and p are two distinct Riesz points in sp(u), show that N(p) C F(h), and F(h) is the direct sum of N(p) and F(h) n F(p). (b) A Riesz operator u is defined as a continuous operator such that all points f 0 in the spectrum sp(u) are Riesz points. For any E > 0, the set of points h E sp(u) such that 1x1 > E is then a finite set { p i ,. . ., p , } ; let pz be the projection of E onto N(pJ in the decomposition of E into the direct sum N(pJ -t F(pJ ( I < i < Y), and let
+ +
+
u =u -
2u
i=l
0
p i . Show that sp(v) is contained in the disk
Problem I ) that lim I1u"Il"" n-m
< E.
151 < E , hence (Section 11 .I,
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ELEMENTARY SPECTRAL THEORY
(c) In the Banach space Y(E), let X be the closed (11.2.10) subspace of all compact operators. Show that, in order that u E Y(E) be a Riesz operator, it is necessary and sufficient that lim (d(u", Y ) ) ' '= " 0. (To prove that the condition is necessary, use n-rm
(b), observing that u" = u" i- w,, where w, is an operator offinirerank, hence compact. To prove that the condition is sufficient, use the result of Problem 3 of Section 11.3, which can be interpreted in the following way: if ilgIl< 4, then either h = 1 does not belong to sp(y u ) or is a Riesz point for y -t u.)
+
5. C O M P A C T OPERATORS IN H I L B E R T SPACES
Let E be a prehilbert space, u an operator in E. We say u has an adjoint if there exists an operator u* in E such that (11.5.1) for any pair of points x,y in E. It is immediate that the adjoint u* is unique (when it exists), and (by Section 6.l(V)) that then (u*)* exists and is equal to u. It is similarly verified that when the operators u and v have adjoints, then u + v, AM, and uv have adjoints respectively equal to u* + v*, Xu*, and u*u*. (11.5.2) If u is continuous and has an adjoint, then u* is continuous and lJu*JJ = JIuJ)in Y(E). Zf E is a Hilbert space, every continuous operator in E has an adjoint. From (11.5.1) and the Cauchy-Schwarz inequality (6.2.4) we deduce for any pair x, y ; taking x = u*(y), we get Ilu*(y)II d llull . llyll for any y E E, which proves the continuity of u* and the inequality JIu*JJ d JJuJ1; the converse inequality is proved by interchanging u and u* in the argument. If E is a Hilbert space and u is continuous, then, for any Y E E , the linear form x-+ (u(x)Iy) is continuous, and by (6.3.2) there exists a unique vector u*(y) such that (11.5.1) holds. From the uniqueness of u*(y), we conclude that u* is linear, hence the adjoint of u. The second statemento f (11.5.2) does not extend to prehilbert spaces. An operator u in a prehilbert space E is called self-adjoint (or hermitian) if it has an adjoint and if u* = u ; the mapping (x, y ) -+(u(x) I y ) = ( u ( y ) I x ) is
then a hermitian form on E; the self-adjoint operator u is called positioe (resp. nondegenerate) if the corresponding hermitian form is positive (resp. nondegenerate); one writes then u > 0. For any operator u having an adjoint, u + u* and i(u - u*) are self-adjoint operators.
5
COMPACT OPERATORS IN HILBERT SPACES
329
(1 1.5.3) (i) If a continuous operator u in a prehilbert space E has an adjoint, = then u*u and uu* are selfadjoint posiiive operators, and j(u*uI( = (Juu*(( llu\lz = 11u*11’. In particular, i f u is self-adjoint, I I U ’ I / = 1 1 ~ 1 1 ~ . (ii) If P is the orthogonal projection of E on a complete vector subspace F (Section 6.3), P is apositive hermitian operator. Conversely, i f E is a Hilbert space, every continuous operator P in E which is hermitian and idempotent (i.e. P2 = P) is the orthogonal projection of E on the closed subspace P(E) (these operators are called the orthogonal projectors in LZ(E)).
(i) The fact that u*u and uu* are self-adjoint follows from the relations (u*)* = u and (uv)* = v*u*; moreover (u*u(x) I x) = (u(x) I u(x)) 2 0 for any x E E, and it is proved similarly that uu* is positive. Further this last relation shows that IIu(x)11’ < IIu*u(x)II . llxll by Cauchy-Schwarz, hence (by (5.7.4)) llul12 < IIu*uII. On the other hand, /Iu*uII < IIu*Il . llull = l\ull’ by (5.7.5) and (1 1.5.2), and this concludes the proof of (i). (ii) If P is the orthogonal projection of E on a complete subspace F, then (P . x l y - P y ) = 0 for x E E, y E E, hence (P . x Iy) = ( P . x IP - y ) = (x I P .y ) , which proves that P is hermitian, and it is positive since ( P x I x) = ( P x I P x) 2 0.Conversely, suppose E is a Hilbert space, and P2 = P = P*; as the then, for all x,y in E, ( P . x I y - P . y ) = ( x I P . y - P ’ . y ) = O ; relation y = P x implies P * y = P2 . x = P * x = y , P(E) is the kernel of 1, - P, hence is a closed vector subspace; furthermore, for any y E E, y - P y is orthogonal to every P . x,in other words to P(E), which proves (ii). (1 1.5.4) If E is a Hilbert space, the adjoint of any compact operator u in E is a compact operator. As E is complete, it will be enough to prove that the image u*(B) of the -
ball B: llyll < 1 isprecompact. Let F = u(B), which is a compact subspace of E, and consider, in the space %,-(F) (Section 7.2) the set H of the restrictions to F of the linear continuous mappings x -+ (x 1 y ) of E into C,where y E B; we prove that H is relatively compact in V,(F). Indeed, we have I(x - x’I v ) ~ < IIx - x’/I by the Cauchy-Schwarz inequality, since llyll < !, which shows that H is equicontinuous; on the other hand F is contained in the ball llxll < IIuII, hence I(x Iy)l < llull for anyy E B and a n y x E F ; Ascoli’s theorem (7.5.7) then proves our contention. Therefore, for any E > 0, there exist a finite number of points y j (1 < j < m) in B such that for any y E B, there is an indexj such that I(u(x) I y - yj)l < E for any x E B. But by (1 1.5.1 ) this last inequality is written I(x I u*(y) - u*(yj))I < E , and either u*(y) = u*(yj) or we can take x = z/llzll, where z = u*(y) - u*(yj);we therefore conclude that I/ u*(y) - u*(yj)II < E , and this ends the proof.
330
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ELEMENTARY SPECTRAL THEORY
Note that the proof that u*(B) is precompact still holds when E is not complete; but it can happen that in a prehilbert space E, a compact operator has an adjoint which is not compact.
(11.5.5) Let u be a compact operator in a complex preliilbert space E , having an adjoint u* which is compact. Then : (a) The spectrum sp(u*) is the image of sp(u) by the mapping 5 + [. (b) For each A # 0 in sp(u), k(A; u) = k(X; u*). (c) I f u = u - 1 * 1 , , then u*(E) is the orthogonal supplement (Section 6.3) of v-'(O) = E(A; u), and the dimensions of the eigenspaces E(A; u) and E(X; u*) are equal. (d) The subspace F(X; u*) is the orthogonal supplement of N(A; u), and the dimensions of N(A; u) and N(X; u*) are equal.
We have u* = u* - X l,, hence (u(x)I y ) = (x I u*(y)) from (11.5.1), and therefore the relation u(x) = 0 implies that x is orthogonal to the subspace v*(E). Now by (11.4.1) applied to u*, u*(E) is the topological direct sum of F(X; u*) and of the subspace u*(N(X; u*)) of N(X; u*), and from linear algebra (A.4.17) it follows that the codimension of u*(E) is equal to the dimension of u*-'(O) = E(X; u*); hence we have dim E(A; u) < dim E(X; u*). But u = (u*)*, hence we have dim E(A; u) = dim E(X; u*); furthermore, the orthogonal supplement of E(A; u) contains u*(E) and has the same codimension as v*(E), hence both are equal, which proves (c). This also shows that for any eigenvalue A # 0 of u, X is an eigenvalue of u*, and as the converse follows from the relation u = (u*)*, we have also proved (a). The same argument may be applied to the successive iterates vh of u, and shows that the image of E by u * ~= (oh)* is the orthogonal supplement of the kernel of vh. Using (11.3.2), (11.4.1), and the relation u = (u*)*, this immediately proves (b) and (d). Theorems (1 1.4.1) and (11.5.5) can be translated into a criterion for the solutions of the equation u ( x ) - Ax = y : (1 1.5.6)
Under tlze assumptions of (11.5.5) :
(a) If2 is riot in the spectrunz of u, the equation u(x) - Ax = y has a unique solution in E for every y E E. (b) IfA # 0 is in the spectrum of u, a necessary and suficient condition for y E E to be such that the equation u(x) - Ax = y have a solution in E is that y be orthogonal to the solutions of the equation u*(x) - Ax = 0.
5 COMPACT OPERATORS IN HILBERT SPACES
331
For a finite dimensional space, this reduces to the classical criterion for existence of a solution of a system of scalar linear equations.
(11.5.7) Let u be a compact self-adjoint operator in a complex Hilbert space E. Then: ( a ) Every element of the spectrum sp(u) is real and k(A) = 1 for every eigenvalue A # 0 of u. (b) If A, p are two distinct eigenvalues of u, the eigenspaces E(A) and E(p) are orthogonal. (c) Let (p,) be the strictly decreasing (finite or infinite) sequence of eigenvalues > 0 , (v,) the strictly increasing (finite or infinite) sequence of eigenvalues p , whence it (11.1.2) the mapping [ -+ (u - [ lE)-' is analytic for follows at once that the mapping 5 + ( I E - tu)-' is analytic for < l/p. Now, for 5 in a sufficiently small neighborhood of 0, the power series m
n=O
runconverges to ( I E - @)-I
in 9 ( E ) (8.3.2.1); by (9.9.4) that power
< l/p. Furthermore, for each r such series converges for every { such that that 0 < r < l/p, if M is the maximum of Ij(lE - tu)-' 11 for 151 = r, the Cauchy inequalities (9.9.5) yield IIu"lI < M/r" < Mp". In particular, if we use (11.5.3),
332
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ELEMENTARY SPECTRAL T H E O R Y
we get here 1 1 ~ 1 1 ~ 0 , and any x E E, llu"(x)112 < l l ~ " - ~ ( x )I!u"+l(x)/l Schwarz). 0
.
5 COMPACT OPERATORS IN HILBERT SPACES
337
+
(b) Suppose E is a Hilbert space, and u is a compact self-adjoint operator. If u(x) 0, show that u"(x) # 0 for any integer n > 0, and that the sequence of positive numbers m, = llu"+'(x)ll/llu"(x)l~ is increasing and tends to a limit, which is equal to the absolute value of an eigenvalue of u. Characterize that eigenvalue in terms of the canonical decomposition of x ; when does the sequence of vectors u"(x)/ llu"(x)11 have a limit in E ? (Use (11.5.7).) 11. Let u be a compact self-adjoint operator in a complex Hilbert space E, and let f b e a complex valued function defined and continuous in the spectrum sp(u). Show that there is a unique continuous operator u such that (with the notations of (11.5.7)), the restriction of u to E(pk) (resp. E(vk), E(0)) is the homothetic mapping y -f(pk)y (resp. y + f ( v , ) y , y + 0). This operator is written f ( u ) ; one has ( f ( u ) ) *=.f(u). If g is a second function continuous in sp(u), and h = f + g (resp. h = f g ) , then h(u) = f ( u ) -t- g(u) (resp. h(u) = f ( u ) g ( u ) ) . In order thatf(u) be self-adjoint (resp. positive and self-adjoint), it is necessary and sufficient that f ( 6 ) be real in sp(u) (resp. f(5) 0 in sp(u)); in order that f ( u ) be compact, it is necessary and sufficient that f ( 0 ) = 0. 12. Let u be a compact positive hermitian operator in a complex Hilbert space E. Show that there exists a unique compact positive hermitian operator u in E such that v2 = u ; v is called the square root of u.
Let E be a separable complex Hilbert space, (e,Jnbl a Hilbert basis of E. Let u be the compact operator in E defined by u ( e l )= 0, u(e.) = e n W l / nfor n > 1 . Show that there exists no continuous operator u in E such that u2 = u. (Observe first that H = u*(E) -is a closed hyperplane orthogonal to el, and that it is contained in H' = u*(E); as H' is orthogonal to x l= u(el), conclude that necessarily x1 = 0; next consider x2 = u(e2), and observe that u(v(e2))= 0, hence necessarily x 2 = hel, where is a scalar; but this implies x2 = 0, hence u(e2)= 0, a contradiction.) 14. Let E be a separable complex Hilbert space, ( e J n b o a Hilbert basis, u the compact positive hermitian operator in E defined by u(eo)= 0, u(e.) = e./n for n > 1 .
13.
The point a =
x (e,/n) does not belong to u(E). Let Eo be the dense subspace of E m
+
n=1
which is the direct sum of u(E) and of the one-dimensional subspace C(eo a ) . Show that the restriction u of u to Eo is a compact positive hermitian operator which is nondegenerate, although its continuous extension d = u to E is degenerate; furthermore, in the canonical decomposition (11.5.8) of the vector eo a E Eo , the summands do not all belong to Eo .
+
Let U be a compact operator in a complex Hilbert space E, and denote by R and L the respective square roots (Problem 12) of the compact positive hermitian operators U * U and UU*,respectively. Show that there exists a unique - continuous operator V in E, whose restriction to F = R(E) is an isometry onto U(E), whose restriction to the orthogonal supplement F' to F is 0, and which is such that U = VR (observe that llUxll= l l R x l l f o r e a c h x ~ E ) . P r o v ethat R = V * U = R V * V , a n d L = VRV*. (b) Let (m,) the full sequence of strictly positive eigenvalues of R, and (a,) a corresponding orthonormal system (Problem 8). If h, = Va,, show that (b.) is a n orthonormal system, and that, for any x E E, Ux - = x m . ( x I a&, where the series on the
15. (a)
right-hand side is convergent (if R,x (11.5.7), that lim IlR n-r m
-
"
= k= 1
"
mA(xIak)ak, show, using the proof of
R,I/ = 0, and apply (a)). Deduce from that result that (cL.)
is also the full sequence of strictly positive eigenvalues of L , and that (b,) is a corresponding orthonormal system. The sequence (m,) is also called the full sequence of singular values of U.
338
XI
ELEMENTARY SPECTRAL THEORY
(c) Let (p.) be the sequence of distinct eigenvalues # 0 of U,arranged in such an order that 1p.l > (pn+llfor every n for which is defined; let d, be the dimension of N(pn), and let (h.) be a sequence such that h, = pI,Ih,( > lh,,+ll for every n for which is defined, and for each k for which px is defined, the indices n for which h, = pk form an interval of N having dx elements. Show that, for each index n such that h,, and a, are defined, N(pk)for 1
< k < r , and
n (h,I < n a , . (Let V be the (direct) sum of the subspaces n
I= I
i=1
let Uv be the restriction of U to V; show that there is in V a Hilbert basis (eJ)lsJs, such that (U(eJ)Ie x )= 0 for k >j ; for n < m, if W, is the subspace of V having e l , ..., en as a basis, let U. be the restriction of U to W,, and let P,,be the orthogonal projection of E on W,. Show that
n lhjI2 is
j= 1
equal to the determinant of U ~ U=, P, U *UP,, and apply Problem 9(a).) (d) Let T be an arbitrary continuous operator in E, and let (y,,) (resp. (6,)) be the full sequence of singular values of UT (resp. T U ) . Show that y. < a. IITll (resp. 6, < a. IITll) for all values of n for which a,, y. and 6, are defined (if S = TU, observe that S * S < IITIIZU*Uand use Problem 8(d)). (e) Suppose T is also a compact operator, and let (fin) be the full sequence of its singular values. Show that n" y J < J=l
(n" a,)( fi6,) J=1
J=I
for all values of n for which
a,, fin, and ynare defined (apply Problem 9(b)). 16. Let E be a complex Hilbert space, (an)a sequence of points of E, (A,) a sequence of real numbers. Show that if the series u(x) = h,(x I a&. is convergent in E for n
every x E E, u is a hermitian operator in E. The convergence condition is always satisfied if the series of general term h,l~anllzis absolutely convergent. If in addition (a,) is a n orthonormal system, the convergence condition is satisfied if the sequence (A,) is bounded. When the h. are 2 0 and the convergence condition is satisfied, u is a positive hermitian operator. 17. Let E be a complex Hilbert space, Eo a dense vector subspace of E; ( x l y ) and //XI/ denote the scalar product and the norm in E. Suppose a second norm //xllois given on E,, for which E, is a Banach space, and is such that, for x E E,, one has llxll < a .Ilxllo, where a is a constant (in other words, the identity mapping lE0 of Eo with the norm /lxllo,into Eo with the norm IlxlI, is continuous). (a) Let U be a hermitian operator in Eo (which a priori is not assumed to be continuous); show that if U is continuous for the norm llxllo, it is also continuous for the norm llxll. (If I/U//o is the norm in 2'(EO) when Eo is given the norm Ilxll,, show that, for every integer n , one has, for x E Eo ,
Use the inequality IIUkxllz< llxll. IIU2k~II(Problem lO(a)).) (b) Let 0 be the continuous extension of U to E, which is a hermitian operator in E. Show that the spectrum of 0 is contained inthespectrumof U(when U isconsideredas an endomorphism of the Banach space Eo for the norm llxllo). (If 5 is a regular value for U,observe that ( U - 5 . l E 0 ) - I can be extended by continuity to E, using (a).) (c) Every eigenvalue h of U is real; if V = U - A . lE0, and if E(h)= V - ' ( O ) is finite dimensional in Eo, V(E,) is supplementary and orthogonal to E(A) in the prehilbert space Eo, and in addition it is closed for the norm llxll,; it follows from (12.16.8) that the restriction of V to V(E,) = F(h) is a linear homeomorphism of
5
COMPACT OPERATORS
IN
339
HILBERT SPACES
F(h) onto itself for the norm I/x/Io.Show that if p = 0 - h . l,, then P-l(O) = E(h) and P(E) = in E (apply (a) to the inverse of the restriction of V to F(h)). (d) Deduce from (c) that if U (or a power of U ) is a compact operator in Eo (for the norm Ilxllo),then 0 is a compact operator. (If (A,) is the sequence of eigenvaluesof U, deduce from (b) and (c) that the subspace of E orthogonal to all the subspaces E(h,) is the kernel of 0.) 18. Let E be an infinite dimensional complex Hilbert space. For a positive hermitian operator T i n E, the following conditions are equivalent: (1) T(E) is dense in E; (2) T-'(O) = {O}; (3) (Tx I x) > 0 for any x # 0 (use the Cauchy-Schwarz inequality applied to (TxJy)); (4) Tis nondegenerate. We say that a continuous operator Uin E is quasi-hermitian if there exists a nondegenerate positive hermitian operator T such that TU= U*T. (a) Show that every eigenvalue of U is real; if V = U - h . l E and if V-'(O) is finite dimensional, then E is a direct topological sum of V(E) and V-'(O), and h is a simple pole of (U - 5 . lE)-'. (Consider the Hilbert space obtained by completing E for the scalar product (Tx I y ) , and apply Problem 17.) a: = 1. (b) Let (01") be an infinite sequence of distinct real numbers such that
"
Let E be the Hilbert sum of a sequence of finite dimensional Hilbert spaces En such that dim(E,) = n (n 2 1). In En, let ( e i J l be a Hilbert basis, U, an operator in En such that Uneln= cciein ei+,,"for i < n - 1 and U,e,, = tL,e,, . Prove that llUnl.il< 2, but for any complex number 5 such that 151 = 1, ll(U, 5 . lEn)-'ll 2 44;. There is a unique continuous operator U on E whose restriction to each En is Un; show that U is quasi-hermitian and that the CL, are its eigenvalues, but that its spectrum contains the circle 151 = 1. (c) Let U be a compact quasi-hermitian operator. Prove that if U # 0, the spectrum of U cannot be reduced to the point 0, the eigenvalues hn # 0 of U are simple poles of (U- 5 . I,)-' and N(h,; U )= E(&; U ) and T(E(h,; U ) )= E(h.; U *). Furthermore, the intersection of the subspaces F(h,; U ) is equal to U -'(O), and the intersection of U-'(O) and U(E) is reduced to 0 (method of (a)). (d) Suppose E is separable, and let U be a compact continuous operator in E having the following property: there is a sequence (h,) of real eigenvalues of U * such that the sum of the E(X,; U * ) is dense in E. Prove that U is quasi-hermitian. (Show that (for a suitable k(n)), and there is in E a total sequence (b.) such that U*b, = hkCn)bn a,(x I b,)b, with suitable a. > 0; cf. Problem 16.) define T such that Tx
+
+
=I n
(e) Suppose E is separable, and let U be a compact operator in E satisfying the following conditions: (1) all the eigenvalues An of U are real and k@.; U )= I for every n ; (2) the intersection of the subspaces F(hn;U)is equal to U-'(O); (3) the intersection of U -I(O) and U(E) is (0). Prove that U is quasi-hermitian (use (11.5.5) and (d)). (f) Suppose E is separable, and let (eJnbobe a Hilbert basis for E. Prove that the operator U defined by Ue2.
= 0,
Ue2,+
=
nfl
+
for all n 2 0, is compact and quasi-hermitian, but the sum U -I(O) U(E) (which is an algebraic direct sum) is not a topological direct sum (cf. Section 6.5, Problem 2). m
(g) With the same notations as in (f), let U be the operator defined by Ueo =
e,/n,
"=I
Ue, = e,/nZ for n 2 1 ; using (e), show that U is compact and quasi-hermitian, but the sum U-'(O) U(E) is not dense in E; conclude that U * is not quasi-hermitian.
+
340
XI ELEMENTARY SPECTRAL THEORY
19. Let E be a Hilbert space, U a continuous operator in E. Suppose there is an element n f 0 in E such that: ( I ) the elements a, = U"a for n > 0 (with Uon = a ) form a total sequence in E; (2) the image of E by the hermitian operator V = U U * is the onedimensional subspace D = K a . Let Eo be a closed vector subspace of E not reduced to 0 and such that U(Eo) c Eo; let Uo be the restriction of U to Eo , and Po the orthogonal projection of E onto Eo . (a) Show that Eo cannot be orthogonal to D . (Observe that for any y orthogonal to D, U y = - U * y , and conclude that if Eo was orthogonal to D, one would have U p = (- I)"U*"yfor every y E Eo; show that this contradicts the assumption that the U"n form a total sequence in E.) (b) Prove that for any x E Eo , U:x = Po U,*x = Po U * X . (c) Prove that the image of Eo by Vo = Uo U z is not reduced to 0, hence is the subspace Do = Po(D) of dimension 1 (use the fact that Poa # 0 and the result of (b)). (d) Prove that the elements U;(Pon)constitute a total sequence in Eo. (Let Fo be the closed vector subspace of Eo generated by that sequence, and F6 the orthogonal supplement of Fo in Eo. Prove that FL is orthogonal to D and that Uo(Fb) = U(Fb) = Fb; conclude as in (a).) 20. Let E be a separable Hilbert space, U a compact operator in E whose spectrum consists of 0 and of an infinite sequence (AJof distinct eigenvalues # 0 such that k(h,) = 1 and that E(hJ is one dimensional for every n. For each n , let a. be an Figenvector of U corresponding to h,, and b, an eigenvalue of U * corresponding to h, (see (11.5.5)); a. and b, are chosen in such a way that (a. I b,,) = I . (a) Let A and B be the closed vector subspaces of E respectively generated by the a. and the b,, B' and A' the orthogonal supplements of B and A, respectively. Show that B' is stable for U , contains U - l ( O ) and that the restriction of U to B' has a spectrum reduced to 0. (b) Suppose the series of general term lh,, . lla,ll~ IIb.11 is convergent. Show then h,(x I b.)a,, where the series is convergent in E; U - l ( O ) that for any x E A, U X
+
+
=c "
then contains A n B'. (c) Give an example in which A n B' is infinite dimensional and U ( A n B') = A n B'. The following method may be used: let E be a Hilbert sum of three Hilbert spaces of infinite dimension F, R, S, having respective Hilbert bases (f"),(r,,), and (.yn); define a sequence (an)in F -1 R such that f. is the projection of a, in F and the closed vector subspace generated by the a, is F t R. To do this, observe that in a Hilbert , closed vector subspace generated by the space H with a Hilbert basis ( e n ) n 3 othe eo en for n > I is equal to H, and take for F R the Hilbert sum of a sequence o f Hilbert spaces all equal to H. Define U such that Unn= h,,a,,, Urn= p,,r,,-l for I? 2 I , Url = 0, Us,, = p , , ~ . + ~ where , the sequences (ti,) and (pn)converge rapidly enough to 0 (cf. Section 11.2, Problem 3). One thus gets B = F, A = F I R, and A n B'= R.
+
+
6. T H E FREDHOLM INTEGRAL E Q U A T I O N
We now apply the preceding theory to the example (11.2.8). We consider here the prehilbert space G of continuous complex-valued functions in I = [a, h], with ( f i g ) =Jabf(t)z) rlt, and the operator U such that U f is the function
6 THE FREDHOLM INTEGRAL EQUATION
341
t -+ @ K ( s , t)f(s) ds.
We say that the operator U is defined by the kernel function K. (11.6.1 ) The compact operator U in G has a compact adjoint which is defined by the kernel function K* such that K*(s, r ) = K ( t , s).
We prove for a d x
< b the identity
which, for x = 6,will yield the result by definition. Both sides of (11.6.1.1) are differentiable functions of x in [a,b ] , by (8.7.3) and Leibniz's rule (8.11.2); they vanish for x = a, and their derivatives are equal at each x E [a, b ] by (8.7.3) and (8.1 1.2), hence they are equal everywhere in [a, b ] (8.6.1 ). We leave to the reader the expression of the criterion (1 1.5.6) for that particular case (the " Fredholm alternative ").
If K ( t , s) = K(s, t ) (in which case the kernel K is called hermitian), the compact operator U is self-adjoint. As the prehilbert space G is separable ((7.4.3) or (7.4.4)), it can be considered as a dense subspace of a Hilbert space (6.6.2), and therefore we can apply to the operator U the results of (11.5.8). We shall denote by (An) the sequence of the (positive or negative) eigenvalues of U, each being repeated a number of times equal to its mu/tiplicity, and ordered in such a way that lAnl b / A n + , [ ; and we will denote by (q,) an orthonormal system in G such that, if the values of n for which An = pk (resp. A,,= v k ) are ni, m + I , . . . , m + r, then q m ,v",+~, . ..,qm+r constitute a basis for the eigenspace E(pk) (resp. E(vk)); we therefore have U ( q , ) = Anqnfor each t i . The q, are called eigenfunctions of the kernel K.
e
(11.6.2)
If K
is a hermitian kernel, the series
c A; d [ d t n
1 A:
is conwrgent and
n
jabIK(&t)12 d s .
a
Indeed, if we apply the Bessel inequality (6.5.2) to the function s -+ K(s, t ) and to the orthonormal system ((p"), we obtain, for any N
n= 1
342
XI
ELEMENTARY SPECTRAL THEORY
i.e.
c 2: N
(11.6.2.1)
fl=
1
lcpn(t)I2
G [IK(s9 t>I2 ds
for every t E I . Integrating both sides in I and using the relations (cp, and (11.6.1 .I) yields the result. The canonical decomposition in be written f
=
c c,
cpn
n
+f o ,
e of any function f
where c,
=
( f I cp,)
I cp,)
1
G (11.5.7) can here f(t)cp,(t) d t ; but, as
=la b
=
E
already observed,,f, may fail to be in G; on the other hand, the series
e,
1 cncpn
converges in the Hilbert space but not in general in the Banach space E = %?,-(I) (the series cncpn(t)will not necessarily converge for every t E I).
1 n
However: (11.6.3)
f (t)
If K is a hermitian kernel, and .f = Ug f o r a function g E G (i.e. K(s, t ) g ( s )ds), then rhe series c, cp,(t) converges absolutely and
=lab
uniformly
tof(t)
We have in
1 n
in 1.
c the canonical decomposition g = 1dncpn + g o ; as
V is a
n
continuous linear mapping of G into E = %?,-(I) (11.2.8), U can be extended to a continuous linear mapping of into E, and Ug, = 0, hence we have -f = Ug = Andncpn,where now the convergence is in E; i.e. the series
1
e
A,, d, cp,(t) converges uniformly to f ( t ) in E; as c, = (fI cp,)
= (Ug I cp,)
=
n
( g I Ucp,) = 1,(g I cp,) = And, , we have proved (11.6.3) except for the statement on absolute convergence. But for any integer N, we have, by Cauchy-Schwarz (for finite dimensional spaces)
and the right-hand side is bounded by a number independent of N, by Bessel's inequality (6.5.2) and (11.6.2.1). (11.6.4) I f K is hermitian, and 1 # 0 is not in the spectrum of U, the unique solution f'of the equation Uf - Af = g , f o r any g E G, is such that
where the series is absolutely and uniformly convergent in I, and dn = ( g I cp,).
6 THE FREDHOLM INTEGRAL EQUATION
343
e
We know that the unique solution of U f - Af = g in belongs to G since G (11.5.6), and by (11.5.11) we have c, = (fl cp,) = l/(A,, - 2). As g /If= Uh we can apply (1 1.6.3), and this proves the result. g
E
+
(11.6.5) Under the same assurnptions us in (11.6.4), the unique solution of U f - /If = g can be written
with
where the series is absolutely and unifornily coniIergerit for (s, t ) E I x I. By the proof of (11.6.3), we have
An dncp,,(t) = Q ( t ) ,the series convergn
ing absolutely and uniformly in I. As 1
A(An - A)
+ -1= A2
An
A(Ai -A)
the formula in (11.6.4) gives
The theorem will follow when we have proved the uniform convergence of the series A: Icpn(s)12: for there is a 6 > 0 such that IA, - A1 2 6 for each n, fl
hence
by Cauchy-Schwarz, and this will prove that the series
is absolutely and uniformly convergent in I x I ; the conclusion then results from (8.7.8).
344
XI ELEMENTARY SPECTRAL THEORY
Now consider the function H(s, t )
=I
b
K ( u , s ) K ( t , u ) du; for each fixed
I~qn(s)qn(t)
t E I, we can apply to it (11.6.3), and we see that H(s, t ) = n
where the series is convergent for any pair (s, t ) E I x I. In particular H(s, s ) = il: Iqn(s)12 for all s E I, and H(s, s) is continuous; by Dini’s n
theorem (7.2.2), the convergence is uniform in I. Q.E.D.
(11.6.6)
If K is hermitian, then
uniformly,for s E I.
With the notations of the proof of (11.6.5), we have (11.6.6.1) n-tm
\
k= 1
uniformly for s E I ; if we evaluate the integral in the statement of (1 1.6.6), using the fact that the qkare eigenvectors of U , and that they are orthogonal, we obtain the expression in the left-hand side of (11.6.6.1), whence the result. -
In general, the series
d,cp,(s)q,(t)
wiI/ not be contlergent for all
n
(s, t ) E 1 x I ; but we have the special result:
(11.6.7) (Mercer’s theorem) Suppose the coinpact operator Udejiied 61) the hertnitian kernel K(s, t ) is posititre. Then we have K ( s , t ) = 1, q,(s)q,,(t), n
where the series is absolutely and uniformly coniiergent in I x I.
We recall that we have here I n> 0 for every n (11.5.9). We first prove that for each s E I , the series I nlqn(s)I2 is convergent. For any s E I, we have K ( s , s) > 0. Otherwise, there would exist a neighborhood V of s in I such that B(K(s‘, t ) ) Q - 6 < 0 for (s’, t ) E V x V. Let q be a continuous mapping of I into [0, 11, equal to 1 at the points, to 0 in I - V (4.5.2). Then we have
6 THE FREDHOLM INTEGRAL EQUATION
345
by (8.5.3). But the left-hand side is (Ucp I cp), and this violates the assumption that U is a positive operator. Remark now that for any finite number of eigenvalues Ak (1 < k d n ) ~ , ( s , t ) = ~ ( s t, ) -
&(Pk(S)fpk(t)is the kernel function of a positive operk= 1
ator U,,, for we have
(un
f
f 1 f) = (uf I f) -k = 1 I(f I ‘Pk)12;
but the right-hand side of that equation can be written (Ug Ig) with fl
g=f-
k= 1
(fI (Pk)(Pk,as is readily verified, hence is positive by assumption.
Therefore, by (5.3.1) it follows from K,(s, s) 2 0 that the series is convergent, and we have
c A,,
Icpn(s)12
1 A,, lcpn(s)12
,< K(s, s) for all s E I. By Cauchy-
n
Schwarz, we conclude that
for all (s, t ) E I x I. Hence, as K(t, t ) is bounded in I, f o r f i x e d s E I, the series A,,cp,(s)cp,,(t)is uiiifornily conrergent for t E I. By (11.6.6), (8.7.8), and
1
-
fl
An cp,I(s)cp,,(t) = K(s, t ) for all (s, t ) E I x I since
(8.5.3), we conclude that n
t -+ IK(s, t ) -
2
1A , l a ) c p n ( t ) l fl
particular, we have K(s, s) = series
1 A,, lq,(~)1~is n
proves that the series
is continuous in I and its integral in I is 0. In
c An
Icp,,(~)1~; by Dini’s theorem (7.2.2) the
n
therefore uniformly contiergent in I, and (11.6.7.1) -
1 A,, cp,(s)cp,(t)
is absolutely and uniformly convergent
n
in I x I, which ends the proof.
Remarks (11.6.8) The result (11.6.7) is still true when we only suppose that U has a finite number of eigenvalues vk < 0 (1 < k < m).For (11.5.7(c)) shows then that in the space FL+,,orthogonal supplement of E(v,) -.. E(v,) in G, the restriction of the operator U is positive, and we apply (11.6.7) to that
+ +
346
XI
ELEMENTARY SPECTRAL THEORY
operator, which, as is readily verified, corresponds to the kernel function K(s, t ) - A,,q h ( ~ ) q h (where t), h runs through all the indices (in finite num-
1 h
ber) such that Ah < 0. The conclusion is then immediate. (11.6.9) We can consider the operator U in a larger prehilbert space, namely the space F, of regulated functions (Section 7.6) which are continuous on the right (i.e. such thatf(t+) = f ( t ) for a < t < b) and such that f ( b ) = 0; for such a function the relation Jab If(t)12 dt = 0 implies f ( t ) = 0 everywhere in 1 = [a, b], for it implies f(t) = 0 except at the points of a denumerable subset D (by (8.5.3)), and every t such that a < t < b is limit of a decreasing sequence of points of I - D. The space G may be identified to a subspace of F, , by changing eventually the value of a continuous function f E G at the point 6; it is easily proved (using (7.6.1)) that G is dense in F, . The argument of (11.2.8) then shows that U is a compact mapping of F, into the Banach space E = gC(I) (and a fortiori a compact mapping of the prehilbert space F, into itself). All the results proved for the operator U in G are still valid (with their proofs) when G is replaced by F, .
PROBLEMS 1. Extend the results of Section 11.6 (with the exception of (1 1.6.7)) to the case in which K(s, t ) satisfies the assumptions of Section 8.1 1, Problem 4 (use that problem, as well as Section 11.2, Problem 5). 2. In the prehilbert space G of Section 11.6, let (J,) be a total orthonormal system (Section 6.5); let
KAs, t )
=
kh(slz)
k= 1
and
HAS) = /ablK.(s, t)l df
(the “nth Lebesgue function” of the orthonormal system (fn)).For any function g E G,
let s,(g)
n
=
(g I f x ) f x , so that s,(g)(x) =
k= 1
1.”
K,(x,
t ) g ( r ) dt
for any x
E I.
(a) Prove that if, for an x o E I, the sequence (H,(xo)) is unbounded, then there exists a function g E G such that the sequence (s,,(g)(xo))is unbounded. (Use contradiction, and show that under the contrary assumption it is possible to define a strictly increasing sequence of integers (n,), and a sequence (g,) of functions of G, with the following properties: (1) let ch = sup IJ:K.(xo,
“
assumption), let dk = c1 -1 c2 sup(ml,. . ., mk-,); then m,
t)gh(t) dt
1
(a number which is finite by
+ . . . + ck-1, let mk ~Kn,(xo,t ) dt, and let qr 2,+l(9, + l)(d, + k ) ; (2) let vkbe a continuous func=
nl
=
Jnu
I
, dt tion such that vk(a)= Tt(b) = 0, Ivk(t)l< I in I and J o b ~ . , ( x o t)Tk(f)
I
mk/2
(see Section 8.7, Problem 8); then g k = ~ , / ( 2 ~-t ( 9I)).~ Then show that the function m
g=
k=l
g k is continuous in 1 and contradicts the assumption: to evaluate the integral
6 THE FREDHOLM INTEGRAL EQUATION
gi
Knk(xO,t ) g ( t ) dt, split g into ih
majorize the two other ones (“method of the gliding hump”).) (b) Show that for the trigonometric system (Section 6.5) in I = [-1, 11, the nth Lebesgue function is a constant h, , and that lim h, = w (observe that
I I
n- m
/A/‘“
(h-l)ln
for 2 < k
+
sin n v t 2 dt 3 sin vt kn
< n). Conclude that, for any xo E I, there exists a continuous function g in I,
such that g(- 1 ) = g(l) = 0, for which the partial sums
.
5 (1’g ( t ) e - i K ” r
h=-n
dt)etkff”/2
-1
of the “Fourier series” of g are unbounded for x = xo (Cf. Section 13.17, Problem 2.) 3. Let g be a continuous complex valued function defined in I = [- 1 , I ] and such that g(- 1) = g(1) = 0; g is extended to a continuous function of period 2 in R. Let K(s, t ) be the restriction ofg(s - t ) to I x I ; ifg(-t) =g(t), the compact operator U defined by the kernel function K(s, I ) is self-adjoint. Show that the functions ~ “ ( t= ) enntt/2/i are eigenvectors of U , the corresponding eigenvalue being the “Fourier coefficient ” a.
= J : l
g(t)e-“”“ dt of g.
Using that result and Problem 2, give examples of a hermitian kernel function K for which the series of general term h , a v . ( t ) has unbounded partial sums for certain values of s and t , and of a positive hermitian kernel function K for which there is a function f e G such that the series
rn
C (fI vn)vn(t) has unbounded partial sums for
fl=l
4.
certain values o f t . Let I = [-2v, 271, and define K(s, t ) in I x I to be equal to the absolutely convergent -sin ns . sin nt for 0 C s < 2n, 0 C t < 2 7 , and to 0 for other values of (s, t ) .=inz in I x I. Give an example of a functionfg G such that in the canonical decomposition off, fo does not belong to G. (The eigenfunctions of K are the functions pnsuch that v,(t)= 0 for -2v < t < 0, y,(t) = n-1’2sin nt for 0 C t < 2n. Take forfa continuous
series
function in I equal to 2 n - t in [0,27r], and show that the series verges everywhere in I, but has a discontinuous sum.)
m
C (f1 ~.)v,(t) con-
n=1
5. With the general notations of Section 11.6, let K be a hermitian kernel defined in
I x I, and let U be the corresponding self-adjoint compact operator in G. Show that for every h > 0, Uhcorrespondsto the hermitian kernel Kh,which is defined inductively by K 1 = K, and Kh(s, t )
=jab w
u)K(u, t ) du.
-
Prove that for h 3 2, Kh(s,t ) = C ht v n ( s ) v n ( t ) ,the series being absolutely and
”= 1
uniformly convergent in I x I. Show in addition that
and that the sequence (Ahfl/Ah)is increasing, and has a limit equal to lh1)’, where h, is an eigenvalue of K of maximum absolute value (use Cauchy-Schwarz).
348
XI ELEMENTARY SPECTRAL T H E O R Y
K be an arbitrary continuous kernel function in I x I, and let U be the corresponding compact operator in G . Let M be a finite dimensional subspace of G such that U(M) c M ; let ( + h ) l d h < n be an orthonormal
6. With the notations of Section 11.6, let
basis of the space M, and write
u+h
=
C
k=l
ahk
4 k . Show that
(For each f E I, apply Bessel's inequality (6.5.2) to the function s +K(s, t ) and the orthonormal system ( # h ) in G.) Let (A,) be the sequence defined (for the operator U ) in Section 11 -5,Problem 15(c). Prove that the series
m
"=l
Ih,12 is convergent, and
(Apply the preceding result to any sum of subspaces N(pk), with the notations of Section 11.5, Problem 15(c).) 7. Give an example of an hermitian kernel K(s, t ) , such that, if U is the corresponding compact operator in G , and V the square root of U 2 (Section 11.5, Problem 12), there is no hermitian kernel to which corresponds the compact operator V . (If there existed such a kernel, Mercer's theorem (11.6.7) could be applied to i t ; take then for K the first example in Problem 3.) 8. In order that the compact operator U defined by an hermitian kernel K(s, t ) be positive, show that a necessary and sufficient condition is that K be (in I x I ) a function of positive type (Section 6.3, Problem 4; to prove that the condition is necessary, write the inequality
JabfT dt K(s, t ) f (s) ds 3 0 Jab
for a function f which is 0 outside arbitrary small neighborhoods of a finite number of points x i of 1 (1 ,< i < n). To prove that the condition is sufficient, use the same method as in Section 8.7, Problem I). Conclude from that property and from Section 6.3, Problem 8 and Section 6.6, Problem 5 , a new proof of Mercer's theorem. 9. (a) A kernel function K(s, t ) defined in 1 x I (with I = [a, b ] ) and satisfying the assumptions of Section 8.1 1, Problem 4, is called a Volterra kernel if K(s, t ) = 0 for s> t . Let M = sup IK(s, t ) l . If U is the compact operator in G corresponding (S,
¶)El
Y
I
to K (Problem l), show that U" corresponds to a Volterra kernel K, such that (Kn(s,f ) l < M"(t - s)"-'/(n - l ) ! for n > 1 and s < t (use induction on n). Deduce from that result that the spectrum of U is reduced to 0, and that for any 4 E C , ll(lE - [ U ) - ' - lE1I < M1 1. Show that for that kernel, the function R(s, 1 ; A) in (11.6.5) is equal to exp((t - s)/A) for s < t , and to 0 for s > t. (Use (8.14.2) to compute U".) 10. Let F, be the prehilbert space defined in (11.6.9) for the interval I = [0, I ] , and let U be the operator defined in Problem 9(b), so that for every function x E F,, y = Ux is the function t - t
Jot
x(s)ds. The space
F+ is a dense subspace in a Hilbert space E
(6.6.2); U is extended by continuity to a compact operator in E, again written U (11.2.9). The closed subspaces Eo of E, distinct from 0 and E, and such that U(E,) c Eo
will be determined in this problem.
7 THE STURM-LIOUVILLE PROBLEM
349
(a) Show (using (7.4.1)) that the operator U satisfies the condition of Section 11.5, Problem 19, with a equal to the constant function of value I in 1; let a. E E, be such that Po a = so a. with so > 0 and I/ao11 =- 1 , so that 0 < so < 1. For every x E Eo , V O X= sb(x1 ao)ao(notations of Section 11.5, Problem 19). (b) For every 5 E C not 0, let f(5) = 1 - si((Uo 5f)-'ao I a,). Prove that for ( # 0, -
f(- n An
the series being absolutely and uniformly convergent in I x I (it is supposed, as we may, that 0 is not one of the A,). We observe that (d) follows from (11.6.3) and (11.7.8) when the additional assumption is made on u’ that it!’ is continuous in 1. To prove (d) in general, let ti (1 d i d wz) be the points of f where M)’ has a discontinuity, and let cli = w’(ti+) - w ’ ( t i - ) . Then the function u = w
m
+ i=
ctiK,, satisfies all the conditions of (d) and in addition 1
has a continuous derivative, by (11.7.6). Using (1 1.7.10.1 ) we conclude the proof of (d). From the fact that the identity mapping of E = +?,-(I) into G is continuous, it follows that for the functions w satisfying the conditions of (d), we can also write MI = c,cp,, the sequence being convergent in the n
prelzilbert space G. To prove (c) it will then be enough to show that the set P of these functions lli is dense in G. Now, for any function u E G, consider the continuous function M’, equal to u in [~+.!-,b-~], m m
to a linear function x + M X + fi satisfying the first (resp. second) bounduy condition (11.7.2) in
and to a linear function in each of the intervals
7 THE STURM-LIOUVILLE PROBLEM
355
We can in addition suppose that at the points a, b, the value of w, is 0 or 1 ; it is then clear that lu(x) - w,,,(x)I < llull
+ 1 in each of the intervals
and therefore IIu - iv,,J2 is arbitrarily small by the mean value theorem; as w, satisfies all conditions in (d), this proves our assertion. Once (c) is thus proved, it is clear that the total sequence (cp,,) must be infinite, and (applying (11.6.2)), (a) is also completely proved. Finally, (e) and (f) follow at once from (11.5.11).
Remark. It is possible to obtain much more precise information on thecp,and A,,, and to prove in particular that AJn2 tends to a finite limit (see Problems 3 and 4).
PROBLEMS
1. Let 1 = [ a , b] be a compact interval in R , and let Ho be the real vector space of all real-valued continuously differentiable functions in I; Ho is made into a real prehilbert space by the scalar product (x 1 y) =Job (x’y‘
+ xy) df .
(a) Show that Ho is separable (approximate the derivative of a function x E Ho by polynomials (7.4.1)); Ho is therefore a dense subspace of a separable Hilbert space H (6.6.2).
(b) If (x.) is a Cauchy sequence in the prehilbert space Ho , show that the sequence (x.) is uniformly convergent to a continuous function u in I, and that if (y,,)is a second Cauchy sequence in Ho having the same limit in H, then (yn)converges uniformly in I to the same function v ; the elements of H can thus be identified to some continuous functions in I, which however need not be differentiable at every point of I. (Observe that for every function x for any function z z’(u) = z’(b)
=
E
E
H o , Ix(t) - x(u)l
< (t - a ) 1 / 2 ( j o b x dt)’” ’2
in I.) Show that,
Ho which is twice continuously differentiable in I and such that
0,(v I z) = -
\
uz“ dt
. a
+
fab
uz dt.
(c) Let a , ,8 be two real numbers, q a continuous function in I. Show that in N o , the function x
+ @(x) =
s.“
(x” -tqx2)df
-
~ ( x ( u )-) P(x(b))’ ~ is continuous. Let A be the
subset of H consisting of the functions x such that
Jab
x 2 dt = 1 (observe that this is not
a bounded set in the Hilbert space H). Show that in A n H o , the g.1.b. of @ ( x ) isfinite. (One need only consider the case a > 0, ,8 > 0. Assume there is a sequence (x,) in
356
XI
ELEMENTARY SPECTRAL THEORY
A n H, such that lim rD(x,) = - m, and, if y,, = n-
dt)
liZ
, lim yn = -1 cu ; consider n-m
m
the sequence of the functions y ,
= x,/y,,
and derive a contradiction from the fact that,
y', dt 0, and on the other hand, there is an interval [a,c] jab a number p > 0 such that Iy.(t)l > p for every n and every point t [a, c].)
on one hand lim
=
n-m
C
I and
E
(d) Let p1 be the g.1.b. of @(x) in A n H o . Show that if (x.) is a sequence in A n Ho such that lim @(x,) = p,,(x,) is bounded in H (same method as in (c)). Deduce from n-t m
that result that, by extracting a convenient subsequence, one may assume that the sequence (x,) is uniformly convergent in I to a function u (which, however, need not a priuri belong to H) (use Ascoli's theorem (7.5.7)). (e) @(x)is a quadratic form in Ho , i.e. one has @(x t y ) = @(x) @ ( y ) 2 'Ux, y ) , where Y is bilinear; for any function z which is twice continuously differentiable in 1
+
and such that z'(n) = z'(b) = z(a) = z(h) = 0,one has 'Y(x, z ) =
-jab
XZ"
-'
dt
+lab
qxz dt ;
z ) can be defined by the same formula for any function v continuous in 1. Show that for any such function z and any real number [, one has '€'(u,
and deduce from that result that one must have
lab+ (uz" - quz
p I u z )dt = 0.
Hence, if w is a twice continuously differentiable function such that w" = qrc - p l u , one has [ab (u - w)z" dt = 0 by integration by parts;conclude that u - w is a polynomial
of degree < 1 (observe that by substracting from u - w a suitable polynomialp of degree 1, there exists a function z such that z" = u - w - p, z(a) = z(b) = z'(a) = z'(b) = 0). Hence u is twice continuously differentiable, satisfies the differential equation us - qu
+ plu
u2 dt = 1 ; furthermore, u'(a)
last statement, express that for any z 2.
E
= 0,
= - au(a),u'(h) = pu(6).
H,, cD(u
+ [z) > p1 J
" (u a
(To prove the
+ [ z ) dt,foranyreal ~
number 6.) (a) With the notations of (11.7.10), suppose first that k l k 2 # 0, and let a = h,/k,, p = - h 2 / k 2 . Show that the q ncan be defined (up to sign) by the following conditions: (1) p, is such that, o n the sphere A : (yl y ) = 1 in G , the function (I) (defined in Problem l(c)) reaches its minimum for y = pl,and that minimum is equal to A, ; (2) for n > 1 , let A, be the intersection of A and of the hyperplanes ( y I pr)= 0 for 1 < k c IZ - 1 ; then F~is such that on A,, CD reaches its minimum for y = rp", and that minimum is equal to A,. (The characterization of y1 follows at once from the results of Problem 1; use the same kind of argument to characterize vn.) (b) If k l = 0, k2 # 0,prove similar results, replacing a by 0 in @, but replacing the sphere A by its intersection with the hyperplane in G defined by y ( a ) = 0. Proceed similarly when k , # 0 and kZ = 0, or when k , = k 2 = 0.
7 THE STURM-LIOUVILLE PROBLEM
357
(c) Under the assumptions of (a), let z , , . . . , z , - ~ be n - 1 arbitrary twice continuously differentiable functions in I, and let B(zl, . . . , z " - ~ )be the intersection of A and of the n - 1 hyperplanes ( y I z x )= 0 ( I < k < n - 1). Show that in B(zl, . . .,zn-,), the function CU reaches a minimum p(zl, .. ., z,- at a point of B(zl, . , . , zn-,),and that h, is the 1.u.b. of p ( z l , . . . , z . - ~ ) when the z , vary over the set of twice continuously differentiable functions in I (the "maximinimal" principle; same method as in (a) to prove the existence of the minimum; the inequality is proved by the same method as in Section 11.5, Problem 8). Extend the result to the cases k , k 2 = 0. 3. (a) One considers in the same interval I two linear differential equations of the second order y" - y I y hy = 0, y" - q2y hy = 0, with the same boundary conditions (11.7.2); let (hi')),(hi2))be the two strictly increasing sequences of eigenvalues of these two Sturm--Liouville problems. Show that if 41 < q 2 ,then hi') < hiz) for every - hb2)l < M for every n (use the maximinimal n,and if I q l ( t ) - q2(t)l < M in I, then principle). (b) Conclude from (a) that there is a constant c such that
+
+
for every 11, with I : h - a . (Study the Sturm-Liouville problem for the particular case in which y is a constant.) 4. (a) Let y be any solution of (11.7.3) in I = [a,h ] for h > 0. Show that there are two constants A, w such that y is a solution o f the integral equation
(*I
y ( t ) = A sin J h ( t
1.
+ w ) -1- I
dh
t
q(s)y(s)sin Jh(t
- s) ds.
Show that there exists a constant B independent of h, such that A' < B(y I y ) (use Cauchy-Schwarz in order to majorize the integral on the right-hand side of (*)). (b) Deduce from (a) that if, in the Sturm-Liouville problem, k 1 k 2# 0 or k l = k2 = 0, then there are two constants C o , C,, such that, for every n, and every t E I lp,,(t)- J2// sin Jhn t 1
< co/n
and
Ipi(t) - 4 211JXn cos JGtl
< c,
with I = b - a
(use (a), and the result of Problem 3(b)). What is the corresponding result when only one of the constants k , , k 2 is O ?
APPENDIX
ELEMENTS OF LINEAR ALGEBRA
Except for boolean algebra (Section 1.2) there is no theory more universally employed in mathematics than linear algebra; and there is hardly any theory which is more elementary, in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. We shall give a brief survey of the concepts and results of linear algebra which are used in this book. For a more complete account we refer the reader to the works of Halmos [I I], Jacobson [13], or Bourbaki [4].
1. VECTOR SPACES
(A.l.1) Throughout this appendix, K is a fixed (commutative) field, whose elements are called scalars. A vector space orer K (or simply a vector space, if there is no ambiguity about K) is a set E endowed with the structure defined by two mappings: (x, y ) --f
x +y
(A,x) + Ax
of E x E into E
(called addition)
of K x E into E
(called scalar multiplication)
having the following properties:
+y + z)
(1.1)
x + ( y + z ) = (x + y ) + z
(1.2)
x+y=y+x
(1.3)
there exists an element 0 E E such that x = 0
(written x
+ x for all x E E l
t Experience shows that, contrary to the opinion of some authors, there is no risk of confusion in using the same symbol 0 to denote the zero elements of all vector spaces (and of K in particular). 358
1 VECTOR SPACES
for each x E E there exists an element - x (A p)x = Ax px ~ ( xy)= + A~ %(/AX)= (Ap)x 1 * x =x
(1.4) (11.1) (11.2) (11.3) (11.4)
+ +
E
+
E such that x
+ (-x)
359
=0
(in these conditions, x, y , z are arbitrary elements of E and 2, p are arbitrary elements of K). Conditions (1.1) to (1.4) express that E is a conmutatiue group with respect to addition. It follows that, if ( x i ) i s is any ,finite family of elements of E, the sum E x i is unambiguously defined. (We recall the convention that
c
isH
xi
= 0.)
is @
+
+ +
From (1.3) and (11.2) we deduce that Ax = A(x 0) = Ax 20, and therefore A0 = 0 for all A E K . From (11.1) it follows that Ax = (A 0)x = 3.x + Ox, so that Ox = 0 for all x E E. Finally, from these two relations and (1.4), (11. I), and (11.2) we deduce that ix + A( -x) = A0 = 0, and Ax + (-A)x = Ox = 0, so that (- J.)x = A( - x) = -(Ax). The elements of E are often called vectors. (A.1.2) An additive group consisting of the element 0 alone is a vector space: the scalar multiplication is the unique mapping of K x ( 0 ) into ( 0 ) . The field K is a vector space over itself, the scalar multiplication being multiplication in K. If K’ is a subjield of K and if E is a vector space over K, then E is also a vector space over K’ if we define the scalar multiplication to be the restriction to K’ x E of the mapping (A, x) + Ax. (A.1.3) If E is a vector space, a vector subspace (or simply subspace) of E is defined to be any subset F of E such that the relations x E F, y E F imply Ax -t~ c Ey F, for all scalars A, p . The restriction to F x F (resp. K x F) of the addition (resp. scalar multiplication) in E is a mapping into F, and therefore F a vector space with respect to these mappings. This justifies the terminology. If (xi)l ;, n,). It is immediately verified that the mapping u so defined is indeed r-linear. _,,
9
(A.6.3) Consider the case where all the E,areequal tothesamevectorspaceE. An r-linear mapping u : E' -+ F is said to be alternating if u ( x l , x , , . . . , x,) = 0 whenever there are two distinct indices i < j such that x i = x j . It follows from this definition that
+ x j , Xi+,, . , x j - I , xi + . . . , x,) * ., xi,. . . , . . . , x,) + u(x,, . . . , . . . , xi,. . . ,x,)
0 = u(x,, . . . , xi-1, xi = u(x,,
I
xj, X j f l ,
* *
xj,
xj,
for all x i E E. ln other words, if we interchange two of the arguments x i , x j in u, the value of u changes sign. Since every permutation 0 of the set { I , 2, . . ., r} may be expressed as a product of transpositions, the following formula is an easy consequence: (A.6.3.1)
u(xu(l),xu(2) . * * 9
9
xo(r))
= ~u ~(x,, x2 2
. . > xr), *
where E, is the signature of the permutation 6. If E is finite dimensional and ( b j ) l s j Q nis a basis of E, then the u(bj,,bj, , . . . , b j r )are zero by definition whenever two of the indices j,, . . . ,j , are equal. By virtue of (A.6.3.1), the values of u(bjl,. . . , bj,) for sequences of distinct indices ( j , ) are determined by those which correspond to increasing
APPENDIX: ELEMENTS OF LINEAR ALGEBRA
374
--
sequences j , < j , < < j , . Conversely, if we assign an arbitrary element c ~ , ~ ~ of , . .F , ~ to each increasing sequence, then there exists a unique alternating r-linear mapping u : E' -,F such that u(bj,,b,, , . . . , bjp) = c j l j ,.,,j , whenever j , < j , < . . . < j , . The verification of this statement is left to the reader. (A.6.4) Consider the particular case of alternating n-linear forms on E", where n is the dimension of E. The remarks above show that such a formf is completely determined by its value f ( b , , b, , . . . , b,,), where (bi) is any basis of E; andfis identically zero if and only iff(b,, . . . , b,) = 0. It follows that, if f o is one of these nonzero forms, then every other alternating ti-linear forms on E" can be written as A fo , uhere A is a scalar. These hypotheses and notations will remain in force for the rest of this section.
Let u be an endomorphism of E. Then there exists a unique scalar det(u) such that, i f f is any alternating n-linear form on En, we haue: (A.6.5)
(A.6.5.1)
f ( u ( x , ) , u(x,), . * * 4 x n ) ) 5
for all choices of x i E E ( 1
= det(u)
.S(X,,x2 . 9
9
*
9
xn)
< i < n).
It is enough to prove this for,fo, in which case the result follows from . . ., u(x,)) is an alternating n-linear form on E". (A.6.4) and the fact that ( x l , .. ., xn)-,fo(u(xl),
The scalar det(u) is called the determinant of
Clearly we have
det( lE) = 1.
(A.6.5.2) (A.6.6)
11.
r f u , u are two endomorphism of E, then det(u u) = det(u) det(u).
(A.6.6.1)
By applying (A.6.5) to the alternating n-linear form (x,,
. . . ,X") +fO(U(Xl),
'
. ,U(X">)
and the endomorphism u, we obtain ~ O ( U ( ~ X I* .) *) , ~(~(xn))) = det(u)f0(4~1),* * * 3
= det(v) det(u)fo(x,,
9
u(Xn>>
. . . ,xJ.
Since fo is not zero, the formula (A.6.6.1) now follows from the definition of det(u t i ) . 0
MULTILINEAR MAPPINGS. DETERMINANTS
6
(A.6.7)
375
det(u) # 0 ifand only if u is bijective.
If u is bijective, it has an inverse u-l such that zi u-' = 1., Hence, by (A.6.6) and (A.6.5.2), we have det(u) det(i4-') = I , so that certainly det(u) # 0. If u is not bijective, then it is not injective (A.4.19), hence there exists b, # 0 such that u(b,) = 0. There exists a basis (b;), of E containing 6 , (A.4.5), and we have fo(b,, . . . , 6,) # 0, whereas fo(u(b,), . . . , u(b,)) = 0. Hence det(u) = 0. (A.6.8) Let ( b , ) l s be a basis of E, and let M ( u ) = ( z j i ) be the matrix of u with respect to the two bases (b;)and (b;)of E (or, as is usually said, tllenzatrix of I I witA respect to the basis (b;)). Since ,fo(h,, . . . , b,) # 0, the formulas (A.6.1 .I)and (A.6.5.1) give (A.6.8.1)
det(4
=
c
EU
%(l)lMU(2)2
..
*
%("),
3
U
where CJ runs through the symmetric group 6,of all permutations of (1, 2, ..., n}. The determinant of the matrix M(u) is by definition the determinant of u. This provides the link between our theory and the classical theory of determinants in its original form. We shall not need to use this latter theory, and we leave the task of transcribing our results into the old-fashioned notation to those readers who are interested in this type of calculation. In applications it is always much simpler to go back to the definition (A.6.5), as we shall illustrate by considering the eigenvalues of an endomorphism. (A.6.9) The definition of the eigenvalues of an endomorphism u is that given in ( l l . l . l ) , except that the field C is now replaced by an arbitrary field K. It follows immediately from (A.6.7) that these eigenvalues are the roots of the equation (called the characteristic equation of u)
det(u - 1 lE) = 0.
(A.6.9.1)
The formula (A.6.8.1) shows immediately that the left-hand side of this equation is a polynomial of degree n in A, with leading coefficient (- 1)". In what follows we shall assume that the field K is algebraically closed, so that det(u - A IE) factorizes into linear factors (A, - A)(& - A) * . (A, - A). (A.6.10) (A.6.10.1)
There exists a basis (b,, . . . , 6,) of E such that
+
u(bJ = A j b j ai.;+,bi+,
+ + ainh, 1 . -
(I
< i < n).
376
APPENDIX: ELEMENTS OF LINEAR ALGEBRA
Conuersely, i f (bi)l
is a basis with this property, then
det(u - A lE) = (A, - A)(A2 - A)
* . . (A, - A).
The proof is by induction on n. By hypothesis, there exists a vector b, # 0 in E which is an eigenvector for the eigenvalue A,; in other words, u(b,) = A, b, . Let us split E into a direct sum Kb, V, and let p : E + V be the corresponding projection (A.3.4). The mapping x -p(u(x)) is an endomorphism of V, and hence there exists a basis (bl, . . . , bnwl)of V such that
+
P(u(bi))=pibi
+ Ui,i+lbi+l +
+ ~ ( ~ , , - ~ b , - , (1 < i < n - 1)
and consequently
u(bJ
= pibi
+~
+ +
( ~ , ~ + ~ bai,n-.lb,,-l ~ + ~
+ ai,,b,
(1
< i < n - 1)
for suitable scalars m i , . We now have
fo(~(b1) - Ab1, . . ., u(bn) - Abn) =fo((/ll
- A)bi
* * . > (Pn-1
+ . + alnb, , *
-1)bn-i
A)b2
( ~ 2
+. + * *
~ 2 b, ,
,.. .,
+an-1,nbn,(An-A)btJ.
If we expand the right-hand side by means of (A.6.1 . I ) and use the definition of an alternating multilinear form, we see easily that the only term which does not vanish is (PI - m
2
- A)
. . . (Pn- 1 -
A W L ,
-
A)fO(b,, . . . , b,),
(P,,-~ - A)(A, - A). This and therefore det(u - A . lE) = ( p l - A)(p2 - A) proves that the p i are (except possibly for their order) the scalars A,, . . . , An-l, and the calculation above also establishes the second assertion of (A.6.10). The matrix of u with respect to a basis satisfying the conditions of (A.6.10) is said to be lower triangular.
(A.6.11) (A.6.11 . l )
For each integer k > 0, we have det(uk - I * 1E) = (A: - A)(Ak, - A)
*
(A: - A).
For it follows from the formulas (A.6.10.1) that U k ( b i ) = A!bi
+
bi+,+ . *
*
+ apb,
and the result therefore follows from (A.6.10).
(1 < i < n )
7 MINORS OF A DETERMINANT
377
(A.6.12) The endomorphism u is a nilpotent element of the ring End(E) and on1.y if all its eigenvalues are zero.
If u is nilpotent, it follows from (A.6.11) that all the eigenvalues of u are zero. Conversely, if all the ,Ii are zero, the formula (A.6.10.1) shows, by induction on k , that uk(E) c Kbk+ + K b , + , + * * + Kb, if k < n, and finally that u"(E) = {0}, that is to say, u" = 0.
-
7. M I N O R S OF A D E T E R M I N A N T
(A.7.1) Let E be a vector space of dimension n over K, and let ( b i ) l d i 6 n be a basis of E. For each subset I of the index set A = { 1, 2, . . . , n } , let E(I) be the subspace of E generated by the 6 , with i E I . Then E is the direct sum of E(I) and E(A - I). If I = { i l , i , , . . . , ir}, where i , < iz < ... < i,, let j , be the bijection of K' onto E(I) such that j , ( e k )= b,, (1 < k < r ) , where (ek)l4 k G r is the canonical basis of K' (A.4.4). Also letp, be the linear mapping of E onto K' such that p,(bi,) = ek for 1 < k < r, and p,(bj) = 0 if j $ I. The kernel ofp, is therefore E(A - I), and the restriction ofp, to E(1) is a bijection of E(1) onto K'. (A.7.2) Let u be an endomorphism of E and let M ( u ) = ( a j i ) be its matrix with respect to the basis (bi) (A.6.8). If I , J are two subsets of the index set A, having the same number of elements r, consider the endomorphism uJ, = p u 0 j , of K'. Its matrix with respect to the canonical basis (e,) of K' consists of those a j i for which i E I and j~ J. The determinant of this matrix (that is to say, det(u J l ) ) is called the r x r minor of det(u), corresponding to the basis ( b J l i Q n of E and the subsets I, J of the index set A. 0
(A.7.3) An endomorphism u of E is of rank r i f and only i f all the s x s minors (where s > r ) in det(u) relatiue to (bi)are zero and at least one of the r x r minors is nonzero.
Let p be the rank of u. With the notation of (A.7.2), we have ujI(K') = pj(u(E(I))), hence (A.4.18)rank(uJ I ) = dim(uJ1(K')) < dim(u(E(1))) < dim u(E) = rank(u) = p . If r > p , we therefore have det(u,,) = 0, by (A.4.19) and (A.6.7). On the other hand, there exists a subset I, of A, containing p elements, such that E(1,) is supplementary to ker(u), and a subset J, of A containing p elements, such that E(A - J,) is sumlementarv to u(E) (A.4.5).
378
APPENDIX: ELEMENTS OF LINEAR ALGEBRA
It follows that u, restricted to E(To), is a bijection of E(Io) onto u(E) (A.4.19), and that p J orestricted to u(E) is a bijection of u(E) onto KP (A.3.5). Hence u~~~~ is bijective and therefore det(uJo,,) # 0 (A.6.7). The proposition follows immediately from these remarks. (A.7.4) With the preceding notations let us now take 1 = J = { 1, 2, . . . , m}, hence A - I = A - J = {m + 1, . . . , n } , and let us suppose in addition that uA-,,I = 0, in other words the matrix M(u) has the form
where X = M(u,,) is an m x m matrix, Y = M(u,,A - ,) an m x (n - m) matrix and Z = M(u,-,, A-I) an (n - m) x (n - m ) matrix (0 standing for the zero ( n - m) x m matrix). Then we have det(M(u)) = det(X) det(Z).
(A.7.4.1)
For if u is the endomorphism of E(1) having X as matrix with respect to (bi)l i 6 m , we have, with the notations of (A.6.10), f(u(b,),* . . > u(bm), u(bm+l), . * . 4bn)) = f ( v ( b l ) , * * * v(b,n), u(b,n+1), . . . 4bn)). 7
9
But the mapping ( ~ 91 *
* * 9
xm)
-tf(xl, * . ., x m
9
u(bm+1),
*
., u(bn))
is an alternating m-linear form on (E(I))m, hence, by (A.6.5), f(v(bl),
*
. ., u ( b m ) ,
. . u(bn)) X ) f ( b i , . . . , b,, , u ( b m + 1 ) , . . ., u(bn))*
u(bm+l)?
= (det
'9
For each j 2 rn + 1, let us write u(bj) = CJ + cy, with c j E E(I), c; By definition of an alternating multilinear form, we have
E
E(A-I).
. ., b m 4 b m + I), . . ., u(bn)) =f(b1, . . ., b m c;+1, . c:)* Let )Y then be the endomorphism of E(A - 1) having 2 as matrix with respect f(b1, *
9
* *
9
9
to ( b j ) m + l Q j Qby n ;definition w(bj) = cj" f o r j e A - I. The mapping ( x m + 1,
* *
9
xn) +f(b1,
*
. ., brn
is an alternating ( n - m)-linear form on (E(A f(b,,
. . . ,b,,
9
Xm + 1,
.. ., xn)
- I))"-",
~ ( b , + ~.). ., , ~ ( b , ) = ) (det Z)f(bl,
hence we get similarly
. . . , b,)
= det 2
which proves (A.7.4.1). By induction on r, we conclude that for any " triangular
7 MINORS OF A DETERMINANT
matrix of matrices ”
where X i j is an m i x mj matrix, we have (A .7.4.2)
det 0 = (det XII)(det X 2 * )
(“ Computation of a determinant by blocks ”).
(det XrV)
379
REFERENCES
[ I ] Ahlfors, L., “Complex Analysis.” McGraw-Hill, New York, 1953. [2] Bachmann, H., “Transfinite Zahlen” (Ergebnisse der Math., Neue Folge, Heft I). Springer, Berlin, 1955. [3] Bourbaki, N., “ Elements de Mathkmatique,” Livre I, “Theorie des Ensembles ” (Actual. Scient. Ind., Chaps. I, 11, No. 1212; Chap. 111, No. 1243). Hermann, Paris, 1954-1956. [4] Bourbaki, N., “ Elements de Mathkmatique,” Livre 11, “Algebre,” Chap. 11 (Actual. Scient. Ind., Nos. 1032, 1236, 2nd ed.). Herrnann, Paris, 1955. [5] Bourbaki, N., “ Elkments de Mathkmatique,” Livre 111, “Topologie gknkrale” (Actual. Scient. Ind., Chaps. I, 11, Nos. 858, 1142, 4th ed.; Chap. IX, No. 1045, 2nd ed.; Chap. X, No. 1084, 2nd ed.). Hermann, Paris, 1949-1958. 161 Bourbaki, N . , “Elements de Mathkmatique,” Livre V, “ Espaces vectoriels topologiques” (Actual. Scient. Ind., Chap. 1, 11, No. 1189, 2nd ed.; Chaps. 111-V, No. 1229). Hermann, Paris, 1953-1955. [7] Cartan, H., “Seminaire de I’Ecole Normale Superieure, 1951-1952: Fonctions analytiques et faisceaux analytiques.” [8] Cartan, H., “Theorie elementake des fonctions analytiques.” Hermann, Paris, 1961. [9] Coddington, E., and Levinson, N., “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. [lo] Courant, R., and Hilbert, D., “Methoden der mathematischen Physik,” Vol. I, 2nd ed. Springer, Berlin, 1931. [ I I] Halmos, P., “Finite Dimensional Vector Spaces,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1958. [I21 Ince, E., “Ordinary Differential Equations.” Dover Publications, New York, 1949. [I31 Jacobson, N., “Lectures in Abstract Algebra,” Vol. 11, “Linear Algebra.” Van Nostrand, Princeton, New Jersey, 1953. [I41 Kamke, E., “Differentialgleichungen reeller Funktionen.” Akad. Verlag, Leipzig, 1930. [I51 Kelley, J., “General Topology.” Van Nostrand, Princeton, New Jersey, 1955. [I61 Landau, E., “Foundations of Analysis.” Chelsea, New York, 1951. [ 171 Springer, G . , “Introduction to Riemann Surfaces.” Addison-Wesley, Reading, Massachusetts, 1957. [I81 Weil, A,, “Introduction a I’ktude des varietes kahleriennes” (Actual. Scient. Ind., No. 1267). Hermann, Paris, 1958. [I91 Weyl, H., “Die Idee der Riemannschen Flache,” 3rd ed. Teubner, Stuttgart, 1955. 380
INDEX
In the following index the first reference number refers to the number of the chapter in which the subject may be found and the second to the section within the chapter. A
Abel’s lemma: 9. I Abel’s theorem: 9.3, prob. 1 Absolute value of a real number: 2.2 Absolute value of a complex number: 4.4 Absolutely convergent series: 5 . 3 Absolutely summable family, absolutely sumniable subset: 5.3 Adjoint of an operator: 11.5 Algebraic multiplicity of an eigenvalue: I I .4 Amplitude of a complex number: 9.5, prob. 8 Analytic mapping: 9.3 Approximate solution of a differential equation: 10.5 Ascoli’s theorem: 7.5 At most denumerable set, at most denumerable family: I .9 Axiom of Archimedes: 2.1 Axiom of choice: 1.4 Axiom of nested intervals: 2.1
B Banach space: 5.1 Basis for the open sets of a metric space: 3.9 Belonging to a set: 1 . 1 Bergman’s kernel: 9.13, prob. Bessel’s inequality: 6.5 Bicontinuous mapping: 3.12
Bijective mapping, bijection: 1.6 Bloch’s constant: 10.3, prob. 5 Bolzano’s theorem: 3.19 Borel’s theorem: 8.14, prob. 4 Borel-Lebesgue axiom: 3.16 Borel-Lebesgue theorem : 3. I7 Boundary conditions for a differential equation: 11.7 Bounded from above, from below (subset of R): 2.3 Bounded subset of R : 2.3 Bounded real function: 2.3 Bounded set in a metric space: 3.4 Broken line: 5.1, prob. 4 Brouwer’s theorem for the plane: 10.2, prob. 3
C Canonical decomposition of a vector relatively to a hermitian compact operator: 11.5 Cantor’s triadic set: 4.2, prob. 2 &-Capacityof a set: 3.16, prob. 4 Cartesian product of sets: 1.3 Cauchy’s conditions for analytic functions: 9.10 Cauchy criterion for sequences: 3.14 Cauchy criterion for series: 5.2 Cauchy’s existence theorem for differential equations: 10.4 MI
382
INDEX
Cauchy’s formula : 9.9 Cross section of a set: 1.3 Cauchy’s inequalities: 9.9 Cut of the plane: 9.Ap.3 Cauchy-Schwarz inequality: 6.2 Cauchy sequence: 3.14 D Cauchy’s theorem on analytic functions: 9.6 Center of a ball: 3.4 Decreasing function: 4.2 Center of a polydisk: 9.1 Degenerate hermitian form: 6.1 Change of variables in an integral: 8.7 Dense set in a space, dense set with respect Circuit: 9.6 t o another set: 3.9 Closed ball: 3.4 Denumerable set, denumerable family: 1.9 Closed interval: 2.1 Derivative of a mapping at a point: 8.1 Closed polydisk: 9.1 Derivative in an open set: 8.1 Closed set: 3.8 Derivative of a function of one variable: 8.4 Closure of a set: 3.8 Derivative with respect to a subset of R:8.4 Cluster point of a set: 3.8 Derivative on the left, on the right: 8.4 Cluster value of a sequence: 3.13 Derivative (second, pth): 8.12 Codimension of a linear variety: 5.1, Derivative (pth) with respect to an interval: prob. 5 8.12 Coefficient (nth) with respect to an ortho- Diagonal : 1.4 normal system: 6.5 Diagonal process : 9.13 Commutatively convergent series: 5.3, Diameter of a set: 3.4 Difference of two sets: 1.2 prob. 4 Differentiable mapping at a point, in a set: Compact operator: 11.2 8.1 Compact set: 3.17 Differentiable with respect to the first, Compact space: 3.16 second, . . . , variable: 8.9 Complement of a set: 1.2 Differentiable (twice, p times): 8.12 Complete space: 3.14 Differential equation: 10.4 Complex number: 4.4 Dimension of a linear variety: 5.1, prob. 5 Complex vector space: 5.1 Dini’s theorem: 7.2 Composed mapping: 1.7 Direct image: 1.5 Condensation point: 3.9, prob. 4 Conformal mapping theorem: 10.3, prob. 4 Dirichlet’s function: 3.1 1 Disk: 4.4 Conjugate of a complex number: 4.4 Connected component of a set, of a point Discrete metric space: 3.2 and 3.12 Distance of two points: 3.1 in a space: 3.19 Distance of two sets: 3.4 Connected set, connected space: 3.19 Constant mapping: 1.4 Contained in a set, containing a set: 1.1 E Continuity of the roots as function of parameters: 9.17 Eigenfunction of a kernel function: 11.6 Continuous, continuous at a point: 3.11 Eigenspace corresponding to an eigenvalue: Continuously differentiable mapping: 8.9 11.1 Convergence radius of a power series: 9.1, Eigenvalue of an operator: 1 1.1 prob. 1 Eigenvalue of a Sturm-Liouville problem: Convergent sequence: 3.13 11.7 Convergent series: 5.2 Eigenvector of an operator: 1 I . 1 Convex set, convex function: 8.5, prob. 8 Eilenberg’s criterion: 9.Ap.3 Coordinate (nth) with respect to an ortho- Element: 1.1 normal system: 6.5 Elementary solution for a Sturm-Liouville problem: 11.7 Covering of a set: 1.8
INDEX
Empty set: 1.1 Endless road: 9.12, prob. 3 Entire function: 9.3 &-Entropyof a set: 3.16, prob. 4 Equation of a hyperplane: 5.8 Equicontinuous at a point, equicontinuous : 7.5 Equipotent sets: 1.9 Equivalence class, equivalence relation: 1.8 Equivalent norms: 5.6 Equivalent roads: 9.6 Essential mapping: 9.Ap.2 Essential singular point, essential singularity: 9.15 Euclidean distance: 3.2 Everywhere dense set: 3.9 Exponential function: 4.3 and 9.5 Extended real line: 3.3 Extension of a mapping: 1.4 Exterior point of a set, exterior of a set: 3.7 Extremity of an interval: 2.1 Extremity of a path: 9.6 F
Family of elements: 1.8 Finer distance, finer topology: 3.12 Finite number: 3.3 Fixed point theorem: 10.1 Fourier coefficient (nth): 6.5 Fredholm equation, Fredholm alternative: 11.6 Frobenius’s theorem: 10.9 Frobenius-Perron’s theorem : 11.l, prob. 6 Frontier point of a set, frontier of a set: 3.8 Full sequence of positive eigenvalues: 11.5, prob. 8 Function: 1.4 Function of bounded variation: 7.6, prob. 3 Function of positive type: 6.3, prob. 4 Functional graph, functional relation: 1.4 Functions coinciding in a subset: 1.4 Fundamental system of neighborhoods: 3.6 Fundamental theorem of algebra: 9.1 1
G Geometric multiplicity of a n eigenvalue: 11.4 Goursat’s theorem: 9.10, prob. 1
383
Gram determinant: 6.6, prob. 3 Graph of a relation: 1.3 Graph of a mapping: 1.4 Greatest lower bound: 2.3 Green function of a Sturm-Liouville problem: 11.7 Gronwall’s lemma: 10.5
H Haar orthonormal system: 8.7, prob. 7 Hadamard’s three circles theorem: 9.5, prob. 10 Hadamard’s gap theorem: 9.15, prob. 7 Hausdorff distance of two sets: 3.16, prob. 3 Hermitian form: 6.1 Hermitian kernel: 1I .6 Hermitian norm: 9.5, prob. 7 Hermitian operator: 11.5 Hilbert basis: 6.5 Hilbert space: 6.2 Hilbert sum of Hilbert spaces: 6.4 Homeomorphic metric spaces, homeomorphism: 3.12 Homogeneous linear differential equation: 10.8 Homogeneous hyperplane: 5.8, prob. 3 Homotopic paths, homotopic loops, homotopy of a path into a path: 9.6 and 10.2, prob. 6 Hyperplane: 5.8 and 5.8, prob. 3 Hyperplane of support: 5.8, prob. 3 I
Identity mapping: 1.4 Image of a set by a mapping: 1.5 Imaginary part of a complex number: 4.4 Implicit function theorem: 10.2 Improperly integrable function along an endless road, improper integral: 9.12, prob. 3 Increasing function: 4.2 Increasing on the right: 8.5, prob. 1 Indefinitely differentiable mapping : 8.12 Index of a point with respect to a circuit, of a circuit with respect to a point: 9.8 Index of a point with respect to a loop: 9.Ap.l Induced distance: 3.10 Inessential mapping: 9.Ap.2
384
INDEX
Infimum of a set, of a function: 2.3 Infinite product of metric spaces: 3.20, prob. 7 Injection, injective mapping: 1.6 Integer (positive or negative): 2.2 Integral: 8.7 Integral along a road: 9.6 Integration by parts: 8.7 Interior point of a set, interior of a set: 3.7 Intersection of two sets: 1.2 Intersection of a family of sets: 1.8 Inverse image: 1.5 Inverse mapping : 1.6 Isolated point of a set: 3.10 Isolated singular point: 9.15 Isometric spaces, isometry: 3.3 Isomorphism of prehilbert spaces: 6.2 Isotropic vector: 6.1 1
Jacobian matrix, jacobian: 8.10 Janiszewski’s theorem: 9.Ap.3 Jordan curve theorem: 9.Ap.4 Juxtaposition of two paths: 9.6 K
Kernel function: 11.6 L Lagrange’s inversion formula: 10.2, prob. 10 Laurent series: 9.14 Least upper bound: 2.3 Lebesgue function (nth): 11.6, prob. 2 Lebesgue’s property: 3.16 Legendre polynomials: 6.6 and 8.14, prob. 1 Leibniz’s formula: 8.13 Leibniz’s rule: 8.11 Length of an interval: 2.2 Limitof a function, limit of a sequence: 3.13 Limit on the left, limit on the right: 7.6 Linear differential equation: 10.6 Linear differential equation of order n : 10.6 Linear differential operator: 8.13 Linear form: 5.8 Linear variety: 5.1, prob. 5
Linkedbyabroken line(points): 5.1, prob. 4 Liouville’s theorem: 9.1 1 Lipschitzian function: 7.5, prob. 12, and 10.5 Locally closed set: 3.10, prob. 3 Locally compact space: 3.18 Locally connected space: 3.19 Locally lipschitzian function: 10.4 Logarithm: 4.3 and 9.5, prob. 8 Loop: 9.6 and 10.2, prob. 6 Loop homotopy: 9.6 and 10.2, prob. 6 M
Majorant: 2.3 Majorized set, majorized function: 2.3 Mapping: 1.4 Maximal solution of a differential equation: 10.7, prob. 4 Maximinimal principle: 11.5, prob. 8, and 11.7, prob. 2 Mean value theorem: 8.5 Mercer’s theorem : 1 1.6 Meromorphic function: 9.17 Method of the gliding hump: I I .5, prob. 4, and 11.6, prob. 2 Metric space: 3.1 Minimal solution of a differential equation: 10.7, prob. 4 Minorant: 2.3 Minorized set, minorized function: 2.3 Minkowski’s inequality: 6.2 Monotone function: 4.2 Morera’s theorem: 9.10, prob. 2 N
Natural boundary: 9.15, prob. 7 Natural injection: 1.6 Natural mapping of X into X/R: 1.8 Natural ordering: 2.2 Negative number: 2.2 Negative real half-line: 9.5, prob. 8 Neighborhood: 3.6 Newton’s approximation method: 10.2, prob. 5 Nondegenerate hermitian operator: 11.5 Norm: 5.1 Normally convergent series, normally summable family: 7.1 Normed space: 5.1
INDEX
0 One-to-one mapping: 1.6 Onto mapping: 1.6 Open ball: 3.4 Open covering: 3.16 Open interval: 2.1 Open neighborhood: 3.6 Open polydisk: 9.1 Open set: 3.5 Operator: 11.1 Opposite path: 9.6 Order of an analytic function at a point: 9.15 Order of a linear differential operator: 8.13 Ordered pair: 1.3 Origin of an interval: 2.1 Origin of a path: 9.6 Orthogonal projection: 6.3 Orthogonal supplement: 6.3 Orthogonal system: 6.5 Orthogonal to a set (vector): 6.1 Orthogonal vectors: 6.1 Orthonormal system: 6.5 Orthonormalization: 6.6 Oscillation of a function: 3.14
385
Precompact set: 3.17 Precompact space: 3.16 Prehilbert space: 6.2 Primary factor: 9.12, prob. 1 Primitive: 8.7 Principle of analytic continuation: 9.4 Principle of extension of identities: 3.15 Principle of extension of inequalities: 3.1 5 Principle of isolated zeros: 9.1 Principle of maximum: 9.5 Product of a family of sets: 1.8 Product of metric spaces: 3.20 Product of norrned spaces: 5.4 Projection (first, second, ith): 1.3 Projections in a direct sum: 5.4 Purely imaginary number: 4.4 Pythagoras’s theorem: 6.2
Q Quasi-derivative, quasi-differentiable function: 8.4, prob. 4 Quasi-hermitian operator: 11.5, prob. 18 Quotient set: 1.8 R
P p-adic distance: 3.2 Parallel hyperplane: 5.8, prob. 3 Parseval’s identities: 6.5 Partial derivative: 8.9 Partial mapping: 1.5 Partial sum (nth) of a series: 5.2 Partition of a set: 1.8 Path: 9.6 and 10.2, prob. 6 Path reduced to a point: 9.6 Peano curve: 4.2, prob. 5, and 9.12, prob. 5 Peano’s existence theorem: 10.5, prob. 4 PhragmCn-Lindelof’s principle: 9.5, prob. 16 Picard’s theorem: 10.3, prob. 8 Piecewise linear function: 8.7 Point: 3.4 Pole of an analytic function: 9.15 Positive definite hermitian form: 6.2 Positive hermitian form: 6.2 Positive hermitian operator: 11.5 Positive number: 2.2 Power series: 9.1
Radii of a polydisk: 9.1 Radius of a ball: 3.4 Rational number: 2.2 Rank theorem: 10.3 Real line: 3.2 Real number: 2.1 Real part of a complex number: 4.4 Real vector space: 5.1 Reflexivity of a relation: 1.8 Regular frontier point for an analytic function: 9.15, prob. 7 Regular value for an operator: 11.1 Regularization: 8.12, prob. 2 Regulated function: 7.6 Relative maximum: 3.9, prob. 6 Relatively compact set: 3.17 Remainder (nth) of a series: 5.2 Reproducing kernel: 6.3, prob. 4 Residue : 9.15 Resolvent of a linear differential equation: 10.8 Restriction of a mapping: 1.4 Riemann sums: 8.7, prob. 1
386
INDEX
Riesz (F.)’s theorem: 5.9 Road: 9.6 Rolle’s theorem: 8.2, prob. 4 RouchB’s theorem: 9.17
S Scalar : 9.1 Scalar product: 6.2 Schoenflies’s theorem: 9.Ap., prob. 9 Schottky’s theorem: 10.3, prob. 6 Schwarz’s lemma: 9.5, prob. 6 Second mean value theorem: 8.7, prob. 2 Segment: 5.1, prob. 4, and 8.5 Self-adjoint operator: 1 1.5 Semi-open interval : 2.1 Separable metric space: 3.10 Separating points (set of functions): 7.3 Separating two points (subset of the plane): 9.Ap.3 Sequence: 1.8 Series: 5.2 Set: 1.1 Set of mappings : 1.4 Set of uniqueness for analytic functions: 9.4 Simple arc, simple closed curve, simple loop, simple path: 9.Ap.4 Simply connected domain: 9.7 and 10.2, prob. 6 Simply convergent sequence, simply convergent series: 7.1 Simpson’s formula: 8.14, prob. 10 Singular frontier point for an analytic function: 9.15, prob. 7 Singular part of an analytic function at a point: 9.15 Singular values of a compact operator: 11.5, prob. 15 Solution of a differential equation: 10.4 and 11.7 Spectral value, spectrum of an operator: 11.1 Sphere: 3.4 Square root of a positive hermitian compact operator: 11.5, prob. 12 Star-shaped domain: 9.7 Step function: 7.6 Stone-Weierstrass theorem: 7.3 Strict relative maximum: 3.9, prob. 6 Strictly convex function: 8.5, prob. 8
Strictly decreasing, strictly increasing, strictly monotone: 4.2 Strictly negative, strictly positive number: 2.2 Sturm-Liouville problem: 11.7 Subfamily: 1.8 Subsequence: 3.1 3 Subset: 1.4 Subspace: 3.10 Subspace of a normed space: 5.4 Substitution of power series in power series: 9.2 Sum of a family of sets: 1.8 Sum of a series: 5.2 Sum of an absolutely summable family: 5.3 Supremum of a set, of a function: 2.3 Surjection, surjective mapping: 1.6 Symmetric bilinear form: 6.1 Symmetry of a relation: I .8 System of scalar linear differential equations: 10.6
T Tangent mappings a t a point: 8.1 Tauber’s theorem: 9.3, prob. 2 Taylor’s formula: 8.14 Term (nth) of a series: 5.2 Theorem of residues: 9.16 Tietze-Urysohn extension theorem : 4.5 Titchmarsh’s theorem: 11.6, prob. 1 1 Topological direct sum, topological direct summand, topological supplement: 5.4 Topological notion: 3.12 Topologically equivalent distances: 3.12 Topology: 3.12 Total derivative: 8.1 Total subset: 5.4 Totally disconnected set: 3.19 Transcendental entire function: 9.15,prob. 3 Transitivity of a relation: 1.8 Transported distance: 3.3 Triangle inequality: 3.1 and 5.1 Trigonometric polynomials : 7.4 Trigonometric system: 6.5
U Ultrametric inequality: 3.8, prob. 4 Underlying real vector space: 5.1
INDEX
Uniformly continuous function: 3.1 1 Uniformly convergent sequence, uniformly convergent series: 7.1 Uniformly equicontinuous set: 7.5, prob. 5 Uniformly equivalent distances: 3.14 Union of two sets: 1.2 Union of a family of sets: 1.8 Unit circle: 9.5 Unit circle taken n times: 9.8
Vector space: 5.1 Volterra kernel : 11.6, prob. 8 W
Weierstrass’s approximation theorem: 7.4 Weierstrass’s decomposition: 10.2, prob. 8 Weierstrass’s preparation theorem: 9.17, prob. 4 Weierstrass’s theorem on essential singularities: 9.15, prob. 2
V
Value of a mapping: 1.4 Vector basis: 5.9, prob. 2
387
Z Zero of an analytic function: 9.15
Pure and Applied Mathematics A Series of Monographs and Textbooks Editors
Samuel Eilenberg and Hyman Baas Columbia University, New York
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