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- u of B* x V* onto B"- is the adjoint of the map x ->- (x, 0) of B into B x V. It follows from Lemma 2 that the norm defined on B* by [u] = inf {[(u, g)]: g E V"} corresponds to the norm defined on B by [x] = [(x, 0)], and that the norms [ ] and II II on B are equivalent. But [x]
=
inf {[(f, 0)]: '[f
=
x}
=
inf UIK(f, 0)11: '[ f
= inf {11(f, Hf)ll: '[I = x} = inf {(~IIII12 + IIHIII')'!2: '[I =
=
x}
x] = {x] .
Also, [u]
=
inf {[('[*u, g)]: g E V*}
=
inf {II K-1-(,[*u, g) II: g E V*}
inf {11(H*g + r-"-u, -g) II: g E V"} = inf UI (H*g - r-"u, g) II: g E V*} = inf {(~-'IIH"-g - r-"-uI1 2+ IlgI12)"2: gE V"-] = {u} . =
This proves Theorem 1. Of the two constituents, ~l '11111 and IIHIII, \vhich enter into the norm {xl = inf {(~IIIII' + IIHIII')'2: r- I=:c} on B, imprecisely speaking the second constituent
IIHIII = sup {II(T - zI)I(z) + x II: z E U
J}
measures the degree of approximation to which one can solve the resolvent equation (T - zI)I(z) = x on Up and the first 1}111II1 measures the size of the approximate solution. A similar comment can be made for the norm ~ ~ on B". Thus Theorem 1, which states that the norms correspond, establishes a relationship between approximate solutions to the resolvent equation for T on the set U and approximate solutions to the resolvent equation for T"- on the set U,. In view of the fact that the spectral manifolds were defined in terms of approximate solutions to the resolvent equation, it is not surprising that Theorem 1 should contain information pertinent to the duality theories defined above. Actually, the duality theory which we obtain as a consequence of Theorem 1 will be less precise than Theorem 1 itself. It therefore seems likely that a complete duality theory for an arbitrary operator will be based upon some version of Theorem 1 itself rather than upon the duality theory which we now proceed to derive from Theorem 1. It would have been possible to derive Theorem 1 in more generality by considering possibilities other than V for the Banach space of analytic functions from U l to J
82
A DUALITY THEOREM FOR AN ARBITRARY OPERATOR
389
B, but this was not necessary for the applications to be made. Let the subset N of B consist of all x such that for each c > 0 there exists f in V with IIHfl1 < c and 7:f = x. Let the subspace M of B* be the closure of the set Mo of all u in B" such that there exists g in V* with H"g = 7:"u. Then N is a subspace and M1. = N. COROLLARY.
We first show that N = [x: {xL --> 0 as r) --> 0]. If x EN f in V with 7:f = x and IIHfl1 < c. Thus {x}~ ~ (1J11fW + C')'/' --> C as IJ --> O. Since c is arbitrary it follows that {X}, --> 0 as 7J --> O. Conversely if {x}" --> 0 as ~'--> 0 then for each c > 0 there exists r; > 0 such that {xL < c, which implies that there exists f in V with Tf = x and (~IIJII' + IIHfll')" < c. Thus IIHfl1 < c, so that Proof.
for each c
> 0 there exists
xEN.
We next show that Mo = {u: {u; 1 is bounded for r; then there exists g in V* with H*g = 7:i-u. Therefore
> O}.
If u E Mo,
is bounded for r; > O. Conversely, assume now that {u}" < c for all positive ~'> Then for each positive integer n there exists gn in V" with (nIIH*gn - r*ull' + Ilgnll') < c'. Thus Ilgnll < c. Since the unit sphere of V" is compact in the weak star topology, there exists a cluster point g of {gn} in the weak star topology. Therefore H*g is a cluster point of {H*gn} in the weak star topology. Since IIH*gn - T*ull' ~ c'n- 1 , it follows that H*g = T*U. Therefore u E Mo. If xENand uEMo, then l<x,u)1 ~ {X}1{U},--> 0 as ~'-->O. Therefore Nand Mo are orthogonal. If x is not in N, then {X}1 does not converge to 0 as 7J --> 0, so that there exists a sequence {r;n) of positive constants converging to 0 and a constant c > 0 such that {xL > c for all n. It follows that there exists a sequence [un} of element; in B'" with {unL < 1 and <x, un) > c. We can thus find a sequence {gn} of elements i~ V* with
As before, there exists a cluster point (g, u) of the sequence Ugn, un)} in the weak star topology of V'" x B*. It follows, as before, that H*g = 7:"u. Thus u E Mo. Since also <x, u) ~ c, the vector x is not in MoL. Thus M if- = N. It follows that N is a subspace and that M 1. = N, as was to be proved. As an easy consequence of this corollary, we obtain the following fundamental theorem.
83
390
ERRETT BISHOP
THEOREM 2. Every operator T on a reflexive Banach space B admits a duality theory of type 4.
Proof. Let F, and F2 be disjoint compact sets. Then there exists an unbounded simple open set U" whose complementary simple open set will be called U., such that F,c U, and F,c U 2 • Define N as in the above corollary. To see that N(F" T)cN, take x in N(F" T) and take c: > O. Let Zo be any point in U2 and let inf {Iz - zol: ZE U,} = c > O. By the definition of N(F" T), there exists an analytic function f from -F, to B such that II(T - zl)f(z) - xll < c: for all z in -F" Since f is analytic on the neighborhood -F, of U, U C, Runge's theorem for vector-valued analytic functions (whose proof is exactly the same as the proof of Runge's theorem in the scalar case) implies that f can be uniformly approximated on U, U C by linear combinations of elementary elements of V,. Therefore f E V. Define f, in V by f, = - f + a(x
+ Tf, zo,
,) .
Then Tf, = -Tf + lim z(z - ZO)-'(X
+ Tf) =
X.
Also,
l1 = sup lIl(T - ZI)f,(Z) + Tf,ll: ZE U,} sup {II-(T - zI)f(z) + (z - zo)-'(T - zl)(x + T f) + T f,ll: sup {ll(T - zl)f(z) - x ll: Z E U1 } + sup {11(z - zo)-1(T - zI)(x + Tf)ll: c: + sup {1I(z - zo)-I(T - zl)(x + Tf)ll: ZE U,} .
IIHf,
= ~
~
E
U1 }
ZE
U,}
Z
Now
Ilx + rfll
= lim
II-[(T -
zI)f(z) - x] II ~ c: .
Thus lI(z - zo)-'(T - zl)(x
+ Tf)11
~ ~
11[(z - zo)-'(T - zol) - 1](x + Tf)11 [c-'II Tll + c-'Izol + 1]c: .
Thus
IIHf,l1
~
[c-'II Til + c-'Izol + 2]c: .
Since c: is arbitrary, x E N. Thus N(FH T)cN. We next show that M(F" T*)cM, where M is defined in the above corollary, It is sufficient to consider u in M(F" T*) such that there exists an analytic function g from -F2 to B* with (T* - zl)g(z) = u
84
A DUALITY THEOREM FOR AN ARBITRARY OPERATOR
391
for all z in -F" since the set of such u is dense in M(F" T*). Since g is analytic on the neighborhood - F2 of U 2 U C, g E V,c V*. By the definition of H* it follows that H*g = ,*u. Thus u E M, so that M(F" T*)cM. Since M 1- = N by the above corollary, it follows that
Since this holds for all operators T, we may replace T by T* and mterchange F, with F, to obtain
It remains to consider open sets G, and G2 which cover the complex plane. There exists an unbounded simple open set U, such that U,cG,U -a(T) and (J,cG" where U, is the bounded simple open set which is complementary to U,. Define V, M, and N as in the above corollary. We first show NcN(iJ" T). Consider x in N, so that for each c > 0 there exists f in V with ' f = x and IIHfl1 < c. This implies that
II(T - zI)f(z)
+ xII =
II(T - zI)(-f(z)) - xii
r- 1 and (T-l - z-l1)g(Z-I) = T-I X • Therefore T-I X E M(Kr-t, T-l). By the above, this implies that T-I X = O. Therefore x = O. Therefore M(L" T) = {O}. Similarly, M(L" Ti-) = {O}. This already shows that T does not admit a duality theory of type 1, since by the above it is not true that
<s x uniformly on U,. If Un} is not uniformly bounded on K, by passing to a subsequence if necessary we may assume that d n +, > n + d n for each n, where
=
d n = sup
~ll/n(z)
Thus the sequence {gn] defined by gn on K. On the other hand,
=
II:
Z
j~+ 1
E -
K} . In is not uniformly bounded
(T - z1)gn(z) = (T - z1)ln+,(z) -- (T - z1)ln(z)
converges uniformly to x - x = 0 on U,. This contradicts condition /3. Therefore {In} is uniformly bounded on K, as was to be proved. DEFINITION 9. The operator T on the reflexive Banach space B will be said to satisfy condition ~/ if for every open set U and every non-zero analytic f from U to B there exists z in U with (T - z1)f(z) O.
'*
THEOREM 4. If T satisfies condition /'3 and if F is an arbitrary closed subset of the complex plane, then (1) M(F, T) = N(F, T), (2) for
87
394
ERRETT BISHOP
each x ~n M(F, T) there exists an analytic function f from - F to B with (T - z1)f(z) = x for all z in -F, and (3) T satisfies conditions a and /. Proof. For each x in N(F, T) there exists a sequence {fn} of analytic functions from -F to B such that II (T - zI)fn(z) - x II < n -1 for all z in -F. By condition (3, {fn} is uniformly bounded on compact subsets of - F, so that there exists a subset, which we continue to call Un}, converging pointwise in the weak topology of B to an analytic function f from -F to B. Clearly (T - z1)f(z) = x for all z in -F. This proves assertions (1) and (2). If F, and F2 are closed and F, c interior F 2 , then N (F" T) = M(F" T)cM(F" T) by (1), so that T satisfies condition a. If T does not satisfy condition /, there exists an open set U and a non-zero analytic function f from U to B with (T - zI)f(z) = 0 for all z in U. Let fn = nf, so that (T - zI)fn(z) -'r 0 as n -'r = uniformly on U. Since T satisfies condition (:J, it follows that the sequence ~nf~ is pointwise bounded. This contradiction proves that T satisfies condition /'. The theorem just proved is related in part to results of Wermer [5]. As a corollary of this theorem, T admits a duality theory of type 1 if T satisfies condition /:J. THEOREM 5. If T+ sat'isfies condition (J, then T admits a duality theory of type 2. If T and T* satisfy condition (3, then T admits a duality theory of type 3.
Proof. Assume that T' satisfies condition {J. Then T", and therefore T, admits a duality theory of type 1. Also, T*- satisfies condition /'. Let G,,···, Gn be open sets covering the complex plane. If M(G" T), . ", M(G a , T) do not span B, there exists u * 0 in B' orthogonal to these subspaces. Since G,,···, Gn cover the complex plane, n7~1 -G, is void. There therefore exist open sets U,,"', Un such that -G,c U, and nr~1 U, is void. Since U, and G, cover the complex plane and since T admits a duality theory of type 1, we see that M((J., T)~ c M( U" T ') .
Thus u € M( Ui , T') for each i. Since T" satisfies condition fJ it follows from Theorem 4 that there exist an analytic function g, from - U, to B* with (T* - z1)g,(z) = u for all z in - U,. If z E - [i, n - U), we have (T* - zI)(g,(z)-g;(z» = O. Since g,-gj is analytic on -uin -(fj and since T* satisfies condition /, it follows that g,(z) = g,(z) for z in - U, n - (fj. Thus if we define the function 9 by g(z) = g,(z) for z in - U" then 9 is uniquely defined and is analytic on U~-l - (f,. Also
88
A DUALITY THEOREM FOR AN ARBITRARY OPERATOR
~95
(T" - zI)g(z) = u whenever 9 is defined. But the sets -l1; cover the plane because n~~l Ui is void. Also g(z) = (T' - zI)-J u -40 as Iz 1---> co. Thus 9 is everywhere analytic and vanishes at infinity, so that 9 = O. Thus u = (T > -- zI)g(z) = O. This contradiction implies that M(G]> T), "', M(G,,, T) span B. Therefore T admits a duality theory of type 2. Assume now that T also satisfies condition /3. We must show that T admits a duality theory of type 3. First consider a bounded open set H such that -H is connected. By Theorem 4 for each x in M=M(H, 1') there exists an analytic function f(x, .) from U =- -H to B such that (T - zI)f(x, z) = x for all z in U. Since l' satisfies condition /-], and therefore condition OJ, the function f(x, .) is unique. For each z" in U define the linear transformation T 0 and II Tzuxn II -4 co as n ---> CD. Since T satisfies condition ;3, it follows that U(x n, zo)} = [T'uxn} is bounded. This contradiction implies that Tzu is a bounded transformation for each Zo in U. If z in U is sufficiently large and if x E M, T,x = (T - zI)-lx EM. Since T,x is an analytic function on U and since U is connected, it follows that T,x EM for all x in M and all z in U. Thus for each z in U the operator T, on M is a right inverse of (T - zI)/A1. To show that T z is the inverse of (T - zI)/M it is sufficient to establish that x = 0 whenever x EM and (T-zI)x = O. To this end, consider x in M with (T-zI)x == O. Define the analytic function h from U to B by h(A) = x + (,\ - z)T"x. Then (T - AI)h(A) = (1' - zI)x
=
0
+ (z
+ (z
- A,)X
- A)X
+ (,\, -
+ (A z)x
=
- z)(T - AI)T"x
0 .
Since T satisfies condition (3, and therefore condition "j, this implies h = O. Thus x = h(z) = O. Therefore (T - zI)/M has an inverse for all z in U, so that (J(T/M(R, T»cH. Now let a covering G]O "', G n of the complex plane by open sets be given. There exists a family [H ik ] , 1 ;;;; i ;:;:: n, 1 ;:;:: k ;:;:: mil of open sets which cover the complex plane such that HikcG i and either Hi" is bounded and -Ri" is connected or Hikc -(J(T). From the preceding paragraphs we see that the M(H'k' T) span B and that (J(T / M(R" , T»cRik
if Hi" is bounded and -Hik is connected. Also a(T/M(H,,,, T»cH", if Rikc -(J(T) because M(Rit> T) consists only of the zero vector in this case. If this were not true, there would exist x -cf' 0 in M(R" , 1') and
89
396
ERRETT
BISHOP
-it.
an analytic function f from to B such that (T - zI)f(z) = x. If we extended the definition of f to -aCT) by defining fez) = (T - zI)-'x for z in -aCT), then f becomes an analytic function defined everywhere and vanishing at infinity. Thus f = O. Therefore x = (T - zI)f = O. Hence a(T/M(ii,., T»cii'k for all i and k. Define M! to be the subspace spanned by the sets M(H'H T), "', M (H""" T). Since -
-
M(Hie , T)cM(G" T) for each k, we have M,cM(G" T). Also M" "', Mn span B. To complete the proof of the theorem we must show that a(T/M,)cG,. As before, with the set H replaced by G" for each x in M,cM(G" T) there exists a unique analytic function f(x, .) from -Gi to B such that (T - zI)f(x, z) = x for all z in -Gi . As before, for each z in -G! define the linear transformation T z from Mi to B by Tzx = f(x, z). As before, T z is a bounded linear transformation for each z in -G t • If the set H,. is bounded and -iii. is connected, we see as before that the spaces M(iiild T) are invariant under T, for each z in -iiik~ -G,. Since otherwise H,. has the property that M(lt, , T) consists only of the vector 0, and since the M(iiik' T) span M, it follows that M, is invariant under Tz for all z in -G,. Thus the operator T z on M, is a right inverse of (T - zI)/ Mi' As before, it follows that T z is the inverse of (T - zI)/M, for each z in -G,. Thus a(T/M,)cGi , as was to be proved. As an immediate consequence of the preceding theorems, it follows that T and T* both admit duality theories of types 1, 2, 3, and 4 whenever T and T* both satisfy condition /3. DEFINITION 10. Let r be a function from the complex plane to the set [0, (;0]. The function r will be said to be a modulus of control for analytic functions if for every open set U and every sequence {fn} of complex-valued analytic functions on U such that Ifn(z) I ~ r(z) for all z in U the sequence {fn} is uniformly bounded on all compact subsets of U. THEOREM 6. Let T be an operator on a reflexive Banach space B. Let the function r defined by r(z) = II(T - zIt'l1 if Ze -aCT) and r(z) = (;0 if Z e aCT) be a modulus of control for analytic functions. Then both T and T'" satisfy condition /3 and therefore admit duality theories of types 1, 2, 3, and 4.
Proof. Since II(T - zIt'l1 = II(T* - zI)-'11 whenever Ze -aCT), it is sufficient to show that T satisfies condition /3. To do this, let U be open and let {fn} be a sequence of analytic functions from U to B such
90
A DUALITY THEOREM FOR AN ARBITRARY OPERATOR
397
that (T - zI)fn(z) -+ 0 uniformly on U as n -+ oo. It is then possible to choose a constant K such that Ilfn(z) II ; 3/4, and If(y) I < 1/4 for all y in Dn(x), where Dn(x) = {y: p(x, y) ;:::: n- ' } and fJ is a metric on C. Then Un is open and Un = M, where M is the minimal boundary of ~l.
n
Proof. If f is any function in 'ill, it is clear that the set a n(f) = {x: x e C, If(x) I > 3/4, If(y) I < 1/4 whenever ye D,,(x)} is open for each n. Since Un is the union of the sets belonging to the class {an(f):f e ~l,
Ilfll;;;;; I} ,
it follows that Un is open. If x e M, by Theorem 1 there exists f in ~( with S (f) = {x}. It is clearly no restriction to assume that Ilfll = 1. Hence If(x) I = 1. Since If(y) I < 1 when y is in the compact set Dn(x), it follows that there exists a positive integer Pn such that If(y) IP n < 1/4 when ye D,,(x). Thus x e (In(fPn). Therefore x e Un. Since this is true for each n, it follows that x e Un. Therefore Men Un. Now consider a fixed x in Un. We must prove that x eM. To this end, we construct by induction a sequence {gn} of functions in ~( having the following properties:
n
(i)
(ii) (iii) (iv)
n
IIgn+! - gnll ;;;;; 2- n + IIgnll;;;;; 3(1 - 2- n -
1
1)
gn(x) = 3(1 - 2- n ) Ign+l(y) - gn(y) I < 2- n - 1 if ye Dn(x) .
We first construct g,. Since x e U" there exists a function that Ilfll ;;;;; 1 and x e (J,(f). Let gl
f in
~{
such
= ~ [f(x)]-lf .
Since If(x) I > 3/4, we have IlgIII;;;;; 3/2 . 4/3 = 2 < 3(1 - 2- 2 ), so that g, satisfies (ii). Clearly g,(X) = 3(1 - 2- ' ), so that gt satisfies (iii). Hence g, satisfies all of the relevant conditions. Assume now that g" "', gk have been chosen to satisfy all of the relevant conditions. Since gk(X) = 3(1-2- k ), there exists an integer }>k such that Igk(y)I (3/4)'/m and If(Y) I ~ hnmCY) for all y in C, or (Case 2) there exists a positive integer n such that for all positive integers m and for all f in ';It either If(x) I ~ (3/4)1/'" or If(Y) I > hnm(y) for some y in C. We shall show that Case 1 is impossible and that Case 2 implies the theorem to be proved. Assume now that Case 1 obtains. Let the positive integer n be given, and choose fin ';!l and a positive integer m such that If(x) I >(3/4)I/m and If(y) I ~ hnm(y) for all y. Write g =fm. Since If(y) I ~ hnm(y) ~ e' /m for all y, we have Ig(y)1 ~ e for all y . .Thus Iigll ~ e < 1. Since If(x)1 > (3/4)I/m we have Ig(x) I > 3/4. Since IfCy) I ~ hnm(y) = bl / m for y in D n, we have Ig(y) I ~ b < 1/4 for y in Dn. It follows that x E Un, where Un is the set defined Un = M, in Theorem 2. Since this is true for each n, we have x by Theorem 2. This contradicts the hypothesis of Theorem 3. Therefore Case 1 is impossible. We are therefore justified in assuming that Case 2 obtains. Thus there exists a positive integer n, henceforth fixed, such that for all positive integers m and allf in ~l either If(x) I ~ (3/4)I/m or I fey) I>hmn(y) for some y in C. Consider now a positive integer m. For each f in ';!( either If(x) I ~ (3/4)I/m or Ilfh- 1 11 > 1, where h = h nm . Thus If(x) I ~ (3/4)I/1n whenever f E ';II and Ilfh- 1 11 ~ 1. Let)S be the Banach space of all continuous functions on C, under the uniform norm, and let ~\ be the subspace Uh- I : f e ~{} of 5B. Define the linear functional cp on l.L~o by defining 1.
«
Let fJ. be in Hx(B).
Suppose that 2 holds. for each g in C.(X) with g L Z.~, PROOF. -
Jg dfJ. 2 But
J
g!
(~ = «gl>1. g dfJ.: g e C.(Xl, gL"'/..sL
Sup) (dfJ.: (e B, Sup) ((x) : ( e B,
P.(S) = Infl
Then
J
so P.(S) L 1. For the converse suppose that 2 does not hold. a function in C.(X) with g > I.~ and Sup I((x) : (
e
«
B,
gl = 1 -
Let g be
Eo
Define the linear functional Lx on B by all (in B.
(5. 1)
L". is a positive linear functional on B (that is, non-negative on non-negative functions) and thus by a standard result (see [10J, p. 22) on the extension of such functionals, there is a positive linear functional L on C.(X) with L(f) = Lx(f) ,
(5. 2)
all (in B,
and L(g) = 1 - t:. By the Riesz representation theorem, there is a p. in H with (5. 3)
L(f) =
Jf dfJ.,
all (in C.(X).
Because of (5. 1), (5. 2) and (5. 3), po is in Hx(B) and P.(S) =
f AS dfJ. < Jg d[l.
=
1-
This completes the proof of Lemma 5. 1.
123
E
~
and If(y)J
! ua.l converges to s(x) if and only if the net I
130
!n( Ua.) l converges to x in the topology of X. If X and the Y.z; are compact Hausdorff, then Y is compact Hausdorff. Suppose now that X and the Y x are compact Hausdorff. Let D be a subspace of Cr(X) and for each x in X, let Bx be a subspace of Cr(Y r)' Let B be the subspace of Cr(Y) consisting of all fin Cr(Y) such that f S is in D, and for each .1': in X, f restricted to Y x is in Bx. If D and all the 87: are closed subspaces and distinguish points, B will be a closed subspace and distinguish points. It is simple to check that the Choquet boundary M(B) of B is 0
(7. 1) We shall now consider a special cas(' of the above construction. Let X be an arbitrary compact Hausdorff sp8.ce and K an arbitrary subset of X. For each .'C in K, let Y x eomist of the one point S r, and for (,Heh .1' in X -K, let Y be the rliscrete topological space eonsisting of the three points rJ') Sec, tx;' Define s: X -?- Y by s (x) = S.c, all x III X. Let D be Cr(X), and if x is in K, let H.r = Cr(Y x)' If x is in X - K, let B be the suhspace of Cr(Y x) consisting of those f that satisfy 7
!
L
f(sJ')
1
= 2 (f(r,)
+ f(tx)).
The construction described above applied to D and the Bx yields a ('losed subspace B of Cr(Y) that distinguishes points. Its Choquet boundary is (7. 1) and is therefore easily seen to satisfy Y - M(B) = s(X - K). Since K was an arbitrary subset of X, this shows that the Choquet boundary can be arbitrarily bad. An example of a bad houndary has also been given by Choquet in [8J. Suppose now that in this example we take X to be the unit interval with the usual topology and K to be the void set. Let ') be Lebesgue measure on X, and let p. be the Baire measure on Y defined by
for all Baire subsets S of Y. Then fJ- is B-maximal. ~ever theless its regular Borel extension p. satisfies P.(M(B)) = O.
131
THE
REPRESENTATIONS
OF
LINEAR
FUNCflONALS
329
This is in contrast to Theorem 5. 3 which shows that a B-maximal measure must be (( concentrated on M(B) » In the sense that p.(S) = 0 for each Baire set S disjoint from M(B). The example shows that thc conclusion of the theorem cannot be strengthened to P.(S) being 0 for each Borel set S disjoint from M(B), even if M(B) itself is Borel. It also shows that the measures p. appearing in Theorem 5. 5 may not be regular. In order to obtain a morp stnkillg example, we take X to be the sub'let of the complex plane
X=rz:[z[..imated on the boundary B of C by polynomials There therefore exists a measure Jl. on B which is orthogonal to all polynomials, but not orthogonal to f. Since
Jl. is orthogonal to polynomials and since - C is connected, we see that M(C). By Theorem 3, there exists dw in H(U) such that Jl. is the boundary measure of dw. By Definition .5, it follows that there exists a bequence (-y,} Jl. £
of compact subsets of U ,,,hich delimits U such that
I.
h dw
---7
J
h dp.
as i
---7 00
for all continuous functionb h on C. Applied to the function f, this gives
1f
dp. = lim
1'"1' f dw =
0,
since fdw is an analytic differential on U. This eontradietb the fact that not orthogonal to f, thereby proving the theorem
Jl. IS
4. Remarks. 1. We have relied heavily, in proving Lemma 4, on Mergelyan's approxi-
mation theorem ([4, 21]) for approximation by rational functiom, on compact sets without iuterior Lemma 4 could be proved equally ,yell by appeal to the general approximation theorem announced ab Theorem 4 of [2] and to a cIabsical theorem of Wah;h [6] on approximation by harmonic functionb. The author wishes to take thib opportunity to point out that Theorem 5 of [1] ib a special case of this theorem of Walsh II. ''Ye have derived Theorem 4 as a bimple consequence of Theorem 3 This raises the question of whether, for a compact set C with connected complement, Theorem 3 can be derived from Theorem 4 That is, it might be suspected that Theorem 3 and Theorem 4 are equivalent, Theorem 3 being a reinterpretation
146
O":l:U
ERKE'rI' HII:!J::lUr
in the dual space of Theorem 4, This does not seem to be the case, so that Theorem 3 represents an essential strengthening of Theorem 4. III. It is natural to ask whether the analog of Theorem 3 holds for more general sets C, provided that H(U) and M(C) are defined appropriately. This seems to be a difficult question. It is clear that some hypotheses on Care necessary. REFERENCES
1. E. BISHOP, The structure of certain measures, this Journal, vol. 25(1958), pp. 283-289. 2. E. BISHOP, Some theorems concerning function algebras, Bulletin of the American Mathematical Society, vol. 65(1959), pp. 77-78. 3. C. CARATHEODORY, Conformal Repre8entation, Cambridge, 1952. 4. S. N. MERGELYAN, Uniform approximation to functions of a complex variable, American MathamaticaI Society Translation no. 101, 1954. 5. O. PERRON, Irrationalzahlen, New York, 1951. 6. J. L. WALSH, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bulletin of the American Mathematical Society, vol. 35(1929), pp. 499-544. 8. A. ZYGMUND, Trigonometrical Series, New York, 1955. UNIVERSITY OF CALIFORNIA BERKELEY
147
A GENERALIZATION OF THE STONE-WEIERSTRASS THEOREM ERRETT BISHOP
1. Introduction. Consider a compact Hausdorff space X and the set C(X) of all continuous complex-valued functions on X. Consider also a subset ~ of C(X) which is an algebra, which is closed in the uniform topology of C(X), which contains the constant functions, and which contains sufficiently many functions to distinguish points of X. Such an algebra ~ is called self-adJ'oint if the complex conjugate of each function in %( is in~, The classical Stone-Weierstrass Theorem states that if 21 is self-adjoint then ~l = C(X). If ~ has the property that the only functions in ~I which are real at every point of X are the constant functions then 21 is called anti-symmetric. Clearly antisymmetry and self-adjointness are opposite properties, in the sense that if ~ has both properties then X must consist of a single point. Hoffman and Singer [2] have studied these two properties and given several interesting examples. The present paper was inspired by their work but it more directly relates to a previous paper of Silov [3]. The purpose of the present paper is to prove the following decomposition theorem for a general algebra ~ of the type defined above. THEOREM. There exists a partition P of X into disjoint closed sets such that ( i) for each S in P the restriction ~s of '2{ to S is anti-symmetric, (ii) if a function f in C(X) has, for each S in P, a restriction to S which belongs to 2l s , then f is in ~, (iii) for each S in P, each closed subset T of X - S, and each c > 0 there exists g in ~l with II g II ;£ 1, with Ig(x) -- 11 < c for x in S, and with Ig(x) I < c for x in T.
Property (ii) of this theorem is the essential new fact of this paper. ,!,he construction given below which leads to the partition P is due to Silov [3], who in essence proved (i) and (iii). Silov proved a weaker property than (ii). Our proofs are different from those of Silov, although the construction is the same. The fact that the Stone-Weierstrass theorem is a special case of the theorem to be proved here is clear. If 21 is self-adjoint then each ~s is self-adjoint. Since ~s is also anti-symmetric, each set S in P consists of a single point. Therefore ~s = C(S). By the theorem to Received June 15, 1960,
777
148
778
ERRETT BISHOP
be proved, it follows that each function in C (X) is in Ill. Thus III = C (X), which is the conclusion of the Stone-Weierstrass Theorem. 2.
Proof of the theorem.
The key step in the proof will be the
following lemma. LEMMA. Let Y be a compact Hausdor.tf space and lB be a subalgebra of C (Y) which contains the constant functions. Let ~ be all real functions in lB. Define Yl 0= y" for Y, and Y2 in Y, to mean that f(Y,) = f(Y2) for all fin !H. Let Q be the set of all equivalence classes for this equivalence relation, so that Q is a partition of Y into disjoint closed sets. Let p. be a finite complex-valued Baire measure on Y and f a function in C (Y) such that
(a)
11p.11~1,
(b)
}gdP.
=
0
(c)
} fdp.
*-
O.
for all g in )S,
Then there exists So in Q and a finite complex-valued Baire measure
lJ on So such that (a,) IllJ II ~ 1, (b,) ~ gdlJ = 0 for all g in ~\ (c,)
In
\
~fdlJ \ ~ \ ~fdP. \.
Proof. It is clearly no loss of generality to assume that)S is closed C (Y). Let 'Y = {gil be a finite set of functions in ~ such that gi ~ 0 for all i,
Xg i = 1 . Let r denote the class of all such 'Y. by writing
Define a partial ordering on
r
{g;} ~ {gil
if there exists a mapping rp of the set of indices j onto the set of indices i such that
for all i. To see that r is a directed set relative to this partial ordering, let {gil and {gil be any two elements of r. Then the set {giga is clearly a common successor of {gil and {gil. Consider 'Y = {gil in r. For each index i let P.i be the measure
149
A GENERALIZATION OF THE STONE WEIERSTRASS THEOREM
779
defined by
for each Baire subset H of S.
Clearly
and kdfl
=
~ ~fdfli .
Thus for at least one value of i with
II fl. II*-O we have
Choose such a value of i and write
It follows that
(a) and (fj)
By the compactness in the weak* topology of the unit sphere of the set of Baire measures on X, it follows that the net {fly} has a cluster point ); in the weak" topology. Let Xo be any point in the support of the measure );, and let So be that member of the partition P which contains Xo. Let X, be any point in X - So. Thus there exists ho in :R with hoexo) *- hoCx,). Let
If the real constants
C,
and
C2
are chosen properly then
It follows that there exists a neighborhood Uo of Xo and a neighborhood U, of X, such that h,(x)
< t,
and
150
X €
Uo
Let A. be continuous real-valued function on the range of h, with 0 ~ A, ~ 1, ;>.,(t) = 0 for t ~ t, A.(t) = 1 for t ~ t. By the Weierstrass approximation theorem, ;>.,(t) is a uniform limit of polynomials in t. Therefore the function h. = A.oh, IS III m. Clearly 0 ~ h. ~ 1, h,(x) = 0 for x in U o, and h,(x) = 1 for x in U,. Define g, = h. and g, = 1 - h.. so that {gJ E r. If 'Y = {gj} is an element of r which follows {g;}, then each vanishes on either Uo or U,. Therefore the support of f1~ is either a subset of X - Uo or of X - U,. Thus the support of v is either a subset of X - Uo or X - U,. By the choice of x o, it follows that the support of v cannot be a subset of X - Uo and is therefore a subset of X - U,. Therefore x, is not in the support of v. Since X, was any point in X - So, it follows that the support of v is a subset of So. Thus v is a Baire measure on So. It is clear from the definition of v and from (0:) and (,8) that (a I) and (e,) are valid. It only remains to prove (b,). Now for each g in )B and each 'Y in r we have
g:
by (b) and the fact that ggi E)B. Passing to the limit gives (b,). This completes the proof of the lemma. Let Q be the class of ordinal numbers whose cardinal numbers are less than or equal to 2~, where f3 is the cardinal number of X. For each (J in Q we define ):)y transfinite induction a partition Pa- of Q into disjoint closed sets. This is to be done in such a way that P u is a refinement of P, for (J > T. The definition is started by defining P, = {Xl'. so that the first partition P, consists of the set X alone. Assume that P, has been defined for all ordinals T < (J. If (J has a predecessor (Jo, let S be any element of P Uo ' and let l! be the set of all functions in ~l which are real on S. Partition S by defining X, == x, for X, and x. in S to mean that !(x,) = !(x,) for all ! in~. Clearly S is partitioned into disjoint closed sets by this equivalence relation. The totality of all sets into which the elements S of PUo are partitioned in this way is defined to be the class P u • If (J has no predecessor, define X, == X 2 , for X, and x. in X, to mean that X, and x. belong to the same element of P, for all T < (J. The equivalence classes of this equivalence relation clearly form a partition P u of X into disjoint closed sets. Thus the classes P u are defined for-
151
A GENERALIZATION OF THE STONE WEIERSTRASS THEOREM
781
all a in Q, and it is clear that P u is a refinement of P whenever a > r. Assume that PU + 1 is a proper refinement of P u for all a in Q, i.e., that P U - 1 -=1= P u • Then P u +! contains a set not in any p. for r < a + 1. Therefore the cardinal number of subsets of X is at least equal to the cardinal number of the set Q. This contradicts the choice of Q. Therefore there exists an ordinal in p such that Pp+! = Pp. We shall show that the partition P = P r satisfies all requirements of the theorem. The fact (i) that 'lls is anti-symmetric for each S in P = Pp is a consequence of the fact Pp = Pr+!. We next prove (ii). Consider to this end f in C(X) such that the restriction of f to S belongs to 'lls for all S in P. Assume that f is not in '2l. By the Hahn-Banach theorem, there exists a bounded linear functional on C(X) which vanishes on m and does not vanish at f. By the Riesz representation thaorem, this functional can be realized as a measure I'- on X. Thus T
~gdl'- = Vdl'-
0 ,
gem
*0.
we may clearly assume that II I'- II ~ 1. We now construct, by transfinite induction, a set Su in P u for each a in Q and a finite complex-valued Baire measure I'-u on Su with
y if
(i) (ii)
f(x)
IIx - yll
> fey)
and
~ K[j(x) - fey)].
Suppose that W is a totally ordered subset of Z; by (i), the net of real numbers IJCx): xE is (bounded and) monotone, and hence converges to its supremum. From (ii) it follows that W is a Cauchy net; by the completeness of E, W converges to a point y in U. By the continuity of f and the norm it follows that y is an upper bound for W. Thus, by Zorn's lemma, there eXIsts a maximal element Xo of Z; since xoE uec, we need only show that xoEbdry C. If not, then Xo is in the interior of C, and there exists a> 0 such that xo+azE C. From the definition of C we see that there exist y in U, x in T and A in [0,1] such that xo+az=Ay+(1-A)X. Thenf(z) ~f(xo) Xo, a contradiction which completes the proof. A possible generalization of this theorem remains open: Suppose E and F are Banach spaces, and let £(E, F) be the Banach space of all continuous linear transformations from E into F, with the usual norm. For which E and F are those such that = II (for some x in E, II xii = 1) dense in £(E, F)? This is true for arbitrary E if F is an ideal in m(A) (the space of bounded functions on the set A, with the supremum norm). Added in proof: If C is a bounded closed convex set, let C' = IJEE* :f(x) = sup lJ(y): yE for some x in C A slight modification of the above argument shows that C' is dense in E*. This solves a problem proposed by Klee in Math. Z. vol. 69 (1958) p. 98.
wl
Ily-xoll (1-A)lly -xii
(l-A)(lIyll +llxll)
T
cl
(1-A)(1
II Til
Txll
I.
BIBLIOGRAPHY
1. R. R. Phelps, Subreflexive normed linear spaces, Arch. Math. vol. 8 (1957)
pp 444--450 2. - - , A representation theorem for bounded convex sets, Proc Amer Math. Soc., to appear. UNIVERSITY OF C~LIFORNIA, BERKELEY
156
MAPPINGS OF PARTIALLY ANALYTIC SPACES.· By
ERRETT
BISHOP.
1. Introduction. We study a mapping f of a partially analytic space K into n-dimensional complex affine space Cn. A partially analytic space K is a Hausdorff topological space which is a countable union of conditionally compact open sets, which is locally connected, and which is endowed, at least partially, with an analytic structure. This means that a countable family {Ma} of subsets of K is given, each with the structure of a complex analytic manifold. An algebra ~ of continuous functions on K, called the analytic functions on K, is also given, with the property that each function in ~ is analytic on each "Ma. The algebra ~r is closed in the topology of uniform convergence on compact subsets of K. The coordinates of the mapping fare required to lie in~. The purpose of this paper is to show that for certain spaces K the mapping f can be chosen to have certain properties. For instance, sufficient conditions are given for there to exist an f which is proper, or which separates points, or which has a certain rank at certain points. Since the complex spaces of Cartan [4J or Behnke and Stein [2J can be thought of as partially analytic spaces, these results apply to complex spaces. 1 Theorem 1 is concerned with the dimensions of the level sets of the mapping f. The statement of Theorem 1 is somewhat complicated; a special case reads as follows. If the level sets of ~ intersect each Ma in a countable set, then f can be chosen so that the level sets of f intersect each Ma in a set of (complex) dimension at most max {d a - n, O}, where da - dim M a. Theorem 1 is closely related to a theorem of Grauert ([6], Satz 12), but the proof given here differs from Grauert's proof and a more gpneral theorem is obtained. The next part of the paper is concerned with analytic polyhedra. With f as above, write
Po-{p: pEUM... lfc(p)1 0 such that
maps U homeomorphically onto the set E={Z=(Z"
. ',Zd): !Z,-fi(P)! 1, then F is said to be of third ClLtegory if the set of all points p in X, X XX 11-1 for which the set of points q in Xn such that (p, q) E X is not of first category is of third category. It is clear that a countable union of sets of third category i" of third category, and that the complement of a set of third category is uen'<e THEOREM 1. Let K be a partially analytic space. For each at, let Ba be a subset of A (.11",) and let hI"" .. , hk", be elements of A (JI",), where le is a fixed positwe integer. Let pir>., 1 < le, con~ist of all p in JI", such that the dimension on the manifold .:If", of ~I U Ba U {h\" .. , P,,} over B", at p is at least i. Let G be the set of all g = (gI, . ,gk) in k~ such that for each at, each i, and each p in pir>. we have
(*)
The dimension on the manifold Ma at p of B", U {g,
+ hI""
.. , g ..
+h
k ,,}
over Ba is at least i. Then k~( - G is of third category tn the le-fold cartesian pi"oduct k~r of ~( with itself.
We show by induction on le that the theorem is true for all positive integers le. We aSbume that the theorem is true for all values of le smaller than the one being considered. The proof of the theorem when le = 1 is the same as the proof of the general inductive step, and so need not be considered separately Having fixed le, let fI,· . ,h be any elements of~. Let Qia, 1 < le -1, be the set of all p in M" such that the dimension on the manifold Ma of ~ U Ba U {hI",.. ,hl..- I a} over B" at pis i. By the mduction hypothesis, the set Go of all (g" . , glv--l) in k-I~ such that for each at, each i (1 < i < le - 1), and each p in Q\, the dimension on the manifold 11{" at p Proof.
166
PARTIALLY ANALYTIC SPACES.
+
+
of B U {g, h' "" . ,g1v--1 hk-1",} over B", is at least i has the property that k- 1m: - Go is of third category. Fix some (g" .. , g1v--1) in Go. Let Qa be the set of points p in Ma for which the dimension at p of 8 1=
m: U
B", U {g,
+ h ' a,
.
,g"-1
+h
k - 1 a}
U {hka}
over
is at least 1. By Lemma 2, the set Z of gIG in ~r such that for each ex and each p in Qa the dimension at p of Ba U {g, h ' a, . , gk hka } over Ba U {g, h1a, .. , gl.--1 P-1",} is at least 1 has a complement of first ., gl.--1 category. We first show that 9 = (g1, .. , gkJ is in G for each 91, in Go and each g.. in Z. Afterward, we show that this implies that km: - G is of third category. We therefore now prove that 9 E G. Consider a point p in some Pia. Let A be the dimension at p of
+
+
+
T1 =
~{U
Ba U {h 1 a,
+
,h".,}
and p. that of Be., so that A- p. :> i because p E pia. at p of
Let v be the dimension
+
Then either A = v or A = v 1 (see [10], p. 43), since T 1 contains one more function than T 2. Consider first the case v = A and i < le. Then v - JJ- :> i, so that p E Qi a. Therefore the dimension at p of Ba U {g1 h1a, .. , gk-1 hk - 1a } over Ba is at least i. Therefore 9 satisfies (*) at the point p. N ext consider the cabe in which either v = A-I or i = le. Thus v-p.:>i-l, so that pE Qi- 1 a if i#l. Therefore the dimension at p of Ba U {g1 h' ", . ,g1v--1 hle- 1a} over Ba is at least i - I ; this clearly also holds if i = 1. Thus to show that 9 satisfies (*) at the point p it is enough to show that the dimension at p of Ba U {g, h 1a,·· ,g .. hka } over Ba U {g, h 1a,· .. , gk-1 hk-1a } is at least 1. Therefore it is enough to show that p E Qa, or that the dimension of 8 1 over 8 2 is at least 1 at p. Consider first the case v = A-1. N ow the level sets of 8 , and 8 2 U m: are the same as the level sets of T1 and T2 respectively, because g1,· .. , g1v--1 are in m:. Thus the dimension at p of 8 1 over 8 2 is at least A- v = 1, so that p E Qa and 9 therefore satisfies (*) at the point p. Consider next the case A = v and i = le. Then A :> p. i = JL le, and as above we see that A is the dimension of 8 1 at p. Since 8 2 contains lc - 1 more functions than Ba and since B", has dimension JL at p, the dimension
+
+
+
+
+
+
+
+
+
167
+
220
ERRETT BISHOP.
of 8 2 at p is at most p,+ (k-l). Therefore the dimension of 8 1 over 8 2 at p is at least p, k - (p, k-l) = 1. Thus P E Qa so that g satisfies condition (*) at the point p in this case and therefore in all cases. It follows that (gl,' . " gk) E G. To see now that "& - G is of third category in k&, note that 1..-1& - Go is of third category in k-l& and for each (gl, " , gk-l) in Go the set of gk in & with (gl,' . " gk) in k& - G is a subset of & - Z and therefore of first category in &. Therefore k& - G is of third category, as was to be proved.
+
+
3. Analytic polyhedra and weak holomorphic convexity. In this section we investigate mappings of a d-dimensional partially analytic space into Cd. Theorem 1 of the last section of course has applications to such mappings, but the purpose of this section is to obtain stronger results, under the assumption that K has certain analytic convexity properties. The basic notion is that of an analytic polyhedron. Definition 5. A subset L of a partially analytic space K is called central if it is the union of some subfamily of the family of structure manifolds of K and if each component of an open set in L is open. Let (fl, " , fn) be functions in & and let L be a central subset of K. Define Po= {p: pE L, I fi(p)
1< 1,1 rN -
(rrl)N
=
r N -1,
so that p is not in Q is N is sufficiently large. Thus if N is sufficiently large Q does not intersect bdry P. Thus each component of Q is eitHer a subset of P or disjoint from P. Since each component of Q intersects S C P, it follows that Q C P, as was to be proved. Now assume that P is prepared. We must show that ReP if N is sufficiently large. To this end it suffices as above to show that R does not intersect bdry P. Assume therefore that (bdry P) n R is non-void for a sequence {Nm} of values of N, where Nm->;oo as m->;oo. Choose a constant c with r l < c < 1 and write where
v=
{p:
I I,; (p) I < c, 1 < i < n}.
169
222
ERRETT BISHOP.
Since (bdry P) n R is non-void, there exists a component G of R - V which intersects bdry P. Now G contains a point qo of the boundary (in L) of V n L, since otherwise G would be a component of R containing no points of S. Let T m be the component of TOm containing qo, so that T meG. Now T m intersects bdry P, for otherwise T m would be a component of R - V and would therefore equal G, contradicting the fact that G contains a point of bdry P. For each p in T m there exists le, 1 < k < n, with \ flc (p) \ :> c. For this value of k we have
r N \ f,(P) \N:> \ (rh(p))N where N
=
N m'
1-\ (rh(p))N f, (p) oF 0
Hence, for pETm,
\ (f.(p) )N(f,(P) )-N -1\
(cr)N -1, and
«cr)N _1)-1
< 2(cr)-N
if N = N m is sufficiently large. Let Dm consist of all complex numbers z such that I zN-ll < 2(cr)-N. For z in D m ,
1-2(cr)-N
< \ z \N < 1 + 2(cr)-N,
so that 1-
2 (cr) -N
i \ arg ZN \, so that I arg ZN I < 4 (cr) -N. There therefore exists an integer j, 0 < j < N -1, such that
I arg z - 2j.".N-1 \ where 1l=1l(]) =max{y(j),m}. Clearly the set T is non-void, since by hypothesis nE T. Clearly also the lemma will be proved if it is shown that 1 E T. To this end, assume 1 ¢ T. Let m be the least integer in T, and choose hl' .., h" as above. We will prove that m - 1 E T and thus obtain a contradiction. The polyhedra Q; with 'Y (j) < m are unreduced, since dim Q; -< dim P J = 'Y (j). Therefore, by applying Lemma 4, we may assume that the polyhedra Q; with 'Y (j) < m are prepared. Let r> ] be such that I hi (p) I < 1 -< i -< 'Y (j), p E 8j, for all values of j. We define
r"
h'm-l= (rhm)N- (rhl)N, h'm = (rh.)N, h'm+l = (rhm+1)N - (rh.)N
h'" =
(rh,,)N -
(rh1)N.
If N is sufficiently large it follows from Theorem 2 that h /1 , · • . , h' fl defi ne an analytic polyhedron Q'; with frame L J such that 8 J C Q'; C Q; C Pi> where .B = max{y(j), m - I } . This shows that m - 1 E T. From this contradiction we conclude that 1 E T, as was to be proved. Definition 6.
A partially analytic space K is called weakly holomor-
2
173
226
ERRETT BISHOP.
phically convex if it is the union of its structure manifolds and if for each compact 8 C K each component of the set S
=
{p: p E K, [ f (p) [ < max q E 8
[
f (q) [, all f in m}
is compact. Definition 7. A continuous mapping of a topological space T, into a topological space T 2 is almost proper if each component of -1 (8) is compact for each compact subset 8 of T 2 • THEOREM 4. Let {Lj} be a finite family of closed central subsets of a weakly holomorphically convex partially analytic space K. Let the dimensions of m on the structure manifolds of K be bounded and have the maximum value n. Let y = y (j) be the maximum of the dimensions of 9f on the structure manifolds of K which are subsets of L j • Then the set of elements f = (f,,' . " fn) in n9f such thai for each j the mapping
fi:
p~
(f,(P),' . ',f'Y(p»
of L} into G'Y is almost proper is dense in nm. Proof. It is permissible to assume that L, = K. Let {8d be a sequence of compact sets whose union is K with 8, C interior 8 i +1 • Let Ti be an open set containing 8 i whose closure is compact and whose boundary does not intersect Si. For each point P in bdry T i , there exists h in W with
[ h(p) [ > 1
> maxqEs,
[h(q) [.
By the compactness of bdry T i , there therefore exist a finite number h" , hm of functions in W such that [hk(q) [ < 1 for q in 8 t and 1 1 for at least one k if P E bdry T i . If we let Q be the union of those components of
Qo={P: pEK,[hi(p)[ max{ I
f,"-l k
(p)
I:
1 [am l-lf m- 1 k(P)I-
~
la r (gr k (p») tr l>m+l-
.
~ 2- r >m>a. r=m+l
r=m+l
Thus we see that (fi) -1 (H) does not intersect bdry Qmi' We therefore have r C Qmj. Thus r is a closed subset of the compact set Qmi' Therefore r is
175
JHut.!'.:'!"!' Hll:!liU.I:'.
compact. It follows that fi is almost proper, as was to be proved. It is also clear that fk can be made arbitrarily close to Ak by taking mo sufficiently large.
4. Special analytic polyhedra and integration. In order to study the local properties of an analytic ployhedron, it is helpful to impose certain restrictions. Definition 8. A reduced analytic polyhedron P in a partially analytic space K with frame L defined by functions fl' . " fn in ~r will be called special if the level sets of fl, . , fn on P are countable and if the set A of all a such that M", C L, .M", intersects P, and dimM",=n has the property that ( U M",) n P is dense in P and that M", n P is open in L for each a in A, 'lEA
It is clear that each component of a special analytic polyhedron is a special analytic ployhedron.
Definition 9. If X is a topological space we define vY, called the k-fold unordered product of X with itself, to be the set of all unordered Ie-tuples of elements of X. '1'he natural map of the k-fold Cartesian product kX of X with itself into",X is denoted by Ox, or Okx if k is not clear from the context. A set in kX is defined to be open if and only if its inverse image under the map Ox is an open set in kX. It is clear that kX is Hausdorff if X is Hausdorff and compact if X is compact. THEOREM 5. Let P be a special analytic polyhedron with frame L in a partially analytic space K defined by the functions f 1," , f n in W. Let En be the subset of On consisting of points whose coordinates are less than 1 in absolute value, and let f be the map
',f,,(p» of P into En. For each a in A let M"'l be the subset of M", consisting of all points in Ma n P at which (fl, . ,fn) is a coordinate set of functions. Then the set M1 = U MIa is dense in P and H = f(P-1I-f 1 ) is closed and nowhere a
dense in En. There exists an integer A > 1, called the multiplicity of P, such that for each z in En-H there exist exactly A distinct points p in M1 with f (p) ~ z. Th e map w from En - H into "P which takes z into the unordered A-tuple of such points can be uniquely extended to a continuous map w of En into "P. The set P has only a finite number of components. Proof.
If a E A the set M 2 a = (M", n P) -MIa is a locally analytic
176
.I:'Al1TIALL:r
A.N AL:r IlL
!!.I:'AU ..... !!.
subset of Ma , since locally it is given by the vanishing of a certain Jacobian. Also M2a=FM(JI.nP because 1I1 2 (J1.=M a nP would imply that locally one of the functions f" ,fn depends on the others, contradicting the fact that the level sets of f = (f,,' " fn) on MOl n P are countable. Thus M 2 a is the union of a finite family {N"'./l} of analytic submanifolds of M a , with dim NOl./l < dim J[ a = n. Since 111, Of. is clearly dense in M and U 111 is dense (JI.
(JI.
(JI.EA
in P, }'1, is dense in P. Now each NOl./l is a countable union of closed sets, as is each 11101 n P, (X ¢ A. Thus f(NOf../l)' 2 E A, and f(M", n P), (X ¢ A arp countable unions of closed sets of topological dimensions < 2n - 2, since the analytic image of a closed set of dimensions < 2n -2 is of dimension < 2n - 2. I t follows that real dim H < 2n - 2. Therefore En - H is connedcd (see r'l] for these theorems) . Since f (.11.) is of real dimension 2n, f( .~[ 1) - II is nOll-void. For each z in En-II let l(z) be the number of points p in 111, with f(p) =Z If l(z) were not iinitt', the set of such points p would have a limit point po in P. Actually Po is in P, since f(po) E En implies by Lemma 3 that po is not in bdry P. If po E JIa for some (X E A, the fact that ."'fla n P i~ open in P implies that the level set of f on Jla containing po also contains the points p which are sufficiently near to Po. Thus this level set has po as a limit point and is therefore not O-cl.imensional; it is therefore uncountable, contradicting the fact that P is a special analytic polyhedron. Thus po is not in JI, C U JI a. Also po is not in P - Jl, SlIlce z is not in H. Thus po - k such that p E H j . If j> k, then p is in Gj - 1 • Therefore
un.
It follows from condition (#) that this can happen for only a finite number of values of i. Therefore 1-1 (F,,) is contained in the union of H k and such H j . Since H j and HI.. are compact, this implies that 1 is proper, as was to be proved. The hypothesis in the last theorem, and also in the following theorem, that compact components Vo of V consist of just one point" which is isolated in V, is in fact unnecessary if K is a complex space. This can be seen from
186
PARTIALLY ANALYTIC SPACES.
239
results of Stein [l1J, which states that if f is an analytic mapping of a complex space K having the property that the components of the level sets of f on K are compact then it is possible to introduce an analytic structure into the class k of such components in such a way that the natural map of K into k is analytic. However, we will not pursue this question here. THEOREM
9. Let K be a weakly holomorphically convex partially analytic
space, the maximum of the dimensions of whose structure manifolds is n. Let each n-dimensional structure manifold be open and let their union Mo be dense in K. Let ~ contain the unit function and separate points of K. For each subset r of ~ let each compact component Va of each level set of V on K consist of just one point, which is isolated in V. Let K be the union of a sequence Sk of compact sets, each having a finite number of components, with Sk C interior Skn. Let J.I be a particular n-dimensional structure manifold of K, such that ~ has rank n at each point of M. Let G be the set of functions (fl'· . . , f2Ml) in 2n+l~ such that
(i) (ii)
f
=
(fl'·
., fn) is almost proper,
1= (fl' . . ,ffl+l)
is proper,
(iii)
the rank of (fl,· .. , f2") at each point of M is n,
(iv)
the functions (fl,· .. , f211+l) separate points of M.
Then G is dense in
2n+l~.
Proof. By Theorem 4, the elements f in n~ which are almost proper are dense Fix such an element f. If V is a level set of f, each component Va of V is compact because f is almost proper. It follows from the hypothes~s of the theorem that Va consists of one point, which is isolated in V. Therefore V is countable, so that all level sets of f are countable. For 1 < i < n let A at consist of all points in M at which rank f < i. Each of these sets is analytic in M and A C A °i+l. Clearly
0.
dim A 0"
< dim M =
n,
since f, whose level sets are countable, must have rank n at some point of lrI. We now show inductively, starting with i = n and working backward, that dim A °i < i. Assume therefore that dim A °in < i 1. Let the analytic set A °i+l be written as the union of submanifolds {Ni+l'Y} of JI, and let {Ni+lll} consist of those Ni+l'Y of dimension i. To show dim A < i, it will be sufficient to show that A 0, does not contain any of the Ni+lll . For if this is the case the inclusion A io C A °in implies that A °i is contained in the union of certain
+
0.
187
240
ERRETT BISHOP.
proper analytic subsets of the Ni+lf3 and of the Ni+1'Y of dimension less than i. so that dim A °i < i. To see that Ni+1f3 q: A °i, note that t has rank i at some point of Ni+1f3, since its level sets are countable and dim Ni+1f3 = i. This completes the proof that dim A °i < i. Let S be a countable dense set in M. For each Nif3, 1 < i < n, choose a point p i f3 in Nif3 at which the rank of t is at least i - I . Let II be the f'et of all tn+1 in ~r such that 1 = t X tn+1 is proper, sueh that 1 has rank i on N at the point pif3 for each i and (3, and such that tn+1 takes distinct ,alues at the points of the set j-1(S). By Theorem 8, there exists tn+1 in H such that
I tn+1(p) -A(p) 1
O. Since by an appropriate choice of the sequrnce {Sd and the functions IXi of Theorem 8 the set H1 can be made to contain any compact subset of K, it follows that H is dense in W. We now choose the functions tn+2, . ,t2n+1 by induction. Assuming that t n+1, . . ,tn+'" haye been chosen, let A ki' 1 < i < n, consist of all points in 1',1 at which t1' . ,tn+'" have rank less than i. Clearly each A\ is an analytic subset of 1',[. Let B~ consist of all points in J[ -A"'n which are identified with some other point of ]1,[ by the functions t1' ,tn+"" Since 1 is proper, the points in K which are identified with a given point pin Bk by the functions t1)' . " tn+k form a compact set L. By the hypothesis of the theorem it follows that L is finite. Let p = P1," ,Pi be the points of J[ n L. Let £ be a small positive number and write
gi =
£-1
(fi - ti (p) ),
1 ubset L of a Stem manifold K. It is a consequence of Theorems A and B of [3] that
I The set L is the set of common zeros of Rome family r of analytic functions on [( II. If L IS a sllbmanIfold of K then every analytic function on Lean be extended to an analytic function on K PrevIOus work on Problems I and II was done by Oka [5], r6], and others. In Section 3 we give new proofs of I and II which are independent of sheaf theory. The proof of I contains an explicit formula for the functions in the class r, and in thiR respeC't gives information which is not available from the sheaf-theoretic proof. Slmilarly the proof of II contains a specific formula for the extension of a function analytic on L to a function analytic on K. This formula makes it clear that II continues to hold tor analytic functions whose values lie in a Banach space, thus answering a question posed by H. Rossi in a conversation with the author. In Section 4 we prove that the first cohomology group of the shea:! of germs of holomorphic functions on a Stein manifold is trivial. Again tb~s result follows from sheaf theory [3]. Previous work was done by Cousin [4], Oka [5], [6], [7J, and others. Since it is the purpose of this paper to proceed independently of sheaf theory, this result is formulated and proved without mention of sheaves or cohomology. As is well-known, the EOlutions to Cousin's first and second problems are consquences of this result. The techniques which we use for solving these problems are all essentially similar. Since all of the problems considered have previously been solved, • Received December 27, 1960. 'This work was supported by the Sloan Foundation and the National Science Foundation
479
191
480
ERRETT BISHOP.
we have not attemptt'd to push the theory to the limit of its generality. It should however be clear to someone familiar with the subject that the
methods employed here are capable of handling the same problems in greater generality. These methods, which seem not to afford so powerful a general approach to these questions as does the theory of sheaves, nevertheless have the advantages of explicitness and simplicity. The gain in explicitness comes from the use of explicit formulas to construct the various analytic functions which arise. These formulas make it clear, for instance, that all of the above mentioned results hold for Banach space valued analytic functions. 2. Preliminaries. In order not to burden the exposition with too much generality (as was done in [2]), we confine our attention to Stein manifolds, merely remarking that there are obvious possibilities for generalizing the results to one type or another of analytic space.
Definition 1. A Stein manifold K of dimension n is an n-dimensional separable complex analytic manifold K having the following properties where A (K) denotes the set of analytic functions on K (a) (b)
For each pair (p, q) of distinct points in K there exists f in ..L (K) with f(p) =l=1(q)· For each p in K there exist 11, ,I.. in A (K) which map some neighborhood of p homeomorphically onto an open set in n-dimensional complex affine space K is holomorphically convex. This means that for each compact subc;et S of K the set
a...
(c)
s=
{p: p E K, I f(p) I <sup
{I I(q) I:
q E S} for all
f
in A (K)}
is compact. We now state, for the special case of a Stein manifold, some of the definitions and results of [2].
Definition 2. Let K be a Stein manifQld and L an analytic subset of K Letf" . ,/k be functions in A(K). Write
Po = {p E L: I f,(p)
I < 1,1
sup{lf(g)l: gET}.
Let C(T) be the 1wifol'mly-normerl Banach 'pare 1'((lued function~ on T. Let ~ be the rlo~w'e of Then the spectrum of the Banach algebra ~I is T. homomorphism rj> of ~r into the compleX' number" T with rj> (f) = f(po) for aU f in ~L
of all continuou., rompleJ'A(K) in the spare C(T). That is, to any continuous corresponds a point po in
The proof of Theorem 4 (Lemma 6 of [2]) was for certain special sets T, but by means of Theorem lone can easily approximate T by such sperial sets and thereby obtain Theorem 4 for T. In order to study a d-dimensional analytic· subset L of a Stein manifold K globally it is sometimes necessary to ronsider a map of K into Cd whose restrirtion to L is almost proper. This ronrept is explained in the following n.efinition.
Definition 4 A continuous map f of a topological space X into a topological spare Y is raIled almo~t proper if every component of {-I (8) is compact for every compact subset 8 of Y.
194
FUNCTIONS OF SEVERAL COMPLEX VARIABLES.
4!M
The following theorem is easily proved from the fact that an analytic subset of a Stein manifold can be exhausted by special analytic polyhedra. THEOREM 5. If L is a d-dimensional analytic subset of a Stein manifold K then K admit., an analytic map into Cd whose restriction to L is almost IJroper. The mapping can be chosen to be non-singular on L at each point of any O-dimensional set of regular points of L.
The final result we shall need guarantees the existence of a function whjch will be used as a convergence factor for constructions involving an almost proper map f of an analytic subset D of a Rtein manifold.
f d+l
THEOREM 6 Let f = (fl, ,fa) be an analytic map of a Stein manifold K into Cd. uho,e 1"estriction to the analytic subset L of K is alm05t proper. Let L have dimension d at each point. Let {Sd be an increasing sequence of compact subsei.s of L, each of which ha.s finitely many components, the union of whose interiors i5 L. Let a be a function from the positi'L'e integers to the po~itive real numben such that
SA C F" = {p E L: [ fi(p) [ sup{1 h(q) I: q E T}. If P E U n L then pEL - F" so that h can be taken to be one of the functions fl" .. ,fd' Thus we need only consider the case p E U -L. It is well known, and a new proof will be given below (Theorem 7), that there e~ists an analytic function hI on an arbitrary open relatively compart set VeK such that h, (p) = 1 and hI vanishes on V n L. N ow since D = jj there exit\ts h2 in W with h2(P) = 1 and I h,(q)h2(q) 1< 1 for all q in D. Now if V is sufficiently large then by Runge's theorem for several complex variable" (a new proof of which will be given below which does not depend on this theorem), the function h,h2 can be uniformly approximated on T U {p} by functions h in W, so that h will have the desired property. ,{'hi" completes the proof that the desired functIOn T can be found.
3. Analytic sets in Stein manifolds. We wish to proye Theorelllb I and II of the introduction. We first prove somewhat weaker results. I, amI Il, below, which are sufficient for most applications. THEOREM I. Let L be an analytic lJubset of a Stein nUlIlifold K. U be an open subset of K whose closure is compact. Let L hare the dimension d at each of its point~. Then
II' III'
[~pt ,{(!IIC
For each Po in U - L there e:riiit~ an analytic fUIl( lion h on (~, defined by (*) beloUJ, which vani~hes on U n L, with h (Po) =1= 0 If in addition L is a submanifold of K then every analyt'ic function u on L can be extended to an analytic function ii on U, gil'en by (a), (d), and (e) below.
Proof· Since K is a Stein manifold and (j is compact there exists an analytic polyhedron Q in K with (j e Q. Thus Q n L is an analytic polyhedron in L. By Theorem 1 there exist functions fl' ,fa in A (K) defining
196
FUNCTIONS OF SEVERAL COMPLEX VARIABLES.
un L CP
an analytic polyhedron P in L with and 1 an allalytic function h on K, defined by (*) and (~) below, lilhich zani,he, on L but doe!:. not vanish at po. If L is a wbmanifold of K then corre8ponding to every analytic function u on L there exi~ts an analytic function II on K with ii (q) = u (q) for 'all q in L. TlI e function fl is given by a conglomeration of formulas which are summarized by * below.
By Theorem 5, there exists an analytic map f = (fl' ,fd) of K into Cd whose restriction 1 to L is almost proper If p E K then J-l(f(p» is countable and consists of isolated points. Let PI, . , pn, be the points of 1-1 (f (p) ), each being counted a number of times equal to ib! multiplicity for the mapping 1- 'rhe multiplicity of a point q for the mapping 1 is defined by considering a special analytic polyhedron Q in L eontaining q defined by multiples of fl' " fd and taking the multiplicity of q for the analytic polyhedron (.); thib multiplicity cloe,> not depend on the ehoiee of Q. Proof.
Let {8d be a bequence of compact subsets of L, each of whil'h has finitely many components, with 8", C 8~+1 for all k and U ~ int 81. = L. For each positive integer lc let HI.. be the union of those components of J-l(rx(lc)£rl) which intersect 8~, where rx(lc) > lc 1 is chosen so that f(81.) C rx(lc)Ed. Thus 8~ C H~. Write
+
for all positive integers k. Now int Hie = HI. n 1-1 (rx (k) Ed) is an analytic polyhedron defined by the functions rx(k)-lfl" ,rx(lc)-lfd' Let Ak be the
199
ERRETT BISHOP.
multiplicity of int H k • Thus for each P in K the number of points in the sequence Pl, P2,· belonging to intHk is either A" or 0, depending on whether f(p) is in a(Je)Ea or not. By Theorem 6, there exists fa+1 in A (K) such that
I frl+l(P) for all p in Gk and all Je.
for all p in G/c. audition
-
(Al'+1
Therefore
If po is any point in K -
1- ' (f (Po».
L, we may choose fd+l so that in
=F exp (frl+1 (g»
exp (fd+l (Po» for all g in
+ Je2 + 1) 1< 1
Write
w = exp(- fd+l) ;"0
that wE A (K) and
lor all p in Gk • Choose JeD::> 2 with f(T) C koErl, where T is a compact sub~et of K to be fixed later. Since {lId is an increasing sequence of sets whose union is K, there exists for each g in '-1(f(T» a smallest le with g E HI... If k> ko then g E Gk - 1 ; in fact, g E int Glv-1 because f(g) E koErlC kEd so that g E 1-' (leEd) n Gk - l C int G1v-1. If k = ko then similarly g E intHko. Thus 00
1-1(f(T» C intHko U
U int Gk • k=k.o
Now for each p in K and k::> 1 define
where the product is taken over all Pi in F k , with F 1 = H 1 and F" = H k - H 10-1 for k> 1. Since at mm,t Ak of the Pi belong to Ih and since w does not ,anish on K, we see that tllf' functions 11 'e are well-defined. We shall f'how that the product h (p)
=
"" II hk (p) k=l
lOnverges uniformly on all compact subsets of K and defines a function h in A (K). Assuming for the moment that this is so, it follows from the fact that P = Pi for some i whenever pEL that h vanishes on L. It follows from the fact that w has values none of which equal w (Po) at the points of 1- 1 (f (Po) ) that h (Po) =F o. Thm; h has all of the desired properties. rt remains to
200
FUNCTIONS OF SEVERAL COMPLEX VARIABLES.
4S!J
prove the 7c2f3IeAIe+l. Let d=exp(-f'd+l), so that dE A (K) and d vanishes nowhere on K. Also
for all p in Gle •
Now if P E U and leo is chosen so large that f(T)
202
C
leoEd.
FUNCTIONS OF SEVERAL COMPLEX VARIABLEI:l.
each Pi in L - H ko is in some G/,. for lc > lco. Therefore the sum .l1 splits into partial sums Ak taken over the sets G/,. for 7c:> lco. Now at most '\/.+1 of the points Pi lie in Gk • It follows from the choice of d that
Thus A
.
2 it P = p. it follows that W(p) =7= O.
By the ehoice of
10
We have thus found a funrtion iv = 17\ on L such that for any C1 in A (L) there exists d 1 in A (K) vanishing nowhere such that there is a funrtion u1 in A (K) which equals uc 1d11V 1 on L. Wp may choose iT-l not to vanish at a O-dimensional subset of L which rontains points in each component of L. Thus the set Ll of zeros of 1 is of dimension at mm,t rl-l. N ow choose iV 2 to have the same propertie;., as 11\ and to be non zero at a O-uimensional subset of Ll containing points ill each analytic component of L 1 • Continuing in this fashion we find 11\, . iCd haying the aboye propertil's and having no common zero on L. By a theorem of ~\.rens r 1] l it follows that there exi",t c1, . ,Crl in A (L) with "
w
By the choire of Wi, there exists d; in A (K) Yallishing nowhere on A (K) such that uCidiWi has an extension from L to an analytic function Ui on K. Write rJ
U=~(di)-1ii,b i=l
, The possibility of using Arens' theorem here was pointed out by Kenneth Hoffman
203
4l:Ji:l
ERRETT BISHOP.
so that u E A (K).
For P in L we have d
U (p)
=
L
(di(p) )-lU(p) c.(p) t4(p )w.(p)
i=1 d
=u(p) LCi(P)W.(p) =u(p). ,=1
This completes the proof of II. The final formula reads •
sel' that the coefficient of each power of X in the polynomial g')'6 in X iR analytic on V')' n VB. Since {W')'}, {h')'6} is a Cousin distribution on P we sre that and
for all y, 8, and TllU& {V')'}, {g')'6} is a Cousin distribution for Ed. By the statement madl' above (whose proof will be outlined below), there exists a resolution {g')'} of this Cousin distribution. Now {Wi'}'} is a covering of P. For l'aeh WI'} dl'fine thl' function F i ,), an Wi')' by (T.
These functiom are the resolution of a Cousin distribution {W;Y}, {Fi')'j~} on P, where F i')'16 is the function
206
4lJ5
FUNCTIONS OF SEVERAL COMPLEX VARIABLES.
on W.'Y n Wl. If for each p in P we write w(f(p)) = p, it follows that
=
{P1'
,p~},
with
P,
~
Fi'YJB(p) =~h-y a resolution on 8 (le 1 , , k t ; 1). We now show by induction on k (t remaining fixed) that the given distribution has a resolution on 8(le 1 , · ,kt;k). L\ssume therefore that this is true for all 7c smaller than the given k. Thu'by the above we see that there exists a neighborhood V 1 of 8(kl' ,k t ; k -1) and a function G1 on V 1 "uch that G1 - Fa IS analytIc on V 1 n U r>. for all IZ. Similarly, there exi;;ts a function Go on some neighborhood V 2 of 8 (le 1 , . , le" le) such that G2 - F r>. is analvtlc on V 2 n U a for all IZ. rfhus G1 - G2 is analytic OIl V 1 n V 2. It is well known and easy to see that therefore there exist open sets 11'1 and 11'2 with
8 (kl'
, k t ; 7c -
1) C Wi C V 1
and 8 (kl'
, kt, k)
C W2 C
V2
/
-I-
and analytic functIOns .11 and A, on W1 and W 2 respectively such that .12 Ai and G1 - G2 are equal on W1 n W 2. It follows that the function B on W1 U W 2 defined as Al G 1 on W1 and A2 G 2 on lV, has the property that B - Fa is analytic on (W 1 U W 2) n (J a for all IZ. Thus the given distribution has a resolution on
+
8(kl'
. ,k t ; 1c)
=
8(kl'
+
, let; k - 1) U 8 (kl'
Thir, ('ompletes thc induction on k
. ,kt, k) C Wi U 11!
It follows that there cxists a resolution on ,k t ) .
8(kl' 'I
209
ERRETT BISHOP.
This in turn completes the induction on f. It follows that there exists a resolution on S (0) = Htl, as was to be proved. Added in proof Let L be an analytic submanifod of dimension d in a Stein manifold K. Let U be an open relatively com part subset of K. Then the contruction used in the proof of Theorem Il, shows that there exists a multiple-valued projection of U into L in the following sense There exists an integer A and an analytic map P-+ {(P"Z,),
ot U into the A-fold unordered produl't An of the "pal'e n = L X C' with itself having the following propertie'l. For earh II in C
}i~or each p in L n (. alld eal'h i with 1 < i < A tor whil'h Iii =1= Ji we have Zi = O. This fact in tum rlearly implies Theorem IT 1, since u ran be extended from L to U by the formula A
u(p)
~
=
ZiU(P.).
i=l
Similarly the proot of Theorem TI ean be L1.'-ed to show the e'l:istenre of certain multiple-valued analytir projl'dions of K into L. UNIVERSITY OF
C \LII'OR'HA,
BERKEL'" Y
REFERE~CES
[11 Richard Alens. "Dense inwI8e limit ring-~.' Mirhiqall Mathematiral Jou1'nal, vol 5 (1938). pp IBn 182 [2] Errett Bishop. ")Iappin,!!"'; of partiall~ analytic ~pa('e~," A~mericaJ1 ·Journal of Mathemati('s. vol R3 (19tH). rp 209 242 [3] Henri Cartan, "Seminaile E. N S ," H)51·32 [4] P. Cousin, "Rur Ie'! fonctions de n vaJiable" complexe~." A.cta Jlathematica, vol. 19 (1895 ) [;)1 Kiyosi Oka, "Sur les fonctions Itnalytique~ de plu~ieur" variable~ I Domaines convexe" par rapport aux fonction~ rationelles," Journal of Science of Hiroshima f"J1iversity, vol 6 (11)36), I'P 245255 - - - , "SUI les fonction" analjtiques de plllsieurs variable~ II Domaines d'holo morphie," .Journal of Science of Hi"o~hima f'niversity, vol 7 (1937), pp 115 130. [i I "flUl les fonction. analytique" de plm,ieUl s variables III Deuxieme prob Ierne de Cousin," .Journal of ScienGe of Hiroshima Universit1/. vol 9 (1939), pp 719
210
PARTIALLY ANALYTIC SPACES.* By ERRETT BISHOP.
1. Introduction. The concept of a partially analytic space was introduced in a previous paper [11. The present paper will refine and extend the theory. To this end it will be necessary to modify the definition of a partially analytic space whIch was given in [1]. Although formally quite different, the modified definition contains only one essentially new feature-local connectedness is no longer required.
DEFINITION 1. A pa1"iially analytic space consists of a separable locally compact Hausdorff space K and a subalgebra ~ of C (K), where C (K) is the algebra of all continuous complex-valued functions on K. The algebra ~r contains the constant functions and is closed in the topology of uniform convergence on compact subsets of K. ~ will be called the set of analytic functions on K. In the previous paper certain structure manifolds were also postulated in the definition of a partially analytic space. The stmcture manifolds are now permitted to occur willy-nilly. DEFINITION 2. A structure manifold J[ of a l'artially analytic space K is a subset of K which has the structure of a complex analytic manifold such that (a) the manifold topology and the subset-of-K topology for J[ are the same, and (b) all functions in ~ are analytic on }1. By manifold we mean connected separable complex analytic manifold. Our first task is to show that the results of [1] remain valid for the more generally defined partially analytic spaces considered in this I)Rper. This can be done by simple modifications of the proofs given in [1]. ,Proofs are only sketched. After this has been done we study a special kind of partially analytic space K, called an embryonic space, which is shown to have the property that the image of K under a proper analytic map of K into a manifold M is an analytic subset of M. This property of embryonic spaces is then applied to obtain simple proofs of two known results, (i) and (ii) below. (i). Let A be an analytic set of dimension k in a manifold M. Let B be an analytic set in the manifold M - A whose dimension at each point of * Received May 11, 1961.
669 7
211
670
ERRETT BISHOP.
A is at least k
+ 1.
We show that jj is an embryonic space. Since the identity map of jj into M is proper, we obtain as a corollary of the proper mapping theorem for embryonic spaces the Remmert-Stein theorem that jj is an analytic subset of M. For this theorem see [9J. JI.[ -
(ii). Let K be a holomorphically convex analytic space and m: a closed algebra of analytic functions on K with respect to which K is holomorphically convex. It is easy to see that the image Ko of K under the natural map of K into Cw. (C denotes the complex numbers) is a separable locally compact Hausdorff space. The algebra m: can be considered as an algebra m:o of functions on K o, and together ~o and Ko constitute a partially analytic space. We show that this space is embryoniC'. From this fact we obtain Remmert';; proper mapping theorem [8J, which states that the image of an analytie bpace under a proper analytic map is an analytic subset of the image space. If p is a point in a partially analytic space K, an n-tuple tl' . ,tn of functions in ~r will be said to be coordinates at the point p if there exists a neighborhood U of p which t = (tl' ., tn) maps properly into an open set V C Cn suC'h that for each h in ~ there exists 9 analytic on V with hjU = 9 0 f. Then K will be said to have rank at most n at p. We show that an embryonic space admits a set of coordinate functions at each point. It follows that an embryonic space can be canonically given the structure of an analytic space in the sense of Serre. Using this fact and the map described above of a holomorphically convex analytic space K onto a canonically associated embryonic space K we show that if ~r is any closed algebra of analytic functions on K with respect to which K is holomorphically com ex then there exists a canonical map cp of K onto an analytIc space Ko such that if ~o denotes the set of functions on Ko such that cp E ~ and if ~Ko denote8 the set of analytic functions on Ko then ~o = ~Ko. This theorem extends work of Stein [10], and has also been proved by Cartan [4], using other methods.
t
to
The theory of the imbedding of analytic spaees has been studied by Remmert, Narasimhan [7], and the author [1]. In this paper we study (Theorem 11) the problem of finding a proper analytic map of a holomorphically convex embryonic space K into complex Euclidean space CN of sufficiently high dimension N which separates certain set of points of K and which locally generates all analytic functions at certain points of K. The imbedding theorems of [1] and [7] then follow as corollaries. The methods used for these constructions are basically those of [lJ but also have something In common with the methods of [7]. The author is indebted to Hugo Rossi for many discussions about the
212
PARTIALLY ANALYTIC SPACES.
671
topics of this paper. Rossi has proved by different methods some results which are closely related to some of the results given below, in particular to Theorem 6 and to Corollary 2 of Theorem 8. 2. Partially analytic spaces. Since a new definition has been given of partially analytic spaces it becomes necessary to redefine analytic polyhedra. DEFINITION 3. A subset L of a partially analytic space K is called a frame if it is contl!ined in the union of some countable family of structure manifolds of K. The dimension of L is the least integer d (possibly d = 00 ) such that L is contained in the union of a countable set of structure manifolds on each of which 2f has dimension at most d. Let fl." ,fn functions in 2f and let L be a frame. Write
Po={p: pEL,
IMp)1
1 has the property that scpn{p·lfi(p)l 1 and assume the theorem true for n - 1. We may aRsume thaVM is En = {(Zl' .. , zn) : I Zi I < 1,1 < i < n} Thus the intersection of H with all but a countable set of the varieties
is a countable union of locally analytic subsets of L e, each of dimension at most n - 2. By the induction hypothesis, f is analvtic on all such Le. Thus f is analytic on 31. COROLLARY.
Under the
a~';Umption~
217
of Theorem 3, let 9 be a continuous
ERRETT BISHOP.
UIU
function on S which is analytic on each };fa and let s be a symmetric poly nomial in variables Z,,' . ., ZII.. For each Z in };f let P" .., PII. denote the points of w(z), counted according to multiplicity. Then the functio'f1 h on Jl defined by h(z) =s(g(pd,' ',g(pll.)) is analytic on };f. Proof. By Theorem 4, it is sufficient to show that h is analytic at each point Zo of };f-H. Let P,o,' ·,PII.° be the points of w(zo). Thus Zo has a neighborhood U such that there exist analytic homeomorphisms ' the set A has no interior in L, the frame A has dimension at most n -7c -1. Since this is true for all Land M it follows that Lk+l and M k+l are frames of dimensions at most n - 7c -1. This com1 we see that (a) is true. pletes the induction. Letting 7c = n To prove (b), notice that for each g in Sk the pair (f,g) identifies some point in Uk with some point in V k. The same property therefore holds for J k. From this (b) is clear and the lemma is proved.
+
COROLLARY.
If L is an n-dimensional frame, if
226
mtieparates points
of L,
PARTIALLY ANALYTIC SPAUEt!.
and if f E ~{n has countable level sets on L then the set T of all g in mn+l such that (f, g) is one-to-one on L ha~ a complement S which i~ strictly of first category in m"+l. Proof. The set L X L - { (q, q) : q E L} can be covered by a countable family {Uk X V k }, where Uk and VI. are disjoint countable unions of frames of dimensions at most n. By the theorem, the complement of the set of all g such that (f,g) (U.) and (f,g) (V .. ) are disjoint is strictly of first category in ~rn+l. From this the corollary follows easily. DEFINITION 11. A point p in a frame L in a partially analytic space K is raIled an analytic point of L of dimenE-ion n and rank at most N if there exists a neighborhood U in L of p and f in 2[N mapping U homeomorphically onto an n-dimensional analytic subset of an open set SeeN such that for each h in mthere exists an analytic function w on S such that
h(p)
=
w(f(p»
for all p in U. Then f is called a set of coordinates at p and U is the corresponding coordinate neighborhood. THEOREM
point of L
8.
Every point p of an embl yonic frame L is an analytic points of K.
if mseparates
Proof. By the corollary just proved, there exist f,. . ,r~ in 2r mapping some neighborhood V of p one-to-one and properly onto an analytic set A C EN. Let H consist of all continuous functions h on A such that there exists g in 2r with g(q) = h(f(q» for all q in U, and let G be the set of germs at z = f(p) of H. Let F be the set of germs of all functions bounded and analytic in some neighborhood of z on the regular points of A, so F consists of the germs at z of all functions analytic' on the normalization of A (see [5J). Let R consist of all functions in some neighborhood of z on A which can be extended to be analytic in some EN neighborhood of z. Thus ReG C F, and G and Fare R-modules. By the remark at the top of p. 291 of [5J, the R-module F is finitely generated. Since R is noetherian, it follows that the sub-module G of F is also finitely generated. Thus there exist gl, . . , gk in 2r such that for each g in 2r there exist h,,' , hk in R with
\
(*)
for all t in A n V g, where V g is some neighborhood of z and 9 is the element in G corresponding to g in 2r. Thus if {8 n } is a decreasing sequence of neighborhoods of z whose intersection is z, we see that for each g in 2r there exists n=n(g) with 8 n C Vg and (h) su p { I f( q) I : q E 8}. Let W separate points of U. Let 1 E W. Let Wo be the uniform closure of W in C (S). Then to every non-zero continuous homomorphism ok- ~ 2-j-10k> 0.
J=k+l
;=k+l
This completes the proof that r is not strictly of first category. The rest of the theorem is proved as in [1]. We turn now to a consideration of certain imbedding properties. By Corollary 1 of Theorem 8 we see that every holomorphically convex embryonic space with a separating algebra ~ of analytic functions can be realized as an
232
analytic space which is holomorphically convex relative to the given algebra ~, and by Theorem 6 in turn we see that the analytic space can be realized as a holomorphically convex embryonic space with a separating algebra ~ of analytic functions. For this reason Theorem 10 holds for arbitrary holomirpically convex analytic or embryonic spaces, and the following theorem also yields information for such spaces.
Let K be a holomorphically conrey analytic space such that 1 E 2! and 2! separates points of K. Let K have dimension at most n at each point, so that by Theorem 2 there e.E'ists an almost proper map f = (fl" . " fn) E 2!n of K into On. Let L 1, . ,Lie be a finite family of frames in X, and U1,' . " Uk open subsets of these respective frames such that there exist integer:, N 1,' • " N k such that each poin t p of Ui, 1 < i < k, is an analytic point of Li of rank at most No. Write THEOREM
11.
N =max{n
+ 1,N""
Then there exists g = (gl" . " gN) in (a) (b)
(c)
,Nk }.
m;N such that
(f,gl" ',gMl) separates points of K, For each i, 1 < i < k, the functions (f' g" . nates at each point of U i , (f, gl) is a proper map of Kinta On+1.
, gs,) are cool'di-
By Lemma 2 the level sets of f on K are all countable. By Theorem 7 the set T 1 of all g for which (a) is not true is strictly of first category in m;N. By Theorem 9 the set T2 of all g for which (b) is not true is strictly of first category in m;N. Thus T, U T2 is strictly of first category. By Theorem 10 there exists g in K - (T, U T 2 ) which has property (c). This completes the proof. Now we would like to improve Theorem 10 by choosing g so that the mapping (fl, ", f,y, gd is proper on certain embryonic frames L in X, where 'Y = dim Land (f 1," , f 'Y) is almost proper on L. Instead of showing that g can be so chosen we show in the next theorem that the map f can be chosen so that g1 automatically satisfies the extra conditions. This theorem will at the same time be a considerable strengthening of Theorem "2:-
pj·oof.
THEOREM 12. Let {Lj} be a finite family of analytic sets of finite dimensions {y = y (j)} in ad-dimensional holomorphically convex analytic space K. Then if 2! is separating there exists f = (fl) ,fa) in 2!d such that
(a) (b)
For each j the mapping fi = (f1, proper, For each j the functions f'Y+1,' . "
233
. , f'Y) of L; into O'Y is almost
f tl
vanish on L j.
692
ERRETT BISHOP.
Sketch of Proof. By decomposing each L J into its homogeneous-dimensional parts we may assume that each Ll is a special frame. By lumping together all L J of a given dimension we may assume that the Ll are special frames L 1, L 2 , ' • • , Ld of dimensions 1,2,· .. , d respectively. Finally, by enlarging the L; appropriately we assume Ll C L2 c· . . C La = K. Now let S be any compact subset of K for which S = Ii. We show by induction on k, 0 < k < d, that there exist hi,' . " hk in W each of absolute value less than 1 on S such that for 1 < j < k the functions h;+l' . , hk vanish on L j and such that h,,' " hj define an analytic polyhedron P j in L; with S n L; C Pj' Since the case k = 0 is trivial, we may assume that hI> ., , h k - 1 have been found with the desired properties. Consider p in K - L k - 1 - S. Thus there exists Wi in W vanishing on L k - 1 with Wi (p) = 1. Since S =B there exists W2 in W with w2(p) = 1 and I W1(q)W2(q) 1< 1 for all q in S. Thus there exist finitely many functions gl, . ,gN in W which vanish on L k - 1 and have absolute values less than 1 on S such that h 1, ' • " h k - 1 , g,,' . " gN define an analytic polyhedron Q in Lk with S n Lk C Q. By the construction used to prove Theorem 3 of [1], we may successively reduce the number N of such functions g, obtaining at the last step functions h/,· . " hk-t', gt' in ~ having the desired properties for the functions hi,' . " hk at the k-th stage. Thus we may assume that the functions hi,' . ,hd can be chosen for each compact set S with S = B. Once this is done the proof of Theorem 4 of [1] can be repeated word for word to construct the desired functions f 1,' " f d. UNIVERSITY OF CALIFORNIA, BERKELEY AND THE INSTITUTE FOB ADVANCI!.D STUDY.
REFERENCES.
[1] E. Bishop, "Mappings of partially analytic spaces," American Journal of Mathe·
matic8, vol. 83 (1961), pp. 209-242. [2] H. Cartan, Seminaire E. N. S , 1951-1952, Paris. [3] - - - , Seminaire E. N. S., 1953-1954, Paris. [4] - - - , Quotient8 of complw analytic 8pace8, International Colloquium on Function Theory, Bombay, 1960. [5] H. Grauert and R. Remmert, "Komplexe Riiume," Mathemati8che Annalen, vol. 136 (1958), pp. 245-318. [6] C. Kuratowski, Topologie II, Warsaw, 1950. [7] R. Narasimhan, "Holomorphically complete complex spaces," American Journal of Mathematics, vol. 82 (1960), pp. 917-1934. [8] R. Remmert, "Holomorphe und meromorphe Abildungen komplexer Riiume," MathematiBohe Annalen, vol. 133 (1957), pp. 328·370. [9] R. Remmert and K. Stein, "tiber die wesentlichen Singularitaten analytischer Mengen," MathematiBohe Annalen, vol. 126 (1953), pp. 263-306. [10] K. Stein, "Analytische Zerlegungen komplexer Riiume," MathematiBohe Annalen, vol. 152 (1956), pp. 68·93.
234
LIlC 1'\.1I1t:llCiUl IVla1.l1Cn1811C81
::society.
A GENERAL RUDIN-CARLESON THEOREM ERRETT BISHOpl
1. Introduction. Rudin [7] and Carleson [4] have independently proved that if S is a closed set of Lebesgue measure 0 on the unit circle
L=\z:\z\ =1\ and if f is a continuous function on L then there exists a continuous function F on
D=
{z: \z\
~
1} F(z) = fez)
which is analytic on D - L such that for all z in S. It is the purpose of this paper to generalize this theorem. Before stating the generalization, we remark that the Rudin-Carleson theorem is closely related to a theorem of F. and M. Riesz, which states that any (finite, complex-valued, Baire) measure on L which is orthogonal to all continuous functions F on D which are analytic on D-L is absolutely continuous with respect to Lebesgue measure dB on L. The proofs of the two theorems show that the results of RudinCdrleson and of F. and 1\'1. Riesz are closely related. We shall state an abstract theorem which shows that the Rudin-Carleson theorem is a direct consequence 01 the F. and M. Riesz theorem. This abstract theorem will permit an automatic generalizdtion of the RudinCarleson result to any situation to which the F. and :\I. Riesz result can be generalized The theorem to be proved reads as follows: THEOREM 1. Let C(X) be the uniformly-normed Banach space of all continuous comple"C-valued functions on a compact Hausdorff space X. Let B be a closed subspace of C(X). Let B-1 consist of all (finite, complexvalued, Baire) measures }L on X such that ffd}L = 0 for all f in B. Let {i. be the regular Borel extension of the Baire meaSltre }L. Let S be a closed subset of X with the property that p.(n = 0 for every Borel subset T of S and every }L in B.i. Let f be a continuous complex-valued function on S-and A a positive function on X such that If(x) 1 for some i. Thus there exists t in B with II t II = 1 and 1. Thus 1 and C > 0 the sum L,;~l II F" 110 t-"" converges, where II P" 110 is defined as above. ( i) (ii)
0
Define the sequence {k i } by k i = 21. Choose the sequences as in lemma 1. Clearly (i) and (ii) are satisfied. Now for Each positive integer n there is a positive integer :i with 2;-1 ~ n < 21. It follows that a", E B j. Thus R.ja", = Q,a.n = an, so that Rma" = a" for all m ~ 21 and therefore for all m ~ 2n. This proves (iii). Now for each n choose J' with 21- 1 ~ n < 21. Thus Proof.
{P",} and
{Ck}
II p.llo
+ kj)l =
~
(1
~
(5n2)1
~
(1
+ 2';)1
(5n2)'" ,
where a = 1 + log, n. From this it follows from elementary calculus that (iv) holds, thereby proving the lemma. LEMMA 3.
Let }2 "'l~O
a,;(n1 ,
••• ,
n~)z~l
.. . z:/JI
••• 'nQSi:!':O
where a = a, and 1 ~ i < co, be a sequence of formal power series with coefficients in a Frechet space F. Let {Ok} be a sequence of positive real numbers. Then there exists a sequence {,sk} with 0 < ,sk < Ok for all k and a sequence {P,,} of mutually annihilating continuous projections of F onto subspaces of dimensions at most 1 such that (a) R"'a,(n,,···, n = a,(n" .. " n whenever m ~ 2i+2n al, where a = a i • n = n , + .,. + n al , and R .. = 2::j~1 P;, (b) P",a,(n,,"', n .. ) = 0 whenever m > 2i +'n"', (c) L.:;:'~1 II P" liD t-"E < co for all t > 1 and C > 0, where II 110 ~s defined as above. al )
al )
Proof. For each- i order the coefficients ai(n" .. " nal) into a sequence {am·~l according to the size of n. We now define a sequence {a k } of elements of F which is an ordering of the totality of the ai(nll " ' , n al )\ For k given let 2i be the largest power of 2 dividing k and let :i = lj2(k2- i + 1). Let a k = ai. Now choose the sequences {,sk} and {P",} as in Lemma 2. Clearly (c) holds. Since (b) is a consequence of (a) we need only check (a). To this end consider a fixed a i (n" .. " n .. ). Now there exists j ~ n with a,(n" "', n = a1. In turn a~ = a k for some k ~ 2i+ln"'. By (iii) of Lemma 2 it follows that Rmak = a k for m ~ 2k and therefore for m ~ 2i+'n"', as was to be proved. al
al)
243
1182
ERRETT BISHOP
We are now prepared to prove a series representation for analytic functions with values in a Frechet space which will be the principal tool in subsequent proofs. THEOREM 1. Let F be a Frechet space and let {Mi} be a sequence of complex analytic manifolds. For each i let CfJi be an analytic function on Mi with values in F. Then there exists a sequence of vectors {b,.} in F and a sequence {P,,} of continuous mutually annihilating projections of F onto one-dimensional subspaces having the following properties. For each i the series L.;:'~l P" 0CfJi converges to CfJi on Mi. For each n we have P"b" = b", so that P" 0 CfJi = CfJ~b", for some analytic function CfJ~ on Mi. For each i the series L.;:'~l CfJ~ converges absolutely and uniformly on all compact subsets of Mi. For each continuous semi-norm II lion F the sequence {II b" II} is bounded.
Proof. For each i let dim Mi = a = ai, so that Mi is coverable by a countable family of analytic homeomorphs r of the unit polycylinder U"
=
{z
= (Z" ••• , z"') : Iz; I < 1, 1
~ j ~ a} •
Thus in the proof of the theorem we may replace the sequence {MJ by the totality of all such r. There is therefore no loss of generality in assuming that each Mi is a polycylinder U'" of dimension a = a i . Let {II Ilk} be a defining sequence of semi-norms on F. Now for each. i the analytic function CfJi has a power series expansion CfJi =
L. 7111:::.0 •••
ai(n" ... , n ..)zfl ... nal-~O
Z:,.
on the polycylinder Mi = U"'. This expansion converges absolutely and uniformly on each compact subset of Mi in each semi-norm II Ilk. By the diagonal process there therefore exist constants Ok > 0 such that the power series for each CfJi converges absolutely and uniformly on each compact subset of Mi in the norm L.;;"-l Ok II Ilk, so that in particular this norm is finite for each coefficient ai(n" ... , n",). Now choose the sequences {Ck} and {Pot} as in Lemma 3 relative to the power series expansions of the CfJi and to the Ok just obtained. Thus the power series for CfJi converges absolutely and uniformly on compact subsets of Mi in the norm II 110 defined above. If some of the projections P,. are zero, these may be omitted from the sequence. Thus for each n there is a vector b" in F with II b" 110 = 1 spanning the range of P". To show that the sequences {P,,} and ibn} have the desired properties~ consider a fixed compact subset T of a fixed Mi. For each n write i"
=
L.
"1+- +n,,=n
max {II ai(n" ... , n,.)zf' ... z:'" 110: z E T} .
244
ANALYTIC FUNCTIONS WITH VALUES IN A
FRECH~l ::'Pl\l.~
111:1,)
By the usual convergence criteria we see that there exist r > 1 and c > 0 such that r",,,. < c for all n. If j is any positive integer let k be the largest integer such that 2i+'k'" < j. Thus for each z in T we have
II PiCfJ.(z) 110
II P
=
~
j ?Ll-!'
2::
-4.-nr»~k
II Pi 110 2:: '''' n~k
a.(n" ... ,
n",)z~1
~ c II Pi 110 ni::;k 2:: r-
... z:'" \1 0
n
Thus L1
= max {~[[ PjCfJ.(z) [[0 : z E ~
c(l - r- ' )-'
2:: r-
k [[
T}
P j [[0 •
:i-l
Now by the definition of k we see that k is the integral part of (j2- i -')' I"', so that k ~ pl'a. for all j sufficiently large. Thus L1 is finite if the sum 2::j~1 r-JP [[ Pi [[0 converges, where c = (2a)-1. By the choice of the sequence {Pj} this series converges so that L1 is finite. Now since [[ b" [[0= I, max {[ CfJ~(z) [ : z E T} = max {II P"CfJi(Z) 110 : Z E T} . Therefore the series L.;~1 CfJ~(z) converges absolutely and uniformly on T. If [[ II is a continuous semi-norm on F then I[ II ~ KII 110 for some K> 0, so that {II b" II} is bounded by K. Finally, we must show that L.;~1 P" ° CfJi actually converges to CfJi (and not to something else). To see this, note by (a) and (b) of Lemma 3 that Rm ° CfJi and CfJi have power series expansions in the coordinates z" ... , z,. which agree up to terms of total order n, whenever m ~ 2i +·n"'. This completes the proof ()f Theorem 1. Before giving the definition of the vectorization of an analytic sheaf, we indicate the terminology to be used, following Godement [5]. A presheaf S on a topological space X assigns to each open U e X a set S(U) and to each open set Ve Ue X a map ryu: S(U)-> S(V) satisfying rWyOryU = rwu for We Ve U. In particular the sam-e-terminology will be used if S is a sheaf, that is, a pre sheaf satisfying axioms (F1) and (F2) on page 109 of [5]. To any presheaf S is canonically associated a sheaf S', and each element f in S( U) gives rise to a unique element in S/( U) which will also be denoted by f. If X is a complex analytic manifold a sheaf S on X is called analytic if it is a module over the sheaf 0 of locally defined analytic functions, that is, jf for each U the set S(U) is an O(U)-module, and if the usual com-
245
ERRETT BISHOP
mutation relations between module multiplication and the restriction maps S (U) --. S (V) and O( U) -. O( V) hold. DEFINITION 1. Let S be an analytic sheaf on a complex analytic manifold M and let F be a Frechet space. Let 0 be the sheaf of locally-defined analytic functions on M and let OF be the sheaf of locallydefined analytic functions on M with values in F, where by definition a continuous function f from an open set U eM to F is called analytic if uof is analytic for all u in F*. Clearly OF is an O-module, i.e., an analytic sheaf. The vectorization SF of S (relative to F) is defined to be the sheaf S ® OF' the tensor product of the O-modules S and OF' This is defined in [5] as the sheaf determined by the presheaf data
U--.S(U)®OAU) ,
where S(U) and OF(U) are considered as O(U)-modules, together with the obvious restriction maps. Note that if T is a continuous linear operator from a Frechet space F into a Frechet space G then the natural homomorphism To of OF int(} 00 induced by T gives rise to a homomorphism T' = 1 ® To of SF int(} So. In particular, if u is an element of F* (and so a continuous linear operator from F into C) then u induces a homomorphism of SF int(} Sa. But Sa is canonically isomorphic to S, in virtue of the canonical isomorphism between the O(U)-modules S(U) ® O(U) and S(U). (See [5] p. 8.) If we identify Sa with S it follows that each u in F* induces a homomorphism u' of SF onto S. DEFINITION
2.
If S is an analytic subsheaf of the Cartesian product
0" we define S;(U) = {fE (OAU»"': uofE S(U) for all u in F*} .
Clearly (OF)'"
S;
so defined is an analytic subsheaf of the Cartesian product
THEOREM
2.
If S is a coherent analytic subsheaf of 0" then to each
p in U c M and each f in S;( U) there exists a neighborhood V of p. functions HI, "', Hk in S( V) and functions G" "', Gk in OF( V) such
that k
ryuf =
L. GmH.,. . -m.=1
Since S is coherent, there exists a neighborhood Vo c U of HIO "', Hk in S( Vo) which generate S at each point of Vo. We may assume that Vo is a compact subset of U. Let Vo::J V1::J v,,::J.'.
Proof.
p and functions
246
ANALYTIC FUNCTIONS WITH VALUES IN A FRECHET SPACt:
HIlO
be a basis for the neighborhoods of p. Let Q be the subset of S( Vo) consisting of all elements in S( V o) which as elements of (O( Vo»" are bounded on Vo. Thus to each h in Q there exists G = (GH " ' , G k ) in (O( V;»k for some i such that the restriction of h to Vi has the form k
h = SG,Hi
•
i=l
By choosing i large enough we may assume that
II Gil,
= sup
{I G,(q) I : q E
V" 1 ~
d~
k}
is finite. Thus if for each pair (i, N) of positive integers we let Q'N be the family of all h in Q such that G can be chosen in (O( V,W with II G II, ~ N, we see that Q = U Q'N and that each QiN is a closed subset of Q, where Q has the norm defined by
II h 110 = sup {I h,(q) I : 1 ~ i
~
n, q E Vol
for each h = (hH "', h,,) E Q c (O( Va»". By the Baire category theorem there exists (i, N) such that QiN has a nonvoid interior. From this it follows as usual that there exists a constant K> 0 such that for each h in Q there exists G in (O(ViW as above with IIGlli ~ Kllhll o. Now consider f as in the statement of the theorem, so that f E S~(U)c(O/f(U»", By Theorem 1 there exists a sequence of vectors {bi} in F which is bounded in each continuous semi-norm on F and a sequence {Pi} of continuous projections on F having one-dimensional ranges such that Si=l Pi of converges uniformly to f on all compact subsets of U and such that for each d we have Pi of = fibi with fi E (O(U»", where Si=l Ifi I converges uniformly on all compact subsets of U. Thus Si~lllfj 110 is finite, since Vo c U. Now for each d there exists u in F* with u> = 1. Thus fj
= uo(fjb j) = uo(PjOf) = (uOPj)of
is in S( U) because f E S~( U) and u ° P j E F*. Thus fj E S( U) for all j. By the above for each d there exists GJ = (G{, .. " GD in (O( Vi»k such that on Vi we have k
fj = m=l ~G~Hm'
with IIGJlli~Kllfjllo. It follows that the series Si=IGJb j converges uniformly and absolutely on V, in each continuous semi-norm on F. Thus the sum of this series is an element G = (G I , " ' , G k ) in (O/f(V,W. Thus in the topology of uniform and absolute convergence on compact subsets of Vi in each continuous semi-norm on F we have
247
ERRETT BISHOP
1186
, f
lim'L..f;b;
=
;=1
t_co
,
k
=
lim'L.. 'L.. G!,.H",b; t-o= ;=1 m-=l
=
i;, (IL~ j;, G!,.b;)Hm
=
'L..GmH"" m=l
k
as was to be proved. The following consequence of Theorem 2 will be useful later. LEMMA 4. If the element f of SA U) has the property that u'f is the zero element of S(U) for all u in F* then f = O.
Proof. By taking a covering of U by small open sets we reduce to the case In which f has a representation
f
k
=
'L.. hJji!~ g, i=I
,
with hi in S( U) and gi in OF( U). Let R be the sheaf on U of relations of h" ... , h k • Thus for each u in F* we see that k
o = u'f = L. hi ® (gi, u) i=I
k
=
'L.. gh induces a group homomorphism of (S( U), DA U»-the free abelian group generated by the elements of the Cartesian product S( U) x OA U)-into (DA U»". It is trivial to check that N(S( U), DF( U»· belongs to the kernel of this map, where N(S( U), DA U» is defined as in [5] p. 8 to be the subgroup of (S( U), OA U» generated by elements of the forms ( i) (XI + X 21 y) - (x" y) - (x" y) (ii)
(x, YI
+ y,) -
(x, YI) -
(x, Y.)
(iii) (ax, y) - (x, ay) where x, x" and x, are in S( U), y, Yll and Y. are in Or( U), and a E D( U). Thus this map induces a homomorphism r-( U) of the quotient (S(U), OAU»(N(S(U), OAU» = S(U) ® OAU) into (OF(U»". It is trivial to check that r-( U) is an D( U)-homomorphism. Now with g and h as above and u in F* we have uor-(U)(h
® g) = uo(gh) = (uog)h E S(U) .
Thus r-( U)(h ® g) E S;( U). It follows that the range of r-( U) is a subset of S;( U). It is now clear that the family of mappings r-( U) induces an O-homomorphism r- of SF into S;. To show that r- is one-to-one we must prove (a) If r-( U)(:£~l hi ® gi) = 0 then each p in U has a neighborhood.." V such that rvu(L.~, hi ® g;) = D. To show that r- is onto we must prove (b) If fE S;(U) then each p in U has a neighborhood V such that Tvu/ = r-(V)(L.~I hi ® gi) for some elements hi in S(V) and gi in DAV). We first prove (a). If we let R be the sheaf of relations on U of hI, ... , hN by the coherence of R there exists a neighborhood Vo of p and elements r , = (ri, ... , rf), ... , r" = (r~, ... , r;;) of R(Vo) which
249
HISIS
ERRETT BISHOP
generate R at each point of Vo.
Now
Thus for each u in F* we have N
L. (uogi)hi = 0 i=l
so that (u 0 gI' .. " u 0 gN) E R( U) for all u in F*. By definition this means that (g" "', gN) E R;( U). Therefore by Theorem 2 we see that there exists a neighborhood V of p and G = (G" "', G,,) in (OAV»n. such that (gI' "', gN) = GIrl + ... + G"r". Thus on V we have
since rj E R(V) for each j. This proves (a). To prove (b) notice by Theorem 2 that there exists a neighborhood V of p, elements h" .. " hN in S( V), and elements g" "', gN in OAV) such that on V we have
This completes the proof of Theorem 3. We state for future reference a version of a theorem of Banach. first giving a definition. DEFINITION 3. If {g,,} is a sequence of vectors in a Frechet space F= the series 2.:.:~I g" is called absolutely convergent if the series 2:.:'=1 II g.. 11 converges for each continuous semi-norm II lion F. Notice that a continuous linear transformation from a Frechet space F to a Frechet space G takes absolutely convergent sequences into absolutely convergent sequences. LEMMA 5. Let a be a continuous linear map of a Frechet space F onto a Frechet space G. Let {gj} be an absolutely convergent sequence from G. Then there exists an absolutely COn1Jergent sequence {fi} in F such that a(f;) = gi for all ~.
Proof. Let {II Ilk} be a defining sequence of semi-norms on F. Since the map a is continuous, we see ([1] p. 40) that for each k the set a{f: Ilfllk 2 1} contains a neighborhood {g: Ilgll~ 21} of 0 in G, where II II~ is some continuous semi-norm on G. Thus for each g in
250
ANALYTIC !"UNCTIONS WITH VALUES IN A l'Kr.\...nJ'.1 ",r....v,,-
G and each k there exists I in F with a(f) = g and for each k choose j = j(k) such that
=
L.llg"ll~
n=J
< 2-
k
III Ilk
~
II g II~. Now
,
so that ~
~
L. L. Ilg .. ll~
'h(z»]h
=
u(z)(l - zh).
By Lemma 5, since }.h(Z) =;cp, we may evaluate both sides of this equality at the point }.h(Z) in Y, obtaining
o=
w(z)(l - zh(>',,(z»).
Since z is not in T, this gives h(}.h(Z» =Z-I. Since h is continuous on Y, by Lemma 5, it follows that h(}.,,(t» =t-- 1 for all t=;6-0 in D". Since by Lemma 7 we see that u(h, z) (f - j(}.,,(z») E'lX for all j in 'lX, to show that u(h, z) is rational over 'lX with a simple pole at }.h(Z), it is sufficient to show that there exists j in ~l such that u(h, z) (f - j(}.h(Z») does not vanish at }.h(Z). To this end, we consider an element j in 'lX with the property that ujE~, where u=u(h, z). By the definition of u, we have (1 - zh)uj
= hf.
so that uj = h(f + zuj). For t in Dh and t=;6-0, this implies by Lemma 5 that
Since h(}.h(t» =t- 1 this gives (*)
if t=;6-0 and t=;6-z. By continuity, (*) is valid whenever t and z are in Dh and t =;6- z. Now letjbe any element in 'lX with j(}.,,(z» =0, j(p) =j(}.,,(O» =;6-0. Let j,,=u"j. Assume thatjnE21 for all nand thatj,,(}.h(z» =0 for all n. It follows by induction from the formula (*) that j,,(}.,,(t» = (t-z)-"j(}.,,(t», for all t=;cz. Thus j(}.h( = j 0 }.h is an analytic function on Dh which is not identically zero such that the function (. -z)-"j(}.I&(·» is analytic for each n. This contradiction shows that our assumption was false. Thus there exists a positive integer n such thatj"_lE~ andj,,_l(}.h(Z»=0 and either j .. EE21 or j"E21 and j,,(}.h(Z» =;6-0. We must have j"E21, sincej.._1E21 andjn_l(}.,,(z» = O. Thus j .. (}./.(z» = (Uj,,_l) (}.,,(z» =;6-0. This is just what was needed to show that u is rational over 21 with a simple pole at }.iI(Z). By the formula (*), we have u(h, z, }.,,(t»::o: (t-Z)-I. Now if P is any point in Y with q=;6-}.h(Z) and q=;6-P, choose j in 21 with j(q) = 1, j(}.h(Z» =0. By the above, we have uj=h(f+zuj). Evaluating at q, we have
.»
264
1962J
ANALYTICITY IN CERTAIN BANACH ALGEBRAS
u(q)
517
= u(q)f(q) = (uf)(q) = h(q)(l + z(uf)(q)) h(q)(l
=
+ zu(q»,
so that
u(q)(l - zh(q» = h(q), as was to be proved. LEMMA 10. Let h be a rational function over the umform algebra ~{ with a simple pole p. The mapping A" is a homeomorphism of DII onto Ell and Ell is an open set in the spectrum Y of ~r. Proof. The mapping A" is one-to-one because we have seen that h(Ah(z»
= Z-1 for all z in D II . We shall show that Ah( U) is open in Y for all open subsets U of D". Since All is continuous and one-one, it will follow that All is a homeomorphism. By taking U = D", it will follow that E" is open. Assume then that A,,( U) is not open, and let z in U be chosen so that A,,(Z) is not in the interior of A,,( U). Since h is continuous on Y and since [h(AII(Z» ]-1 = Z is in D", it follows that t = [h(q) ]-1 will be in DII whenever the point q in Y -AII( U) is near enough to A,,(Z). Choose such a point q. Since qEEAII(U) we have q~p and q~AII(t). By Lemma 9 it follows that
t- 1 = h(q) = u(h, t, q) [1 - th(q)]
= u(h, t, q)[l - tt-IJ = 0, a contradiction. Thus A,,( U) is open, as was to be proved. DEFINITION 11 AND LEMMA 11. Let ~ be a uniform algebra and ~o the set of functions rational over ~, so that ~ c~o. We define A =A(~), called the analytic part of Y, to be the set of all points P in the spectrum Y of ~ such that there exists h in ~o with a simple pole at p. Thus the poles of any function in ~o lie in A. The set A is open in Y and can be given uniquely the !ltructure of a Riemann surface in such a way that all functions h in mo are analytic on A except for a finite number of poles which with their multiplicities coincide with the poles and multiplicities of h when considered as a function rational over ~r. A point p in A is said to be a zero of order k of a function h in ~o if the analytic function h on A has a zero of order k at p. For each p in A there exists g in ~ with a zero of order 1 at p. If h is in ~o, if thE' points PI, ... ,P.. in A with multiplicities kl' ... , k.. respectively are the pol~ of '1, and if hl, ... ,h.. are elements of ~o having simple poles at Pl, ... ,P.. r,pectively then h can be written in the form (*)
h =
"
k,
i=l
i-I
:E :E aii(hi)! + f,
where the aii are constants and f is in ~. The set ~D is a subalgebra of C(X),
265
J[Cu(h, 'Yh.(q))][d[uCh, 'Yh(q))]-l]q = rI>[u(h, 'Yh(q))]dh- 1(q), =
and is therefore an analytic differential. To prove the formula [*], we avail ourselves of the representation (*) of Lemma 11. Thus in proving [*] it suffices to consider functions h", where h has a simple pole in V, and functionsfin~. Now iffE~, both sides of [*] vanish, the left side by the hypothesis on cp and the right side because fdw~ is analytic on VUC. Thus we consider h with a simple pole at a point p in V. Since hndw~ is analytic on VUC except at p, we may replace the contour C by any simple contour about p. Thus in proving [*] we may choose C to be a simple closed rectifiable curve lying in E~ and surrounding the point p. Using the representation obtained above for dw~ in E~, we now compute, letting B be the curve in D~ corresponding to the curve C in E~,
Ie
hndw~
=
Ie h"(q)rI>(u[h, 'Yh(q)])dh- 1(q)
= IBz-"rI>(U(h, z))dz =
=
rI> [IBz-nu(h, Z)dZ]
rI> [iz-nh(1 - Zh)-ldZ] = rI> [EIBzk-nhH1dz]
= rI>[271'"ihn] = 271'"irl>(h"). This proves [*] and thereby completes the proof of Lemma 12, since the fact that dw~ is unique clearly follows from [*]. LEMMA 13. Let ~ be a uniform algebra with spectrum Y whose analytic part is A and whose Silov boundary is X. Let U be an open set in A, and let B be the boundary of U. Let 581 consist of all rational functions over ~ whose poles lie in
270
1962J
ANALYTICITY IN CERTAIN BANACH
ALGl'..tlK1\.;:'
U, and let lB be the closure in C( Y - U) of lB 1• Then Y - U is the spectrum of lB and the Silov boundary of lB is a subset of XUB. to
Proof. Consider any element}.. in the spectrum of lB. The restriction of }.. is some point p in Y, so that
~
X(j) = f(P)
for all f in ~. Assume that PE U. Choose h in fin ~l withf(P) =0, (jh)(p)~O. Then
o~
(Jh) (p)
=
X(jh)
~o
with a simple pole at
= X(J)X(h) = o· X(h)
=
O.
This contradiction proves that PE Y - U. For any h in lBl choose g in g(p) ~o and ghE~. Then g(p)X(h)
=
X(g)X(h)
=
X(gh) = (gh)(p)
P and
~
with
= g(p)h(p),
so that }'(h) =h(p). Since this is true for all h in lB1 it is true for all h in lB. Thus Y - U is the spectrum of'S. We now show that the Silov boundary of lB is a subset of XUB. Consider a point p in Y - U - (XUB). Since X is the Silov boundary of ~ there exists a bounded linear functional CPo on C(X) such that
rpo(j)
= f(P)
for all f in ~. Define the bounded linear functional cp on C(XU {p }) by q,(J)
=
CPo(J) - f(p)·
Thus cp(j) = 0 for all f in ~. Let V be any relatively compact subset of U whose boundary C consists of a finite number of disjoint rectifiable simple closed curves. Let lBy consist of all functions in lB1 whose poles lie in V. From [*J of Lemma 12 it follows that rp(h) = (211'i)-1
fe hdwq,
for all h in .\BY. If we define the bounded linear functional CP1 on C(C) by
is an arbitrary bounded linear function on C(X) which annihilates ~, it follows that fg-lE~L To complete the proof that g-l is rational over ~, choose giE~, 1 ~i~n, such that g; has a simple zero at Pi and g.(PJ) r! 0 for i r! j. Write f = (gl)k\ . . . (gn)kn. By the above we have fg-lE~. Since (jg-l)g=f and g have zeros of the same order at PI,"', Pn, it follows that (jg-l)(p;)r!O for 1~i~n. This completes the proof that g-l is rational over ~. 3. Conditions for analyticity of the spectrum. In this section we derive conditions which imply that certain points in the spectrum of a uniform algebra belong to the analytic part of the spectrum. Somewhat more exact conditions could be given, by refining the techniques employed here, but the
272
1962j
ANALYTICITY IN CRRTAIN BANACH
ALu~~KI\.~
added generality which would be obtained does not seem to justify the attendant complication of the proofs. LEMMA 15. Let 2l be a uniform algebra with Silov boundary X and spectrum Y with analytic part A. Let g be a function in ~ which vanishes at a finite set PI, . . . , Pn of points in Y, all of which lie in A and are simple zeros of g. Let A = inf { I g(x) I : xEX}. Let f in ~ have the properties f(Pl) = 1 and f(Pi) = 0 for 2 ~i ~n. Then there exists a neighborhood Fl of PI in A which g maps homeomorphicallyonto z: I zl (l+ ~ K)X- ~ Kx=x>z
so that go(p)~z. Next consider the case /g(P)/ ~(1+K/2)x, so that /h(P)/ i, k>j, and g'(p') EEg(X k ). Now ~. is a homeomorphism of some neighborhood V in Si of p into Vi. Thus rPlc. o~. gives a homeomorphism of V into Y k • Since ~; is a homeomorphism of some neighborhood W of q into V;, rPk; o~; is a homeomorphism of W into Y k • We may assume that P"E V and q"E W for all n. For each f in ~ we have SO
fa rPki O >"i =10>... =f'OUi
on
v.
10 rPM a >..; = 10>"; = f' a
on
w.
Similarly, U;
I n particular,
l(rPki[>".(p,,))) = j'(u.(p ..)) = j'(p,:)
= j'(q,n =
f'(u;(q,,)) = l(rPk;[>";(q,.)])
for all 1 in ~. Thus rPki~.(P .. )) and rPk;(~;(q,,)) are the same point y" in Y". Now g(rPlci[~i(P)])=g~i(P))=g'(P')EEg(Xk) so that rPk.(~.(P))EYk-X" for all n sufficiently large. Since Tk is isolated in Y,,-X k it follows that y"E Y lc -X" - Tk =Ak for all n sufficiently large. Fix such a value of n. There exists a neighborhood V" of p" in S. which rPlc. 0 ~. maps homeomorphically into Ale. Since Si and Ak are Riemann surfaces it follows from invariance of domain (see [4, p. 95]) that rPki a ~i maps V" homeomorphically onto an open set in Alc containing y". Similarly rPk; o~; maps some neighborhood W" of q,. homeomorphically onto an open set in Alc containing y". We may assume that rPk'~'(V,.)) =rPlc;(~;(W,.)). Thus
fJ = (rPlc; a >..;)- a (rPlc. 0 >"i) is a homeomorphism of V .. onto W". Now
1 a >..j a fJ = 1 0 rP-;; a rPlc; a >... = 10 rPlc; a >..; = 10 >..; on V" for all 1 in
~.
It follows that
289
ERRETT BISHOP
p..
== fJ(P .. ) =
[March
(q,J..j a Xj)-y" = q...
Th us U i(P,,) = U j(q.,,), or P; = q; . This contradicts the fact that (p; , q;) E T and thereby establishes the truth of (3) of Theorem 2. It remains to prove (4) of Theorem 2. We first show that the closure of Sf is a compact subset of S' for each i. Since the set
Rf
=
ui(R i )
where
Ri
=
Xi(A i )
is dense in Sf, it is enough to show that the closure of Rf is compact. Let Ip; } be a sequence of points of R!. Let P.. =Ai(U-;(P;», so that PnEA i . We may assume, by passing to a subsequence if necessary, that {Pn} converges to a point P in Vi. Choose m >i with g(p) EEg('¥m). Then cPmi(P) EEXm. We have the following diagram
q,,,,i
Xi
Xm
Si~ Y i - - Ym--Sm,
and P.. -P in Y i as n- O(). Since cPm; is continuous, cPmi(P .. )-cPm'(P) as n- 0() , so that cPmi(P .. ) EArn for all n sufficiently large, say for all n. Thus for each n there exists a unique point tn in Sm with Am(tn) =cPmi(Pn). Let ql, . . . , qJ.. be those points in Sm with Am(qj) =cPmi(P) , 1 ~j ~ k. Thus to each open set V in Sm containing the points ql, . . . , qk corresponds a neighborhood Vo of cPmi(P) in Y m with A;;; (Vo) C V. Thus tnE V for all n sufficiently large. We may therefore assume, by passing to a subsequence if necessary, that tn converges to one of the points ql, . . . , qk, call it t. Let W be a neighborhood of tin Sm mapped homeomorphically by Am into Y m. Take W so small that A",(W t}) CAm' so that Am is a homeomorphism of W t} onto an open set in Ym • We may assume that t .. E W for all n so that cPmi(P~) =Am(tn)EAmCW). Now PnEA i and cPmi(Pn) EArn. Since cPm' maps a neighborhood of Pn homeomorphically into Am, we may assume by invariance of domain that cPmi maps a neighborhood V .. in Ai of p.. homeomorphically onto a neighborhood of cPmi(Pn) in Am. We may assume that cPmi( V n ) CAm(W p}). Thus A;;; a cPm; maps Vn homeomorphically onto a neighborhood W .. of tn in Sn and A; maps V .. homeomorphically onto a neighborhood Wn of A"t(P .. ) in Si. Therefore the map
-I
-I
-I
on W" maps W" homeomorphically onto W n • If f is any function in ~ then
fa Xi = fa q,mi a Xi = f a X", a fJ Therefore
so that
P': = Ui(X"t(pn» = um(t,,).
290
on W".
1962]
ANALYTICITY IN CERTAIN BANACH ALGEBRAS
54,s
Therefore {p,: } converges to the point um(t) in S'. Thus there is a convergent subsequence of {p,: }. Therefore the closure of Sf is compact. Consider any compact subset K of S'. We have seen above that K ca('Yi) for some i. Therefore K ca('Yi). The set H
= Sf U U('Yi)
is a subset of a('Yi) because it was shown above that Sf ca('Yi). Since Sf is compact H is compact. The set
L=KnH is also compact. Clearly
K contains all
points p such that (p, q) E T for some
q in L. We shall complete the proof by showing that conversely if PEK-L then (p, q) E T for some q in L. Consider pin K -L. Thus P E a('Yi) - H. I t follows that 1J'(p)
I ~ sup{ I/(q) I: q E
'Yi}
for all / in ~. There therefore exists qo in Y i with 1'(p) = /(qo) for all f in ~. Thus either qoEX i or qO=Ai(ql) for some ql in So. In the first case let ql be a point in 'Yi with 7r i(ql) = qo, so that in the first case q = U(ql) EH and J'(q) = /(ql)
= /(qo) = /'(p)
for all/ in ~. Thus (p, q) E T. In the second case let q =Ui(ql). Thus IlES: CH and l' (q) = /(Ai(ql) = f(qo) =1' (p) for all f in ~. Thus (p, q) E T. Thus in either case there exists q in H such that (p, q) E T. Also qEK because 1'(q) = 1'(p) for all/in~. Therefore qEKnH=L. This completes the proof of Theorem 2. \Ve end with a result which completely describes the uniform closure of an algebra of analytic functions on a compact subset of a Riemann surface. THEOREM 3. Let K be a compact subset of a Riemann surface S. Let ~ be a holomorphically complete algebra of analytic functions on S. Let .\B be the uniform closure of ~ on K. Let Y be the spectrum of .\B. Let (S', ~') be the extension of (S, ~) described in Theorem 2, and (1 and T the maps there described. Let M be the union of L = u(K) and all of those components of S' - L which are relatively compact subsets of S'. Then (a) .\B is isomorphic to the uniform closure of~' on M, via the maps U and T. (b) For each q, in Y there exists p in M with q,(f) =f(p) for all / in .\B, where .\B is considered as a subalgebra of C(M). (c) The linear space ~ is of finite codimension in the space .\B o oI7ilt-t;ontinuous functions on M which are analytic at interior points of NI.
Proof. From Theorem 2 it is clear that for each f in ~ the uniform norm of fan K and the uniform norm of T(f) on L are equal. From this (a) follows readily.
291
[lVlarcn
Now by Theorem 2 there exists a compact set D CM such that for each
q, in Y there exists p in D with q,(j) = J(P) for all J in ~'. By (iv) of Theorem 2 and the principal theorem of [1] it follows that 58' is of finite codimension d in 581, where .$8' is the uniform closure of ~' on D and where .$81 is the set of all continuous functions on D which are analytic at interior points of D. Assume that D-M is not a finite set. Thus there exist distinct points PI, ... ,Pd+1 in D-M. Since DCM, for each i there exists a finite measure lJ.i on M such that O. -lJ.i.l~', where 0. is the point mass at Pi. Now considered as linear functionals on ,SZh these measures O;-lJ.i are all linearly independent because by Runge's theorem there exists Ji in .$81 which has the value 1 at Pi and 0 at the other p's and is arbitrarily small on M. But since these d+ 1 measures annihilate 58' we have a contradiction. Thus D - M is finite. Thus if cP in Y does not have property (b) above then cf> corresponds to a point P in D - M. Since p is isolated in D an easy argument shows that cf> is isolated in Y. By a theorem of Silov it follows that cf> is in the Silov boundary of 58. This contradiction shows that (b) is valid for all q, in Y. Finally (c) follows from Theorem 2 and the principal theorem of [1]. REFERENCES
J. Math. 8 (1958), 29-50. 2. C. Chevalley, Theory of Lie groups, Princeton Univ. Press, Princeton, N. J., 1949. 3. E. Heinz, Eine elementarer Beweis des Satzes von Rad6-Behnke-Stein-Cartan ilber analytischen Funktionen, Math. Ann. 131 (1956), 258-259. 4. W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N. J., 1948. 5. L. H. Loomis, Abstract harmonic analysis, Van Nostrand, New York, 1953. 6. S. N. Mergelyan, Uniform approximation tofunctions of a complex variable, Amer. Math. Soc. Transl. 101 (1954). 7. J. Wermer, Function rings and Riemann surfaces, Ann. of Math. 6'1 (1958), 45-77. 8. - - , Rings of analytic functions, Ann. of Math. 6'1 (1958), 497-516. 9. - - - , An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150. 1. E. Bishop, Subalgebras of functions on a Riemann surface, Pacific
UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA
292
of the American Mathematical Society.
THE SUPPORT FUNCTIONALS OF A CONVEX SET BY
ERRETT BISHOP AND R. R. PHELPS The following well·known separation theorem is basic to the considerations of this paper. SEPARATION THEOREM. Suppose that A and B are convex subsets of a real Hausdorff topological vector space E, and that the interior of B is nonempty and disjoint from A. Then A and B can be separated by a hyperplane, that is, there exists a continuous linear functional f =t= 0 on E such that supf(A) ~ inf f(B).
This theorem is a geometric version of the Hahn·Banach theorem. Its proof can be found in any of several texts, for instance in [3, p. 417]. An immediate corollary is the following support theorem. If C is a convex subset of a real Hausdorff topological vector space E, if x is a point in the boundary of C, and if the interior of C is nonempty, then there exists a hyperplane which supports C at x, that is, there exists a continuous linear functional f =t= 0 on E such that f(x) = supf(C). We refer to such a functional f as a support functional of C, and x is called a support point. Note that if f is a support functional of C then every positive multiple of f is also a support functional of C. The assumption that C has interior points is a strong one, but some can· dition is indispensable to the validity of the support theorem. Indeed, V L. Klee has shown [61 that there exists a bounded closed convex subset of a dense subspace of a Hilbert 5pace which has no support points. In the same paper Klee asked whether every bounded closed convex set C in a Banach space has at least one support point In this paper we answer Klee's question affirmatively. We show that the support points of C are actually dense in the boundary of C. This is shown to be true even if C is not bounded. Still assuming that E is a Banach space, we then prove (if C is bounded) that the support functionals of C are dense in the dual space E*. If C is not necessarily bounded, we show more generally that for each f in E* which is bounded on C and each E > 0 there exists a support functional g of C with Ilf - g II < E. In fact it is shown that g can be chosen to strictly separate C from any bounded set X which is strictly separated from C by f. We also show that every hyperplane which intersects the boundary of C contains a support point of C. Examples are given to show that these theorems fail in certain more general situations. The methods of this paper derive from a previous paper [1], in which a proof was indicated of the fact that the set of support functionals of a bounded closed convex set in a Banach space is dense in the dual space. By extending and simplifying the method of [1], we have been able to improve this 27
293
ERRETT BISHOP AND R R. PHELPS
28
result and to obtain proofs of the related theorems mentioned above. Although all our theorems are stated and proved for spaces over the real field, they may easily be formulated and extended to spaces over the complex field by applying them to the underlying real space (obtained by restricting multiplication to real scalars) and to the real parts of complex linear functionalso These formulations are analogous to that given in [3] for the separation theorem. Throughout this paper we will restrict our attention to normed spaces E and the convex sets under consideration will be assumed to be proper and nonempty. We write U for {x. II x II ~ I}, if fE E*, then Ilfll is defined to be supf(U). The proofs of the existence of support points and support functionals are based upon showing the existence of certain support cones for a given closed convex set. We will say that a subset K of E is a convex cone if K is a convex set and AY E K whenever y E K and A ;?; O. If X is a set containing the point xo, and if K is a convex cone such that K + Xo is disjoint from X - {xo}, then we say that K + Xo supports X at Xo. Suppose, now, that K has nonempty interior, that C is convex, and that K + Xo supports C at Xo. Then, by the separation theorem, there exists a nontrivial g in E* such that supg(C) ~ infg(K + xo). It is easily verified (since Xo is in both K + Xo and C) that supg(C) = g(xo) = inf g(K + xo), i.e., g supports C at Xo. Thus, to show the existence of support points and support functionals for C it suffices to find support cones of C which have interior points The cones which we will use for this purpose are all of the following type DEFINITION.
Itf is an element of E* of norm one and if k > 0, then K(f, k)
=
{x- Ilxll ~ kf(x)}.
Clearly, K(f' k) is a closed convex cone; furthermore, if k > 1, then the interior of K(f, k) is nonempty To see this, choose x in E such that II x II = 1 and f(x) > k- 1 • Since f and the norm are continuous, and since II x II < kf(x), it follows that this (strict) inequality is true for all points in a neighborhood of X. The fact that closed convex subsets of a Banach space admit support cones is a consequence of the following lemma. LEMMA 1. Suppose that X is a complete subset of a normed linear space E, that f in E* is of norm one and is bounded on X, and that k > O. If Z EX, then there exists a point Xo in X such that Xo E K(f' k) + z and K(f, k) + Xo supports X at XO.
PROOF We partiaJly order the set X by means of K; that is, x >- y means x - y E K. This, of course, is equivalent to saying that II x - y II ~ kf(x - y). It is easily seen that if there exists a maximal element Xo in X, then K + Xo supports X at Xo. To obtain the conclusion that such an Xo exists for which Xo >- z, it suffices to apply Zorn's lemma to the set Z of those x in X for which x >- z. Suppose, then, that W is a totaJly ordered subset of Z. The set {f(x) x E W} is a bounded monotonic net of real numbers (using W as
294
29
THE SUPPORT FUNCTIONALS OF A CONVEX SET
our directed index set), hence it converges to its supremum. This implies that it is a Cauchy net, and since x, y in W implies that II x - y II ~ kff(x) - fey)]' say, we see that W itself is a Cauchy net in Z. Now, Z = X n (K + z); since K is closed, Z must be complete and therefore W converges to an element of y in Z By continuity of f and of the norm, it is simple to verify that y >- x for all x in W, i.e, W has an upper bound in Z. Zorn's lemma then applies and our lemma is proved. The ideas we have developed so far enable us to prove the density of support points. THEOREM 1. If C is a closed convex subset of a Banach space E, then the support points of C are dense in the boundary of C. PROOF. If z is a point of the boundary of C and e > 0, choose y in E ~ C such that II y - z II < E/2 and choose (by applying the separation theorem to any convex neighborhood of y which is disjoint from the closed set C) f in E* such that Ilfll = 1 and supf(C) ~ fey) By Lemma 1 there exists Xo in C such that Xo EO KCf, 2) + z and KCf, 2) + Xu supports C at Xo Applying the separation theorem to C and K + Xo shows that Xo is a support point of C. Furthermore, since Xo - z E K and Xo E C, we have II XU - z II ~ 2[f(x o) - fez)] ~ 2[f(y) - fez)] ~ 2 II y - z II < e, which completes the proof. To prove the density of support functionals we first prove that if k is large and if g is of norm one and is non-negative on KC!, k) then f and g are close together. This result is precisely stated in Lemma 3; the latter follows from a known lemma [7] which is proved again here for the sake of completeness. LEMMA 2. Suppose that E > 0, Ilfll = 1 = Ilgll, and that Ig(x) I ~ E/2 whenever f(x) = 0 and II xii ~ 1. Then either Ilf + gil ~ E or Ilf - gil ~ e. PROOF. By the Hahn-Banach theorem we can choose h in E* such that g on rl(O) and II h II = sup Ig(U n rl(O)) I. Then, by hypothesis, II h II ~ E/2. Furthermore, since g - h vanishes on f 1(0) there exists a real number a such that g - h = af. Hence Ilg - af II = II h II ~ E/2. Assuming a ~ 0, we will show that Ilf - g II ~ e (Otherwise, the same proof applied to (-a)j would show that Ilf + gil ~ e) If a ~ 1, then a-I ~ 1 and
h
=
Ilg - fll = II (1 - a-I)g Also,
0'
= II
af II
~
a-I ~
1-
Hence II g - f II
II g II
~
+ a-I(g - af) II
+ II g - af II (1
+ Ilg
2 II g - af II
~
1-
I
+ a-I Ilg - afll .
so
- af!l)-I Ilg - afll
~ E.
(X-
If 0
~
~
Ilg - afll .
a < 1, then -
= II g - af II + 1 - a = II g - af II + II g II - II af II ~ 211 g - af II ~ E ,
II g - f II ~ II g - af II
+
---------
II (1 - a)f II
which completes the proof. LEMMA 3. SUPposfl that 0< e: < 1, that Ilfll = 1 If g is non-negative on KC!, k), then Ilf - g II ~ E:.
295
=
Ilgll and that k > 1
+ 2/E:.
30
ERRETT BISHOP AND R R. PHELPS
PROOF. Choose x in E such that II x II = 1 and f(x) > k-I(l + 2/c), and suppose that y in E is such that fey) = 0 and II y II ~ 2/c. Then II x ± y II ~ 1 + 2/c < kf(x) = kf(x ± y), so x ± y E K and hence g(x -+- y) ~ O. This implies that Ig(y) I ~ g(x) ~ II x II = 1. Clearly, then, Ig(y) I ~ c/2 whenever fey) = 0 and Ily II ~ I, so by Lemma 2, either Ilf + g II ~ c or Ilf - g II ~ c. Choose z in E such that II z II = 1 and fez) > max (k I, c). Then z E K so g(z) ~ 0 and it follows that II f + g II ~ (f + g )(z) > c, and our proof is complete. We could now easily prove our density theorem; to include unbounded sets C, it would be formulated somewhat as follows. If f is bounded on C, then there is a support functional of C which is arbitrarily close to f. With little care, however, we can do considerably more than this, the result is expressed as follows. If f strictly separates the set C from a bounded set X, then there is a support functional of C which is arbitrarily close to f and which strictly separates X and C More precisely. THEOREM 2. Suppose that C and X are subsets of a Banach space E, that C is closed and convex and that X is bounded and nonempty. If c > 0 and if f in E*, Ilfll = I, is such that supf(C) < inff(X), then there exist g in E*, Ilgll = I, and Xo in C such that Ilf-gil ~ c and g(xo) = supg(C) < infg(X). PROOF. Let r = supf(C), 11 = inff(X) and choose fJ such that r < fJ < 11. Consider the neighborhood V of X defined by X + (11 - fJ)U = V. This is a bounded set, and since inf f( U) = -I, we have inf f( V) = inff(X) - (11 - fJ) = fJ. Let a = 1 + 2/c and choose z in C such that r - fez) < (2ar l (fJ - r) Let M be larger than 2- I(fJ - r) and sup{lly - zll:yE V} and let k = 2aM(fJ - r)-I. (Note that k > a > 1.) Choose, by Lemma 1, a point Xo in C such that K(f, k) + Xo supports C at Xo and Xo - z E K. We will show that V c K + Xo. Indeed, if y E V, then II y - XO II ~ II y - z II + II XO - z II < M + II XO - z II ;£ M + kf(xo - z) ;£ M + k[r - fez)] < M + k(2a)-I(fJ - r) = 2M < 2aM = k(fJ - r);£ kf(y - xo). By the separation theorem there exists g in E*, Ilgll = 1, such that sup g(C) = g(xo) ;£ inf g(K + Xo) ;£ inf g( V) = inf g(X) - (11 - fJ) < inf g(X). Since 0 ~ inf g(K) and k > 1 + 2/c, it follows from Lemma 3 that Ilf - gil;:::; c. A well known variant (and corollary) of the separation theorem substitutes compactness for interior [5]: If Band C are disjoint convex subsets of a locally convex space, with B compact and C closed, then there exists a hyperplane which strictly separates Band C, that is, there exists f in E* such that supf(C) < inff(B) This result can be improved if E is a Banach space, giving the following corollary of Theorem 2. COROLL"RY 1. If Band C are disjoint convex subsets of a Banach space E, with B compact and C closed, there exist x in C and g in E* such that g(x) = sup g( C) < inf g(B). An obvious corollary of the above variant of the separation theorem states that a closed convex subset of a locally convex space E is the intersection of all the closed half· spaces which contain it, that is, if x rt C then there exists fin E* and a real number c such that the half-space H= {y:f(y);£ c} contains C but not x. We say that the half-space H supports C if C cHand
296
THE SUPPORT FUNCTIONALS OF A CONVEX SET
f(y) = c for some y in C. from Corollary 1.
31
The proof of the following corollary is immediate
COROLLARY 2. If C is a closed convex subset of a Banach space, then C is the intersection of all the closed half-spaces which support it. COROLLARY 3. If C is a closed convex subset of a separable Banach space E, then C is the intersection of a countable number of its supporting closed halfspaces. PROOF. Let {x n};;' 1 be a dense subset of E - C and let d" be the distance from x" to C. By first applying the separation theorem to C and x" + d"U, then applying Theorem 2 to C and x" + 2- 1 d"U, we can find g" in E* such that g" supports C and supg,,(C) < infg,.cx" + 2 ld.U), for n = 1,2,3, .... Suppose, now, that XE E --- C and let d be the distance from x to C. Choose x" such that II x - x" II < 3- 1d Then if y E C, II y - x .. II ~ II y - x II - II x - x" II > d - 3- 1 d = (2/3)d Thus, d" ~ (2/3)d and hence II x - x" II < 2-'d.. , which shows that g,,(x) > supg,,(C) and completes the proof. If C is bounded, then every f in E * is bounded on C, so we obtain another corollary to Theorem 2. COROLLARY 4. If C is a bounded closed convex subset of a Banach space E, t hen the support functionals of C are dense in E *. If fE E*, we say that f attains its norm provided there exists x in U such that Ilfll = f(x). Since U is closed and convex, the proof of the following corollary is immediate from the previous one. (This result was first proved in [1].)
COROLLARY 5. If E is a Banach space, then the set of fin E* which attain their norm are dense in E *. In all the above results we could have dropped the hypothesis that E be a complete normed space provided we assumed that C itself be complete. This follows from the fact that we applied Lemma 1 only to the set C. The following result shows, however, that Corollary 4 (and hence Theorem 2) must fail in an incomplete space. THEOREM 3. If E is an incomplete normed linear space, then there exists a bounded, closed convex subset C of E having nonempty interior such that the suPPort functionals of C are not dense in E *. PROOF. Imbed E as a dense subspace of its completion F and identify (in the obvious way) E* and F*. Since E =1= F, there exists x in F- E suCh--that II x II = 1. By applying the support theorem to x and the unit ball of F, we can find f in F* such that Ilfll = 1 =f(x). Let D = {y.YEF,llyll ~ 1 and fey) = O} and let C' in F be the convex hull of D and x, so that C' is the set of all elements of the form z = ..lx + (1 - ..l)y, where y ED and ..l E [0,1]. It is easily verified (using the compactness of [0, 1]) that the convex set C' is closed; it also has nonempty interior. (If liz - (1/2)x II < 1/8, then 0 < fez) < 1.
297
32
ERRETT BISHOP AND R R PHELPS
Define y by z = f(z)x + (1 - f(z))y; it is not difficult to verify that y E D so that z E C'.) Let C = C' n E; then C is convex, closed and has nonempty interior relative to the (dense) subspace E. Clearly xe C, we will show that if gE E* and g(z) = supg(C) for some z in C, then Ilf - g II ~ 1/2. Write z = AX + (1 - A)Y as above, since z E C, we have z *- x so that A < 1. Hence (since sup g(C') = sup g(C)) it follows that g(x) ~ g(z) = (1 - A)g(y - x) + g (x) and therefore g(y-x)~O. Now 2~llxll+llyll~llx-yll~f(x-y)=1 so that 1 ~ (g - f)( y - x) ~ II g - f II II y - x II ~ 2 II g - f II This shows that no functional in E* which is within 1/2 of f can support C, and completes the proof. Note that Theorem 2 may also fail if we drop the assumption that the set X be bounded. Indeed, if E is a separable nonreflexive Banach space, a theorem of James [4] asserts the existence of f in E* such that f(x) < 1 = Ilfll for all x in U. Let X= {y.f(y) ~ 2} and let C = U; then supf(C) < inf f(B), but if g separates C and B, then g must be some real multiple of f, and hence cannot support C. Another example (which is closely related to the preceding one) shows that Theorem 2 may fail if we drop the hypothesis that f strictly separates X and C. For, if E is a separable nonreflexive Banach space, choose f in E* as above and let C = U and X = {x: II x II ~ 2 and f(x) ~ 1}. Then X and Care disjoint bounded closed convex sets having nonempty interior, and supf(C) = 1 = inff(X). Suppose there existed g in E* and x in C such that g(x) = sup g(C) ~ inf g(X). (Thus, g(x) = II g II.) Let A be the convex hull of x and X; since X has nonempty interior, so does A. Furthermore, it is not difficult to see that infg(A) =g(x) = Ilgll Since XEC we have f(x) < 1, and we can choose z in C such that fez) > 1 - 4-'[1 - f(x)1 Let y = 3-'(4z - x). Then y is in the interior of X, indeed, II y II ~ 3-'(4 II z II + II x II) = 5/3 < 2, while f( y) = (4/3)f(z) - (1/3)f(x) > (4/3) - (1/3)[1 - f(x)] - (1/3)f(x) = 1. Since x E A and y is an interior point of A, we conclude that (3/4)y + (1/4)x = z is an interior point of A. But z is also in C, so there must exist a point w in A which is in the interior of C. This is impossible, since if wE A then g(w) ~ II g II, while g(w) is less than II g II at the interior points w of C. The first example given above shows that the conclusion to Theorem 2 fails if X is unbounded, even though it is assumed that C itself is bounded and X is a linear variety (i.e., a translate of a linear subspace). If we assume that X is a finite dimensional variety, however, we get a valid result (which is actually a corollary to Theorem 2). More generally, we can assume that X is a reflexive variety, that is, X = x + M for some x in E and some sub· space M of E, where M is a reflexive Banach space under the induced norm. COROLLARY 6. Suppose that C is a bounded, closed convex subset of the Banach space E, that X is a reflexive linear variety in E, that E > 0 and that lor I in E*, Ilfll = 1, we have supf(C) < inf I(X). Then there exists g in E*, Ilg II = 1, and Xo in C such that supg(C) = g(xo) < inf g(X) and Ilf - g II < E. PROOF.
X = x
Since I is bounded below on X, it must be constant on X; writing as above, we then have f(M) = O. Let E, = E/M; under the
+M
298
THE SUPPORT FUNCTIONALS OF A CONVEX SET
33
usual factor-space norm, E, is a Banach space and f can be regarded as an element of norm one in Et. Let C, be the image of C in E, under the canonical mapping of E onto E" and assume for the moment that C, is closed in E,. Regarding X as a point in E" we have supf(C,) < f(X), so by Theorem 2 there exists gin E,*, II g II = 1, and Xu in C, such that sup g(C,) = g(Xu) < g(X) and Ilf - gil < e. Since C, is the image of C, there exists Xo in C such that Xo = Xo + M. We can define g on E in the obvious way; then II gil = 1 and Ilf-gil < e in E*; since supg(C) =g(xu) 0, and if z is an element of the boundary of C such that f.cz) = for each i, then there exist.., a support point Xu of C such that II z - XU II < E: and fi(XO) = 0, i = 1,2, .. , II This dual result is valid (even without the assumption that C be bounded) and is a generalization of Theorem 1 In order to omit reference to the specific functionals f" ., fn, we will work with the subspace N = {x f.(x) ~ 0, I = 1,2, . ,Il}, N is of finite deficiency, and every subspace of finite deficiency ha;, the same form as N. We first prove a useful lemma.
°
LEMMA 4. Suppose that N is a closed ~llhsl)([c(' of flJlitl' (/l'ficiClu;y in E, that C is a collvex subset of E, ami that Xu is a support poiut oj C n N ill the suhspace N. Then Xu is a support point oj C.
PROOF Let n denote the deficiency of N, the proof will proceed by induction on n It is obviously true if 11 = 0, suppose, then, that ~;->o and assume that the result is true for subspaces of deficiency n - l. Let g be a nontrivial functional in N* such that g(xo) = sUPf{\C n N) Note that if C c N, then any functional which vanishes on N supports C at xo, so we can assume that there exists an element y in C ~ N. Let N' be the linear span of Nand y, and let C' be the convex hull of C n N' and {x x EN and g(x) ~ g(xo)}. Then C' has nonempty interior relative to N', C' contains C n N',
299
34
ERRETT BISHOP AND R R PHELPS
and Xo is in the boundary of C' By the support theorem, then, support point of C' in N' and therefore it is a support point of C N'. By the induction hypothesis, Xo must be a support point of C.
Xo
is a
n N' in
THEOREM 4. Suppose that N is a closed subspace of finite deficiency in the Banach space E, that C is a closed convex set in E, that € > 0, and that z in N is in the boundary of C. Then there exists a support point Xo of C such that Xo E N and II z - XO II < E. PROOF. There are several cases to consider. First, if C is contained in a proper closed subspace of E, then any functional which vanishes on this sub· space supports C at each of its points, so z itself is a support point of C. Assuming that C is not contained in a proper subspace, we consider whether z is in the boundary (relative to N) of C n N. If it is, then by Theorem 1 there is a point Xu in N which is a support point of C n N such that II z - XO II < € By Lemma 4, Xo is a support point of C in E. Finally, suppose z is not in the boundary of C n N, then there exists a neighborhood (relative to N) of z which is contained in C n N We will show that z itself is a support point of C in E. There exists a point y in E such that the segment [y, z[ == {x: x = AY + (1 - A)Z, 0 < A ~ I} is contained in E ~ C, this follows from the fact that N has finite deficiency and that z is in the boundary of C. Let N' be the linear subspace spanned by Nand y, and note that N is a hyperplane in N'. We will show that N supports C n N' at z, and hence (by Lemma 4) z is a support point of C in E. It suffices to show that the open half·space {x + ry- xE Nand r > O} in N' is disjoint from C. Suppose that C contained a point x + ry of this half-space Since z is in the interior (relative to N) of C n N, there would exist a point lV of C and A in ]0, 1[ such that z = Ax + (1 - A)lV. Hence the triangle with vertices z, lV, and x + ry would be in C and (as can be easily shown) would also contain a point of [y, z[, a contradiction. The apparent duality between this theorem and Corollary 6 leads one to conjecture that the theorem might still be true if it is merely assumed that E/N is reflexive (rather than finite dimensional). The theorem fails, however, under this weaker hypothesis; there is a well known example (see, e g., [2, p. 160]) of a compact convex set C in a Hilbert space H and a line N in H such that C n N = {(2)}, but (2) is not a support point of C (even though H/N is reflexive). All four of our lemmas have valid analogues in more general topological vector spaces, although we do not know whether this is true of the theorems themselves. (These questions will be the subject matter of another paper) For instance, it is unknown whether a closed convex subset of a complete locally convex space must have any support points. (For those special classes of closed convex sets C which are known to have support points, it is known that the support points are dense in the boundary of C, e.g., if C has nonempty interior, or if C is locally weakly compact [5].)
300
THE SUPPORT FUNCTIONALS U1' A
l..UNV~.I\. =>~l
''''
BIBLIOGRAPHY
1. E. Bishop and R R Phelps, A proof that every Banach space is subreflexive, Bull Amer. Math Soc. 67 (1961), 97-98. 2. N. Bourbaki, Espaces vectoriels topologiq1J.es, Chapter V, Hermann, Paris, 1955 3. N. Dunford and J Schwartz, Linear operators, Part I, Interscience, New York, 1958. 4. R C. James, Reflexivity and the supremum of linear functionals, Ann of Math (2) 66 (1957), 159-169 5 V L Klee, Convex sets in linear spaces, Duke Math 1. 18 (1951), 443-466. 6. , Extremal structure of convex sets. II, Math Z. 69 (1958), 90-104. 7. R. R. Phelps, A representation theorem for bounded convex sets, Proc Amer Math. Soc. 11 (1960), 976-983. UNIVERSITY OF CALIFORNIA, BERKELEY
301
HOLOMORPHIC COMPLETIONS, ANALYTIC CONTINUATION, AND THE INTERPOLATION OF SEMI-NORMS By ERRETT BISHOP
(Received September 19, 1962)
1. Introduction
We are interested in an algebra of analytic functions mon a complex analytic manifold M. Associated to each compact set K eM, there is an algebra semi-norm II 11K on 21:, defined by
Ilf 11K
=
max {If(p)
I : p E K} .
The totality of such semi-norms makes ~'( into a locally convex topological algebra. We are interested in the spectrum S of this algebra. Does it possess a holomorphic structure? Here S is defined as the set of all continuous non-trivial homomorphisms of ~l into the complex numbers C. This is an abstract approach to the problem of finding what might be called an envelope of holomorphy or holomorphic completion for M relative to the algebra 21:. This problem has been completely solved in case ~ consists of all holomorphic functions on M, and M admits a locally bijective holomorphic map into some complex affine space Cn. The theory in this case is due to Cartan, Thullen, and Oka, and an elegant presentation can be found in Rossi lID]. The problem has also been completely solved when M is one-dimensional. The theory in this case is essentially due to Wermer, and an inelegant presentation can be found in [4]. One of the purposes of this paper is to derive general results about S, which are sufficiently powerful to give the known special cases just mentioned. There are other important semi-norms on m. These arise from interpolation of the semi-norms II 11K with certain other semi-norms on the algebra 21:. The interpolation of semi-norms has attracted much attention in recent years. The first part of this paper is devoted to an elementary study of this process. We then undertake the study of the algebra ~l, topologized however by a certain family of interpolated semi-norms. Such interpolated semi-norms are important in studying questions of analytic continuation. Our first general result, Theorem 1 below, imprecisely speaking, states that the spectrum S' of ~ with this new topology can be regarded as nearly a finite covering of Cn. Our next general result, Theorem 2, implies the theorem of CartanThullen-Oka mentioned above. It also implies (Corollary 2) a necessary 468
302
and sufficient condition that two analytic functions be analytic continuations of each other. Due to the novelty and possible interest of such a condition, we have taken care to make the statement readable without any knowledge of the rest of the paper. The last section of the paper is devoted to applying our techniques to the study of S in case M has dimension 1, thus recovering in a much more natural fashion some of the results of [41 or lI3]. 2. Locally convex topological algebras
We begin with some standard definitions and facts. DEFINITION 1. Let m: be an algebra over the complex field. A function II II: x --> II x II from ~ to the non-negative real numbers is called an algebra semi-norm if (i) Ilx + yll ~ Ilxll + Ilyll (ii) Ilxyll ~ Ilxllllyll (iii) Ilexll = lelllxli for all x and y in m: and all e in C. An algebra ~l which is endowed with such a semi-norm is a normed algebra. As is well known, an algebra semi-norm on ~ is completely determined by the set U= {XE~: Ilxll < I}. DEFINITION 2. Let ~l be an algebra over the complex field which has the structure of a topological space. Then m: is called an I.c. (locally convex) topological algebra if ( i) subtraction and multiplication induce continuous maps of m: x m: into ~l, (ii) the map (c, x) --> ex of C x ~l into ~{ is continuous, (iii) there exists a basis of the neighborhoods of 0 consisting of convex open sets U which are closed under multiplication. Now if /I II is an algebra semi-norm on an algebra ~1, then the sets U" = {x: II x II < lin} are a basis for the neighborhoods of 0 of some I.c. topology for ~L More generally, any family of semi-norms on m determines an I.c. topology for 21, and every I.c. topology for 21 arises from some family of semi-norms, which may be taken to be the family of all semi-norms on mcontinuous with respect to the given topology. DEFINITION 3. The spectrum S of an I.c. algebra mconsists of allnontrivial (i.e., not identically zero) continuous homomorphisms of ~l into the complex numbers. To each x in 2{ there exists a mapping x of S into C defined by x(rp) = rp(x). The mapping x --> x of m: into a family of complex-valued functions
303
470
ERRETT BISHOP
on 8 is called the Gelfand representation. ~ is called semi-simple if x = 0 only when x = o. In this case the map of ~ onto the family of functions x is bijective, and we sometimes use the same symbol x for both x and x. The weak topology of 8 is the weakest topology in which each of the functions x is continuous. If ~ is a normed algebra with unit, then 8 is compact in the weak topology. In this case either" = 0 for all x in ~ or, by passing to an equivalent norm, we may assume that the norm of the unit is 1. If ~ is an I.c. algebra with unit 1, we therefore admit only those continuous semi-norms II II for which 11111 = 1. lf 8 is the spectrum of the I.c. algebra ~ and II II is a continuous semi-norm on ~, then 8(11 II) will denote the subset of 8 consisting of all cp having the property that I cp(x) I ;;;; II x II for all x in~. Since each cp in 8 determines a semi-norm defined by II x II = I cp(x) I, we see that 8 = U 8(11 II), where the union is taken over all continuous semi-norms on ~. DEFINITION 3. Let II 111 and II 112 be semi-norms on the complex vectorspace ~, and let 0 be a constant, 0 ;;;; 0 ;;;; 1. Then II IIf IIi-a is defined to be that function on ~ whose value at x is
x"
*
II x IIr
* II
x 1I~-8 = inf {E~~I
II Xi IIf II Xi iii-a: x
= E~~I x.} .
'*'
LEMMA 1. Under the hypotheses of Definition 3, II IIf II II~-B is a semi-norm, and if II lis is any semi-norm on ~ with the property that II x lis ~ II x IIr II x II~-B for all x in~, then II lis ~ II IIf II In-B. PROOF. For each g > 0 we can find Xl' ••• , x" and Yl> ••• , Ym in ~ such that EXi = x, EYi = Y,
'*'
* *
E IIx,llflixiln-B;;;; g + IIxlif IIxll~-B, E II y, IIf II y, m- B ;;;; g + II y IIf II Y m- B • Thus x + Y = E Xi + E y" so II x + Y IIf II x + Y m- B ~ E II Xi IIf II Xi m- 8 + E II Yi IIf II Yi m- B ~ E 2g + II x !If II X II~-B + II y IIf II Y m-8 •
*
*
*
*
m-
8 is a semi-norm. With is arbitrary this shows that II IIr II II lis as stated, we see that II x lis;;;; E II Xi 113;;;; E II Xi IIf II Xi 8 whenever x = E Xi' so clearly II lis ~ II IIf II lIi-8 • lf II Ill> II 112> and II 113 are three semi-norms on ~, we may define for instance
Since
g
m-
*
*
*
II x II~ II x IIr II x II! = inf {E;=l II Xi II~ II Xi IIr II Xi III: x
304
=
r;;=1 Xi} ,
HOLOMORPHIC COMPLETIONS
':I:I.L
whenever a + (3 + 'Y = I a I + 1(31 + I'Y 1= 1, and prove the analogue of Lemma 1. We see no reason to hope that log (I (c,!(y) I) for all y in r, we have hE Uu so that ~hdf1. ~ O. Thus
o ~ ~log I a-If I d,u = ~log If I d,u -
log a
so that ~ log If I d,u ~ log a = log If(x) I for all fin ~, as was to be proved. We shall have need of the following somewhat stronger result. LEMMA 31 , With the hypotheses of Lemma 3, let {X;}~~l be a finite family of compact subsets of r, and au ... , a" be non-negative constants with a 1 + ... + a" = 1. For 1 ~ i ~ n, let II Iii be the semi-norm defined by
Ilf Iii
= max {If(y) I: y E Xi}
.
Let x be a point in X with the property If(x) I ~ Ilf Ilr1 • Ilf II~J·
.... Ilf II:n
jar all j in m:. Then the Jensen measure ,u of Lemma 3 can be taken to have the property ,u(XJ ;:;; ai' 1 ~ i ~ n. PROOF. Define U1 as in the proof of Lemma 3. Let Uz consist of all h in Co(X) with a(h)
== a 1 max {h(y) : y C Xl}
+ ... + an max {h(y) : y EX,,} < 0 .
Assume there exists h in Ul n U2 • Then there exists f in ~ with If(x) I~ 1 and r > 0 such that rh(y) ;:;; log If(y) I for all yin r. It follows that
o ~ log If(x) I ~ allog IIfl11 + ... + a" log Ilfll" ~ ra(h)
•
Thus a(h) ~ 0, contradicting the fact hE Uz, so that Ul n U2 is void. As above, there exists on r a Jensen measure ,u for x, having the additional property ~hd,u < 0 whenever a(h) < O. Assume ,u(Xi ) < a i for some i. Let OJ be the function on r which is a i -1 on Xi and a i on r - Xi' Then if h is any function in Co(X) with h(y) < w(y) for all y in r, it follows
307
474
ERRETT BISHOP
~hd,u < O. But is arbitrarily close to ~Wd,u. Thus
that a(h)
< o.
o ~ ~Wd,u =
Therefore
(a, - l),u(XJ
h may be chosen so that
+ a.(l- ,u(X;)) > (a; -
l)a,
+ a,(l- a;)
~ hd,u
= 0 .
This contradiction shows that ,u(X,) ~ a, for all i, as was to be proved. Our interest in Jensen measures arose from the desire to prove the following lemma. LEMMA 4. Let ~ be a (not necessarily closed) subalgebra of C(X) with unit, and let II II be the uniform norm on ~. Let {X'}~=l be a finite covering of X by compact sets. For each i, let II II; be the semi-norm on ~ defined by
Ilfll, = sup {If(x) I: x EX;} . Then if cp is a homomorphism of ~ into C which belongs to the homomorphism space S.(II II) of ~ relative to the norm II II, then cp is in the homomorphism space S.(II II.) of ~ relative to the norm II II;, for some i. More precisely, if II II' is a norm on ~ with II II ~ II II' such that IcpU) I ~ Ilf W(Ilf 11'),-9 for allf in~, then for some i, we have IcpU) I ~ Ilf II~/" (Ilf 11')1-9/,., all f in ~. PROOF. By the hypotheses, there exists an extension II 110 of the norm II II on ~ such that IcpU) I ~ IlfilD for all f in~. Now II 110 = II W (II 11,),-9 for some semi-norm II II' on ~ and some () in (0, 1]. Replacing II II' by max {II II, II II'} if necessary, we may assume that II II ~ II II'. Let S be the spectrum of ~ under the norm II II'. Since II II ~ II II', weseethatforeachxinXthehomomorphismCPz:f-f(x) of ~ into C belongs to S. Thus K, = {cpz: x E Xi} is a compact subset of S for 1 ~ i ~ n. Clearly the norm II II, on ~ is the uniform norm on K;, when ~ is considered as an algebra of continuous functions on S, and the norm II II on ~ is the uniform norm on the subset K = {cpz: x E X} of S. Since
*
for all f in ~ and all positive integers k, by taking k'b roots and letting k - =, we obtain I cpU) I ~ II f
W(II f
11")1-9
where II II" is the uniform norm on S. By Lemma 3, there exists a Jensen measure ,u for cP on S such that ,u(K) ~ O. Thus ,u(Ki ) ~ ()/n for some i. Thus for every f in ~ we have
nULUMu.n.r.n.l\.J \.JVlur .LI.I:l.t.&. ... v ..... lo.J
log I qJU) I ;;;; ~IOg If I d,u ;;;; fJ/n
max {log If(Y) I: y
E
Xi}
+
(1 - ~) log Ilfll" .
Thus I qJU) I ;;;; Ilf IW" -
Ilf 113
with Ilfll. and IIfl13 as above, for every holomorphic function f on K3 which vanishes to order \, at each of the points P" ... , Pk. PROOF. Let Ui C K" 1 ~ i ~ k, be a neighborhood of Pi which admits a biholomorphic map cp, = (cp,,, ... , CPi") onto {(Zl' ... z,,) : I Zi I < 1, 1 ~ i ~ n}, with CPi(P.) = O. Let V. = {p E Ui
:
I rp.j(p) I
Q< (II
11')'-0
for some compact L c M. By our assumption,
II IlL ~ II II~ >Q< (II 11")'-0> for some semi-norm II II" on 21 and some a E (0,11. Thus c II 110 ~ (II II~ >Q< (II 11")'-,")9 >Q< (II 11')' -0 ~ II II~e >Q< (II 11"),'-0> 0 >Q< (II 11')'-0 ~ II II~ >Q< (II where Ilfll'" = max {llfll', Ilfll"}. This implies that II 110 is continuous on the saturation of the elementary analytic algebra ~1. Thus under our assumption, the two saturations are the same, and therefore the two homomorphism spaces are equal. Thus, to finish the proof, consider a compact set L eM. Let J be any compact subset of 1VI whose interior contains L. Since M is connected and K has interior points, we see from the three regions lemma that there exists (J in (0, 11 such that
II IlL
~
II
II~ >Q
... , f,n, this proves the lemma. COROLLARY. If II 110 is an extension of the nonn i I I[ on an elementary analytic algebra ~T, then Lemma 6 remains valid with II II replaced by [[ [[0. PROOF. We have II 110 = [I W II Iii -e for some II IlIon mand some () in (0, 1]. Choosing CI so that I[fi [II ~ C" 1 ~ i ~ m, we have, with F as above
'*'
\
II F(j" ... , fm) [[0 ~ II F(j" ... .f",) W[I F(j" ... .fm) 11:- 11\ ~ (crIll ~dm)er"e(d, ... dm)c~' Idm),-e ~
(ecc,)dl+ +dm(rey-,
thus proving the corollary.
313
480
ERRETT BISHOP
As a consequence of this corollary and of Lemma 5, it follows that, if is an algebra of analytic functions of dimension n on a complex analytic manifold M, and if KcM is compact, then for f1' ... ,j", in ~ there exist r in (0, 1) and c > 0 such that for all families d lO • • • , d ... of positive integers, there exists a unit polynomial F of degree (d lO • • • , d",) with II F(f1' .•• , f",) 11K ~ Cd1 + ~dmr\ with A. as above. This can be expressed by saying that flO • • • , fm are approximately algebraically related on K. (These relations are meaningful only when m ~ n + 1.) ~
LEMMA 7. Let k be a positive integer, a a positive number, and f a unit polynomial in one variable of degree k. Write
S = {z : I z I ~ 1, If(z) I ~ a k} ,
and let So be the projection of S onto the real axis. Then the Lebesgue measure of So is at most 24a. PROOF. Write f in the form fez) = C(z - t1) ... (z - tt)(l - (tt f-1)-l Z)
•••
(1 - (tk)-lZ) ,
where I ti I ~ 2 for 1 ~ i ~ t and I ti I > 2 for t < i ~ k. Thus for z in S we have 11 - (ri)-l z I ~ 1/2 for t < i ~ k, because I z I ~ 1. Therefore
I C II z - r1 I ... I z -
tt I ~ a k 11 - (tt+1)-lZ 1-1 ... 11 - (tk)-IZ 1-1 ~ (2a)k .
Also the sum of the absolute values of the coefficients of f is at most
I C 1(1 + I r1 J) ••• (1 +
Itt 1)(1
+ I tt~l 1-1) ... (1 + Itk 1-1)
Since f is a unit polynomial it follows that I C 13 k Combined with the above this gives
I z - tl I ... Iz for all z in S. Now let a lO Then
Ix
•••
~
I C 13 k • 1, or I C I ~ 3~
k•
rt I ~ (6a)k
,at be respectively the real parts of t,,· .. ,tt.
- a 1I ... I x - at I ~ (6ay
for all x in So. Let f.1- be Lebesgue measure on the real axis. We may aSl"ume that f.1-(So) > 0. We map the real axis into the closed interval [0, 2] by defining
ip(x) = 2[f.1-(So)]-1f.1-({Y: Y E So, y ~ x}) for all real x. Clearly ip is a continuous monotonely non-decreasing map of the real axis onto [0, 2]. It is also clear that I ip(x1) -
ip(x 2 ) I ~ 2[f.1-(So)]-1 I Xl
-
x2 1 •
If we let (31" •• ,(3t be respectively the numbers ip(a1 ),
314
••• ,
ip(a t ), it follows.
nULU.LVlu.nrnll..J \..JV.LU,C
J...,t.£.l..l..lV.L'~
that
I cp(X)
- fi,
I ••. I cp(x)
-
fit I ~ 2t[,u(8 )]-t I x - a , I ... I x - at I ~ (12a)k[,u(8 )]-k , 0
0
for all x in 8 0 , Because cp is constant on each of the open intervals complementary to 8 we actually have cp(8 = [0,2]. Therefore 0 ,
0)
for all y in [0, 2]. By the Tchebyscheff approximation theorem [8] it follows that (12a)k[,u(8 o)]-k ~ 2-t~' ~ 2- k-' ~ 2- k • Thus ,u(80)k ~ (24a)k, as was to be proved. We now establish a numerical measure of the smallness of certain sets which will be useful later. DEFINITION 10. A subset 8 of the plane C' is negligible of order e ~ if the projection of 8 onto every straight line in C' has Lebesgue outer measure at most e. Negligible will mean O-negligible. DEFINITION 11. (By induction). A subset 8 of C"', for n > 1, is said to be negligible of order e ~ if the subset T of consisting of all Zn in C' such that the set
°
°
8(z",)
=
{(z" .. " z,,_,)
C"
E
C"-' : (z" .. " z,,)
E
8}
is not an e-negligible subset of C"-" is e-negligible. The proof of the following lemma, which we do not give, is by simple induction on n. LEMMA 8. Let 8, C C'" for 1 ~ i < and let 8 i be ei-negligible. Then U 8, is f.-negligible, where f. = E e,. DEFINITION 12. Let nand d be non-negative integers. A set 8 C Cn has property per, d) if < r < 1 and if there exists a unit polynomial F in n variables, of degree d, such that I F(z" .. " z,.) I ~ rd for all (Z" ... , z"') in 8. 0),
°
LEMMA 9. If 8 c E" for n ~ 2 has property Per, d), and if a > r, the set T, consisting of all z" in E such that the subset 8(z,,) = {(z" ... , Z"-l) E c"-' : (Z" .. " z,,) E 8} of E'1I.-' does not have property p{ra-" d), is negligible of order 24a, where E = {z : I z I ~ I}. I PROOF. Since 8 has property per, d) there exists F as in Definition 12. Thus there exists a monomial z7" ... z:,:,-' whose coefficient in F is a unit polynomial f in the variable z" of degree d. Thus by Lemma 7 we
315
482
ERRETT
BISHOP
see that the set U
= {z,,:
Z"E
E, If(z,,) I ~ ad}
is negligible of order 24a. Now if z" E E- U, the polynomial Fo of degree d in n - 1 variables defined by Fo(z" ... , Z.,,-l) = F(Z" ... ,z,,) has one coefficient whose absolute value is at least aa. Thus by normalizing Fo to become a unit polynomial, we obtain a unit polynomial F, of degree d in n - 1 variables such that I F,(zj, ... , Z.,,-l) I :::::; a-lIra
for all (ZlI ... Zn-l) in S(z.,,). It follows that S(z.,,) has property P(ra " d) for all z." in E - U. Therefore T c U. This proves the lemma. LEMMA 10. Let F be a unit polynomial of degree d in n variables, and r > O. Then the set
S
= {z E En : I F(z) I ~ r(!}
is negligible of order 24rlln. PROOF. The proof will be an induction on n. For n=l the proposition reduces to Lemma 7. Assume therefore that n> 1. Taking a = r'ln. and applying Lemma 9, we see that the set T, consisting of all Zn. in E such that S(z.,.) does not have property P(r II"!, d), is negligible of order 24 r l/n. Consider z." in E - T, so that S(z,.) is a subset of C,,-l with property P(r'-('/"'l, d). By the case n - 1 of the proposition we are proving, it follows that S(zn.) is negligible of order 24(rl-(l/"'J)'/(n-lJ = 24r'I". By Definition 11 it follows that S is negligible of order 24r'ln , as was to be proved. 1-
LEMMA 11. If the set SeC'" is c-negligible for all c > 0, then S is O-negligible. PROOF. By Definition 10 the statement is clear for n = 1. Assume therefore that n > 1 and that the statement is true for all smaller n. Let T be the set of all z." in C such that the set S (zn) of Definition 7 is not O-negligible, and let TE be the set of all z." such that S(z,,) is not cnegligible. By the induction hypothesis T = Uoo T E • Thus T = Ui-k T2 J, for all positive integers k. By Lemma 8 it follows that T is k 2- J = 2-k+ l negligible. Since k is arbitrary, we see that Tis O-negligible, as was to be proved.
L:7-
THEOREM 1. Let f" ... ,f.,,+l belong to an n-dimensional elementary analytic algebra~. Let II 110 be an extension of the norm II II on ~. Let S be the spectrum of~, under the norm II 110, and let So be the set ((1fJ(f,) , .. " 1fJ(f,,+1») : 1fJ E S}
316
C
Cn+! ,
HOLOMORPHIC COMPLETIONS
the joint spectrum oj j" ... , j"." For each (Z" ... , z,,) in e", let SO(Z" ... , z,,) = {Z"+I : (Z" ... , Z" '1) E So}. Then SO(Z""" z,,) is finite except jor a negligible set oj values oj z" ... , Z", PROOF. If the constant E > 0 is sufficiently small, then the joint spectrum of Ej" ... , Ej",,, which is just ESo, is a subset of En' I. Thus by replacing j" ... , j,,~, by Ej" ... , cj"" if necessary we may assume that So c E" '. By Lemma 6, for each pair (d, e) of positive integers, there exists a unit polynomial Fin n + 1 variables of degree (d, ... , d, e) such that
where
a=
e"n •
Thus for each (Zl> ... , Zn-l-') in So, there exists cp in S such that
I F(z.,· " Z,,-I-,) [ = [F(rp(f,), .. " (P(fn-!-,») [ =
I rpF(f" "', f"t,)
If d and e are sufficiently large, say d it follows that
[~ ~
[I F(f" "', f"t,) do and e
[F(z" .. " z,,_,) [ ~ r~'1
~
[[" ~
C,uIHr,/·1 •
eo, and r 0 is in (r, 1)
•
Now there exists a power of Z"tl whose coefficient in F is a unit polynomial Gin z" ... , Zn. By Lemma 10, [ G(Zl> ... , z,,) I ~ r~'/2 d8 except for (Z,' ... , z,,) belonging to a negligible subset T = T(d, e) of E" of ordera = 24r~/'''. Thus for each (z" .. " z,,) in E" - T we have a polynomial fn in Z"-I-,, of degree e, obtained by substituting the given values of Z,' .. " z" into F, such that
I fo(z,,+,) I ~
r~s
for all Z,,~, in So(z" ... ,z,.) and such that at least one coefficient of fa has absolute value at least r~'/2JdS. Thus by normalizing fo we obtain a unit polynomial f of degree e such that I/(z,,+,) [
~ r~'/'JdS
for all Z,,~, in So(z" ... , z,,), for (z" ... , z,,) in E" - T(d, e). For a fixed value of e, consider the set He= n:~do u:~'" (E" - T(d, e)). If (z" ... , z,,) E He ,then for an infinite number of values of d there exists a unit polynomial 1 of degree e in one variable such that [/(z,,+J [ ~ r~'/2JdS for all Zh' in So(z" ... , z,,). By taking a limit of such polyn'Qmials as d ------+ 00, we obtain a unit polynomial F of degree e in one variJble such that F(z,..,.,) = 0 for all z,,+, in So(z" ... , z,,). Thus SO(Z" ... , Zft) has at most e elements for all (z" ... , z,,) in He On the other hand, En - He = U:~do mT(d, e). Now n;~ T(d, e) is negligible of order a~ since
n:o
..
317
484
ERRETT BISHOP
each T(d, e) has this property. Thus En - He is an increasing union of sets which are negligible of order a and is itself therefore negligible of order U e From all this it follows that the set of (:1: " ••• , :1:,,) in E" for which 8.(:1: 11 • • • , z .. ) is infinite is negligible of order a for all e ~ e.. By Lemma 11, this set is negligible, as was to be proved. COROLLARY 1. Under the hypotheses of Theorem 1, 8,(Z" ... , :1:,,) is finite except for (ZII ... , z,,) belonging to a subset T of C" of Lebesgue measure O. PROOF. It is easy to see that the set T, where 8 0(zll ... , z,,) is infinite, is an Fus and is therefore Lebesgue measurable. By Theorem 1, T is negligible. By Fubini's theorem and induction, it is clear that a measurable negligible set has measure o. Thus T has measure 0, as was to be proved. COROLLARY 2. Let ~ be an algebra of analytic functions of dimension n on a complex manifold M, normed by the norm II 11K corresponding to some compact K eM. Let II II. be an extension of II 11K, and 8 the spectrum of ~ relative to this norm. Then any n + 1 elements f" ... J"+l in ~ have the property that, for almost all (z" ... , z,,) in C n, the set {rp(f.. +l) : rp E 8, rp(f;} = z,' 1 ~ i ~ n} is finite. PROOF. Since K is compact, K is a finite union K = U K i , where K belongs to some component of M. By Lemma 4, it follows that 8 c U 8" where 8, is the spectrum of ~ relative to some extension of the norm I! IIKI" By Lemma 5, 8i c T i , where T, is the spectrum of ~ relative to some extension of some norm which makes ~ into an elementary analytic algebra of dimension at most n. By the above corollary, it follows that our proposition is true if 8 is replaced by T i • Since 8 c UTi, the proposition follows.
4. Manifolds with global local coordinates An algebra m of analytic functions on a complex analytic manifold M will be said to have global local coordinates if there exist all ... , a .. in ~ which are a local coordinate system at each point of M. To each analytic functionf on Mthe derivatives (af)/(aa,), 1 ~ i ~ n, are well defined and are analytic functions on M. If these derivatives are in ~ whenever f is in ~, we say that ~ is closed with respect to derivations by the given coordina tes. DEFINITION 13. Let ~ be an algebra of analytic functions on a complex analytic manifold M which is closed with respect to derivations by given global local coordinates all ... , a... For each f in ~ and each n-tuple
318
tlULUMUKt'tlllJ l.,UlVlr LJ!o llUl'll.:l
Ik, k" be the element of ~ obtained by differentiating I ki-times with respect to ai' 1 ~ i ~ n. Then .1(, the restricted class of semi-norms on ~, will consist of all algebra semi-norms II II on ~ such that there exists c > 0 and an algebra seminorm II II, on ~ such that
(k" ... , k,,) of non-negative integers, let
Illk, kn II
~
1llll,ck,+ +k"k,!··· k,,!
for all n-tuples (k" ... , k,,) of non-negative integers and all I Yl be the class of all semi-norms on ~ of the form
II II
=
II
II~ >Qc
in~.
Let
II 11;-9,
where () E (0, 1], K is a compact subset of M, and II 110 E.1(. Then the I.c. topological algebra determined by the norms Jl on the algebra ~ is called the lJartial saturation of %{, and its spectrum is called the restricted homomorphism space of %{. THEOREM 2. Under the hypotheses of Definition 13, the restricted homomorphism space H 01 %{ can be given the structure of a complex analytic manifold in such a way that (a) the functions in mare analytic on H, (b) for each II II in Jl, S(II II) is a compact subset of H, (c) the natural mapping p - > CPP of Minto H defined by CPp(f) = f(p) for all f in ~'(, is complex analytic and takes M onto an open subset of H which intersects each component of H, (d) the functions ai, ... ,a~ on H defined by a7(cp) = cp(a;), 1 ~ i ~ n, are global local coordinates on H. PROOF. Let 5 consist of all non-trivial homomorphisms cp of %{ into C such that there exists c > 0 such that there exists an algebra semi-norm II II, on %{ with
I CP(fk, k.) I ~ III II, Ck1L -k"k,!··· k,,! , for all f in %{ and for all k" ... ,k". (Thus cp E S if and only if the seminorm f -> I cp(f) I is in :R..) For each such cp consider the power series ~k
1
kn CP(fk 1 k11 )(k,! ...
k,,!)-'z~'
...
z~"
This series converges and defines an analytic function 1 ~ i ~ n, with
Il(z" ... , z,,) I ~ ~k,
k"
Ilfll,hc)kl+ rk"
~
.
1 for I Zi I < c-',
Ilfll,(1- 'Yct".", \
where'Y = max {I z,l, ... , I z" I}. By Leibnitz's rule for the derivative of a product, for each fg = h in ~ we have
hkl
k71
=
~O:>;ISk, [Ir~, {k;!(ji!(k i
319
-
j'}!)-'}J;, ;"gk,-i,
kn-;J·
486
ERRETT BISHOP
Substituting, we obtain h(zl1 ... , zn) = L:k, kn CP(hk, k)(k,!··· kn!)-'z~' ... z~n --
"L-ii 1
."
I n L-Iml
mn
. CP( fj 1 in)r:p(gm 1
= l(Zl1
(J" 1-, ••• J' n"'m I",
... m n",)-1
mn)Z{1-m1 ••• z~nf mn
... , Zn)g(Zl1 ... , Zn) .
Thus for each Z = (z" .. " zn) with IZi I < c-', 1::::; i::::; n, the map cp,: 1----+ l(z" .. " zn) of ~l into C is a homomorphism, reducing to cp when Z = 0. From the definition of 1, it is clear that
h= where h = whenever I
(BjBz,)k[ ... (BjBz n)knl,
Ik, kr' Thus, since 1 is analytic for I Zi I < c-', 1 ::::; i ::::; n, < c' there exists a constant ~ > such that for all I in ~ 11k, kJZ) I = I (BjBz,)"l ... (BjBZ n )"nl(z) I ::::; IIIIID~kl- t knk,! ... k n! ,
°
whenever I Zi I ::::; I, 1 ;;:; i ::::; n, where D consists of all z with I Zi I ::::; 10 for 1 ::::; i ;;:; n, for some constant 10 with I < II) < c-'. It also follows that iii = Zi + cp(a;) , 1 ::::; i ;;:; n. It follows that each cp, is an element of S. Define a subset U of S to be open if for each cp in U there exists t > such that cpz E U for all Z with IZi 1< t, 1::::; i;;:; n. It is then easy to check that, if cp E Sand t > 0, the set
°
{cp, : I Zi I
< t,
1 ::::; i ::::; n}
is an open subset of S. Each I in ~ gives rise to a function 1* on S defined by I*(cp) = cpU). It follows that S is a complex analytic manifold with global local coordinates at', ... , a:. Now if cp E H, there exists II II in Jl with cp E S(II II). By the definition of 'JZ, there exists () in (0, 1], a compact subset K of M, and II 110 in fR with II II = II li~ II 11:-°. Let L be any compact subset of M whose interior includes K. By standard theory we see that there exists c > 0 such that for each I in ~ we have
*
Il/k,
kn 11K;;:;
III IlL kl!
... kn!c k,"- -kn.
It follows that II 11K E fR. Since it is trivial to check that the interpolated semi-norm of every two semi-norms in fR belongs to fR, we have II II E fR. Hence cp E S. Thus He S Consider now any norm II II in Jl. Then S(II II), as the spectrum of a normed algebra with unit, is compact in the weak topology. Let T,
320
4~S'{
HOLOMORPHIC COMPLETIONS
denote S([[ [I) with the weak topology. There is a second topology which [i) inherits as a subset of the manifold S. Let T2 denote S([[ [i) with this topology, so that T. c S. If a net {CPu} converges to cp in T .. clearly cp.,.(f) converges to cp(f) for all f in 'it, since the functions in ~ give rise to analytic functions on S. Thus {CPu} converges to cp in T ,. Therefore the natural bijection A.: T. -> T, is continuous. To show that A. is actually a homeomorphism, consider a net {CPu} in T, converging to cp in T ,. We must show that {CPo-} -, cp in T 2 • To this end, we must look at [[ [[ more closely. As above, we see that [[ [[ = [[ [[t'*' I[ [[~-B, and that [ifk , k"l[~[[fk1 k"l[~-B ~ [[f[[~[[f[[t-Bk,! ••• k n!c", -k" S([[
L
for an appropriate constant c and an appropriate semi-norm [[ [[, on ~. Because of the definition of CPZl it follows that for all cp in S([ [ [[) and all Z with [Zi [ ~ "( < c- 1 for 1 ~ i ~ n we have
I~
[cp,(f)
for all f
in~.
(1-
c"()-n[[fll~l[f[[t-B
Applying this to fN, and then taking Nth roots, we have
I cpz(f) I ~
(1 - c,,()-nIN Ilf Ilf Ilf I[t- B
which in the limit gives
I cpz(f) [ ~ Ilf II~ Ilf lit-B. '*' II lit-B), for all cp in S([I II) and all Z with IZi 1< c-
1, Thus cp, E S([I II~ 1 ~ i ~ n. Now since {CPu}->Cp in T " we have {cpu(a;)}->cp(a i ), 1 ~ i ~ n. Thus for all large enougha, sayfora >a o , we have Icpcr(aJ-cp(aJI «1/2)c- 1 • Also, since CPu E S(II II) for all a, there is a neighborhood Vcr of CPo- in S which is mapped biholomorphically by a7, ... , a= onto the set
{(Z" .•. , zn) E Cn : I Zi -
CPu (a;) I
= {y E
Vcr: I y(a i )
-
1 ~ i ~ n} ,
Ilf,*, II I[t-B).
and we have just shown that Vu c S([[ Thus for all a > a o the open set Uu
< c-"
cp(a i )
I
aD for all i, and the UUt are mutually disjoint. By the Baire category theorem, there exists f in ~o such that for i *- j the set
'*'
Qij
= {z E Q : (hu,;Cz»)(f) = (hu;(z»)(f)}
is not all of Q. Since Q;j is an analytic subset of Q, it has Lebesgue measure O. Thus U Q;j has Lebesgue measure O. On the other hand, for all z in Q - U Q;j, the set
{v(f) : y
E
S(II Ilf
* II
11:- 0 ), yea;)
= Zi, 1 ~ i ~
n}
is infinite. Since ~'(o is an algebra of analytic functions on interior J, and L is a compact subset of J, this contradicts Corollary 2 to Theorem 1. Therefore only a finite number of the sets Urr for a > aD are distinct. Since each Urr is contained in some compact subset of Vu, each Uu is relatively compact in S. Thus U Uu , which is actually a finite union, is relatively compact in S. Thus the net {9?u} has at least one cluster point y in S. Since the map T 2 -+ TI is continuous, \'(11) = 9? Therefore 9? = y. Thus 9? is the unique cluster point of {9?u} in S, and {!Prr} lies in a relatively compact set. Thus {9?rr} converges to 9? in T 2 • Thus \, is a homeomorphism. Now let So consist of all those components r of S such that there exists pin M with 9?p E r. Assume, as we shall show below, that H = So. Then (a) and (d) are trivial consequences of the same statements for S. Since ai, ... , an are global local coordinates on M, and ai, ... , a: on S, the fact that p-+!pp maps M into an open subset of So which intersects each component of So is trivial. From this (c) follows. Finally, for each II Ii in Jl, S(II II) is compact in the subset-of-S-topology, as we just proved, so that (b) is true. Thus it only remains to prove that H = So. We first show So c H. Let Mo denote the image of M in So under the map p -+ 9?p. If 9? is any point in So, let K be a compact subset of M with non-void interior whose image Ko in Mo contains an interior point of the component of So containing!p. By the three regions lemma, there o exists a compact set LcS o and (j E (0, 1] such that 19?(f) I ~ II! II~ II! for all! in~. As we have seen above, the norm f -+ II! IlL on ~ is in !R.
m-
322
489
HOLOMORPHIC COMPLETIONS
Thus cp E H. Hence So e H. To show that He So, it is sufficient to show that S(II II) e So for all elements II II = II II~ II 11~-9 of Jl. By replacing II 110 by max {II 11K, II liD} if necessary, we may assume II 11K ~ II 110. Write (J = (j/n, and define II II, = II II~ II II~- 0 such that
Ilfi II:i CI < c, for 1 ~ i
~
Ilfi II~!I'-CI)
< c-'
k. Thus, by replacing the functions f"
... ,fk by the functions
ft", ... , f:~, we may assume that (*)
for 1 ~ i ~ k. Now in [3] a fundamental construction is given for approximating an analytic polyhedron such as P in an n-dimensional manifold by an analytic polyhedron defined by exactly n functions. The construction goes as follows. First the functions f" ... , fk are changed slightly to put them in general position (a concept which need not concern us here). Checking this construction, we see that this can be done so that the inequalities (*) remain valid. Then the functions f" ... , fk are replaced by the functions g, = ft - ft', ... , gk-l = fk~' - ft', for some integer \N which may be taken to be arbitrarily large. If N is large enough, then gJ .. ,gk-l also enjoy property (*), with of course a different value of c. Thus step by step, we reduce the number of functions, finally obtaining exactly n of them. Thus we may assume that P is defined by exactly n functions, f" ... , J '" each of which enjoys property (*).
323
490
ERRETT BISHOP
Let Q be the closure of P, and B the subset of the boundary of P on which the fi all have absolute value 1. Let f: P --+ (f,(P), ... ,fn(P» be the natural map of S into en. For each positive integer :\, let Io.Q be the :\'-fold symmetric product of Q, consisting of all unordered :\'-tuples of elements of Q. Now it is easily seen (for instance, as in [3]) that for some A, there exists a continuous map w of the closed unit polycylinder En = {(Z,' ... , Zn) : I Zi
I~
1, 1
~
i
~
n}
into Io.Q, having the following properties. For each Z in En, let w(z) be the unordered :\-tuple {PH' .. ,p,,J. Then f(pj) = z, 1 ~ J' ~ A, and for almost all Z in En the points PH .. " p" are distinct. For each P in Q, let PH ... , p", with P, = p, be the points of w(f(p». By changing f" ... .fn slightly, if necessary, we may assume that Po occurs only once in w(f(po», i.e., that!!! .. ',!n are local coordinates at Po on S. Now by [31, there exists a certain measure f! on B such that for each function h analytic on S we have
~rr: (Ji(~) 1
- !i(P»)
'h(~)df!(~)
+ ...
= h(Pl)
for all P in P. Let g be any element of ~r with g(p) in w(f(po». Then by the above we have
~[rr;=l (Ji(~) = L:~
1
-
fi(p»)-llh(~)[rr~_
2
+ h(Pn)
,
*" g(po) for all p *" Po
g(~j»)Jdf!(~)
(g(p) -
h(p;) rrj-r'i (g(p) - g(pj») .
But this last expression equals h(p)g(p), where g(p) = It~2 (g(p) - g(Pi») .
Define, for all positive integers N, hN to be the function on S given by the integal hN(P)
= ~rr;-l[L:; 0 f(~!~,Jrr~-2(g(P)
-
g(~,,»)h(~)df!(~).
Thus hN(po) --+ h(Po)g(po) as N --+ co. Now hN lies in ~r, it is actually a < S (-I)/(1-cr) , 1= < ~. = < n, we . I 'In f 1 , " ' , f an d g. S'Ince Ilfi II 0 = poIynomIa see that II hN 110 ~ GoS(-nN)/(l cr) for all N and some Go> O. Also, since Ilfi 11K ~ s*, 1 ~ i ~ n, for all p in K we have n1
I hN(P) - h(p)g(p) I
-I- JI[rrn
i-I
",N fi(P)j i....Jj=O fi(~)J+l -
. rr~=2 (g(p) -
rrni=l i....Jj=O "'~ fi(P)i ] fi(~)j-l
g(~k»)h(~)df!(~) I ~
324
G,SN/cr
HOLOMORPHIC COMPLETIONS for all N and some C l > O. We apply these remarks to the function h which is defined to be 0 on So and 1 on S - So. Thus hN has the properties IlhNllo
~ Col;-nN/ll-a-l,
IlhNllK
~ ClCN/a-,
and h Apo)
Since Po
E
~
fj(Pn) =1= 0
as
N~ co.
as
N~ co.
II), we have on the other hand II hN II~ II hN II~-e
S(II
I hN(Po) I ~
This contradiction shows that S(II II) c Su, thereby finishing the proof of Theorem 2. We now derive two corollaries. The first is a known result, whose recent formulation in the language employed here is due to Rossi [10]. COROLLARY 1. (Cartan-Thullen-Oka). Let M be a complex analytic manifold with global local coordinates a" ... ,an> and let ~r be the algebra ()f all analytic functions on M. Then the spectrum of ~ is a Stein manifold. PROOF. If we define H as in Theorem 2, clearly the spectrum S of ~ is a subset of H. If rp E S, then rp E S(II 11K) for some compact K eM. As in the proof of Theorem 2, it follows that if L is a compact subset of M containing K in its interior then S(II 11K) is contained in the interior of S(II IlL). Thus we see that S is an open subset of H, and therefore a complex analytic manifold. Let J be any compact subset of S, and define
J =
{rp
E
H: I rp(f) I ~ max
{I y(f) I : of E J}, for all f in
~.} .
Now to each point rp in J there exists, as we have seen, a compact set L c M with rp in the interior of S(II IlL). Since J is compact, it can be covered by finitely many sets S(II IlL). Thus there exists L with J c S(II IlL). It follows that J c S(II IlL). Thus J is a compact set. Since by definition of S the functions in ~l separate points of S, since au ... ,an give rise to global local coordinates on S, and since J is compact for each compact J c S, S is a Stein manifold, as was to be proved. COROLLARY 2. Let Kl and K. be compact sets in C", each component of each having non-empty interior, and Ul and U. be open se1s with Kl c Ul , K. cU.. Let f and g be analytic functions on Ul and U. respectively. For every polynomial F, let I F I denote the polynomial obtained by replacing each coefficient of F by its absolute value. Then g
325
492
ERRETT BISHOP
is an analytic continuation of f if and only if there exist constants () and c, < () ~ 1, c > 0, with the following property. For every polynomial F(zl' ... , z,,' f, ... , fkl k",···) in Z" ••• , Z,,' f, and the various partial derivatives fkl k" of f we have
°
II F(zlI ... , z .. , g, ... , gk, k.. ,···) 11K. :s:; II F(z 11 ... ' nZ J f ... 'fk k71.' ... ) liBXl 1 k1 x [I F I (c, ... , c, c, ... , C + H"+'kl !··· k .. !, ... )]'-B . J
PROOF. Assume that g is an analytic continuation of f. Then U, and U. can be realized as open subsets of a connected complex analytic manifold M such that Z,' ... , z" extend to global local coordinates on M, and such that there is an analytic function h on M which equals f on U, and g on U.. By the three regions theorem, there exists a compact subset L of M and () in (0, 1] with
II 11K. ~ II II~, II Ill-B. Let J be a compact subset of M with L c int J. Thus there exists a constant c > 0 such that Ilhk, k"IIL ~ c k,+ -tk .. k,! ... k .. ! IlhlI J . (*)
for all k" ... , k... In addition we may choose c so that II h IIJ ~ c and II z, IlL ~ C, 1 ~ i ~ n. It follows that for any polynomial Fin Z" ••• , z.. , h, and the hk, k.. we have II F(z" ... , z .. , h, ... , hk, k.. ,···) IlL k1 < ... k' ... ) === IFI(c " ... c " c··· C + +k .. -llk' l' n"' J
•
Substituting the function F(z" ... , z .. , h, ... , hk, k..'···) into (*) and using the estimate just obtained gives the desired result. Now assume that there exist c and () with the given property. Let m be the algebra of all analytic functions on U, of the form F(z" ... , z,,' f, ... , fk , ~, ... ), with F as above. Clearly mis closed with respect to partial derivations by the global local coordinates Z" ••• , z". Thus, by Theorem 2, H is a complex analytic manifold. Clearly U, is naturally imbedded in H as an open set. By Theorem 2, each component of H contains points in Ul • Thus to show that g is an analytic continuation of f, it is enough to show that K. is naturally imbedded in H in such a way that the function on H corresponding to f equals g on K,. By the definition of H, this amounts to saying that the function II lion '2l defined by
..
IIF(z" ···,z.. ,f, ···,fk l k.. , ···)11 0== II F(z" ... , z", g, ... , gk l k .. , · · · ) 11K.
326
HOLOMORPHIC COMPLETIONS
is an algebra semi-norm on ~ which is less than some algebra semi-norm of the class 'J7. By the hypothesis of the existence of () and c, this will be true if the function II 110 defined on ~ by II alia = inf {I F I (c, .. " c, c, .. " ck ,+ +kn.L'k,!··· k n!, ... )} , where the inf is taken over all F with F(z" "', zn,f, " ·,fk, kn , · · · )
=
a,
is an algebra semi-norm which belongs to $.. Clearly II lin is an algebra semi-norm. Thus we need only check that II 110 E SR. Let J be a compact neighborhood of the point Zo = (c, "', c) in en. Let \, be the function c n;-, (c' + 1 - cz.)-" which is analytic on J, if J is taken to be sufficiently small. For each a in ~r write Iiall, = inf{IIIFI(lz,l, "', Iz"l, 1\'1, "', I\'k, kJ, ••• )I\.1}' where the inf is taken over all F as above. Thus II norm on ~l. We see also that
II a 110
II, is an algebra semi-
= inf {I F I (z~, ... z~, \'(ZO), •. " \'1., k,,(ZO) , .•. )} ,
taken over F as above. But since ZO belongs to the interior of J, there exists Co > 0 such that, for all functions (3 analytic in a neighborhood of J, we have
I(3k,
kn(ZO)
I ;:;;
c~,-
+knk,!··· k,,! II (311.1 .
Consider now fixed non-negative integers j" .. " jn. For each polynomial F(z" .. " z., f, •• " fk, kn , " ' ) , let HF be the polynomial such that we have formally
( ~)h a Z,
... (~)jnF(Z ".•. , zn" f a Z"
... " fk n' k
••• )
= HF(z" "', z,,' f, .•• ,fk, kn " " ) '
It is clear that I HF I ;:;; H IFI , in the sense that each coefficient of I HF I is not larger than the corresponding coefficient of H IFI • Thus for all a in ~ we have
II a;, ;" 110 ;:;; inf {I HF I (z~,
"', z~, \'(ZO), •• " \'k, k,,(ZO) , ••• )} ;:;; inf {HIFI(z~, "', z~, \'(ZO), "', \'k, k,,(ZO) , ... )}
\
;:;; cg'
r
+J"i,!'" in!
\
x inf {III F I (z" "', Zn' \', ''', A,k, k","') \I.1} ;:;; C6'+ +J"i,!'" i,,! II a II, , where the inf is taken as before.
This shows that
327
II 110 E $., and so
494
ERRETT BISHOP
finishes the proof. 5. The
case of a Riemann surface
We begin with an elementary lemma in the style of § 3. LEMMA 12. Let f, and f. belong to the one-dimensional elementary analytic algebra W. Let S be the spectrum of~, and So c C' the J'oint spectrum of f, and f2' For each e: > 0, let V(e:) be the set of all z, such that So(z,) is not negligible of order e. Then V(e:) is finite, and there exists c > not depending on £ such that Vee) has at most - clog c elements. PROOF. As in the proof of Theorem 1, we may assume S" c E2. rly Lemma 6, there exists a constant r, < r < 1, such that for all large enough positive integers d and e there exists a unit polynomial Fin tW() variables with
°
°
! F(z" z,) I ;:; r
de
for all (zu Z2) in So. For each z, in E, let m(z,) be the maximum of the absolute values of the coefficients of Zo in the expansion of F(z" zJ} in powers of Z2' Thus, for each Z" m(z,)-'F(z" .) is a unit polynomial of degree e in one variable, and
I m(z,)-'F(z"
Z2)
I -;:;
m(z,)-'r de
for all Z2 in Sb,). By Lemma 7, So(z,) is negligible of order ::4r a m(z,y-,je, Thus if z, E V(e:) we have
so that
It follows that
i F(z" Z2) for z,
E
Vee) and
I ;:;
(e
+ 1)(24e:- r 1
rl )"
I z21 -;:; 1. Now write F in the form F(z" Z2) = E~~Ji(Z2)Z~ ,
where the fi are of degree e. At least one of the fi is a unit polynomial. For this value of i the set T
=
{Z2 : i fi(Z2)
I ;:; 25- p}
is 24/25 negligible. Thus E - T is non-void. Choose l;; in E - T. Then max {Ifi(t)
I: 0
~
i ~ d}
328
=
A. ~
25- 0
•
HOLOMORPHIC COMPLETIONS
495
= A, -'F(z"
n is a unit polynomial
Hence the polynomial f defined by f(z,) and for all z, in V(c) we have If(Z,)
1= 1F(Z" ~
s)
1A,
-1
~
(24· 25c- r y(e t
d
\.
-'(24c- ' r d Y(e
+ 1)
+ 1)
.
Choose d so large that 24· 25c'r d < 1, i.e., d to be the greatest integer less than - clog c, for some large c > o. Keeping d fixed and letting e--> =, we obtain a unit polynomial of degree d in z,' whose roots contain the elements of V(c). This completes the proof. COROLLARY. Under the hypotheses of Lemma 12, 8 o (z,) is totally disconnected for all but a countable number of values of z,. PROOF. If 8 o(z) is not totally disconnected, then for some c > 0 it is not c-negligible. Thus the corollary follows. It would now be possible to derive essentially all of the resuits of [4], using the techniques developed above. For simplicity we restrict ourselves to the following theorem. THEOREM. Let M be a metrizable Riemann surface and 21: an algebra of holomorph'ic functions on M which is not identically constant on any component of M. Let K be a finite union of disJ"oint analytic closed CUTrves lying on M. Let 8 = 8(11 :IK). Then S - K can be given the structure of a Riemann surface, with certain discrete identifications, so that the functions in ~r are holomorphic on 8 - K. PROOF. Consider any point Po in 8 - K. It will be enough to work locally and show that there is an analytic structure on 8 of the required type in some neighborhood of Po. Now consider two elements f, and f. in 21:, such that the equations fl(P) = f,(po), f.(p) = f,(po) have no solution p in K. Such functions are easily found. Let w be a complex parameter, and write H(w)
= {p E K: f,(P) + wf.(p)
=
f,(po)
+
wfJp,,)} .
Thus if H(w) is not void there eXIsts p in K with (,(p) - f,(po)
=
W(J2(P) - f,(po») .
If this is true for all w then
(J,(p) - f,(p,,»)ff,(p) - f.(Po)]-1
\
assumes all complex values as p varies over K. This is impossIble, because the function in question is analytic, and K is a union of analytic arcs. Choosing therefore an appropriate value of w, we obtain f = f, + wf. in ~ for which f(po) does not belong to f(K).
329
496
ERRETT BISHOP
For each z in C let 8(z) = {p E 8: f(p) = z}, i.e., 8(z) = f-l({Z}), where f -leT) ~ {pE 8:f(p) E T} for any any set Tc C. We need the following lemma. LEMMA 13. Let r consist of all z in C - f(K) such that 8(z) has at most k points. Let Zo E C - f(K) have the property that 8(z,,) has at least k components. Then 8(z) has at least k components for all Z sufficiently near to ZOo If 8(zo) has exactly k components and, if for each neighborhood V of zo, the set r n V has positive Lebesgue measure,then for all z near enough to Zo the set 8(z) contains exactly k point';, and there exists a closed disc D about Zo such that f-l(D) consists of exactly k components J, each of which f maps homeomorphically onto D. If a is the inverse map of D onto J, then goa is analytic interior to D for each g in m. PROOF. Let L l , • • • , Lk be distinct components of 8(zo). We can find a closed disk D c C - f(K) with center Zo such that there exist distinct components J lI • • • , J~ of f -leD) with L; c J" 1 ~ i ~ k. Let '2T(f-l(D» be the set of all uniform limits on f-l(D) of functions in ~L If cp is any continuous non-trivial homomorphism of ~T(f-l(D» into C, then cp induces a continuous non-trivial homomorphism of ~1(K) into C, so that there exists p in 8 with cp(g) = g(p) for all g in 21. Since f(p) = cp(f) E D, it follows that p Ej -leD). Thus f -leD) is the spectrum of ~T(f-l(D». Let J be any component of f-l(D), and m(J) the set of all uniform limits on J of functions in m. If cp is in the spectrum of m(J), by the argument just given there is p in j-I(D) with g(p)=cp(g) for all g in m. If p Ef-I(D)-J, by a theorem of Silov-Arens-Calderon [2] there exists g in S21 which is close to 1 at p and close to 0 on J, so that I cp(g) I > II g II.T' This contradiction shows that p E J. Thus J is the spectrum of ~(J). Let a(f -leD»~ be the Silov boundary of ~T(f-l(D», and let a(J) be defined similarly. Then by the local maximum modulus principle ([9] or [12]), a(f -leD»~ c bdry f -leD) = f -l(B), where B = bdry D. If g is any element of S2T, by the local maximum modulus principle every component of
H = {p Ef-I(D) : I g(p) I ~
II g II.T}
intersectsj-l(B). Thus J n H intersects f-l(E). Therefore a(J)cj-l(B). Assume that there exists Zl in D such that z, r:;f(J). If this is the case, then we may assume that also d, = dist (ZlI B) > d 2 = dist (Zl,j(J») . Now f - Z, is in ~1 and does not vanish on the spectrum J of S2T(J). Thus (f - 2 1)-1 E S2T(J). But I (f - Zl)-l I attains the value d 2' on J but is bounded by d;' on f-I(B). This contradicts the inclusion a(J) c f -l(B),
330
497
HOLOMORPHIC COMPLETIONS
and thereby shows that f maps each component J of f-'(D) onto D. In particular, S(z) has at least k components for all z in D. Assume now that S(zo) has exactly k components and that, for each neighborhood Vof zo, the set l' n Vhas positive measure. Using Fubini's theorem, we may shrink the disk D, necessary, so that 1'0 = l' n B has positive one-dimensional Lebesgue measure. Consider Zl E 1'0' Then there are at least k points in S(ZI), one each in S(ZI) n J i , for 1 ~ i ~ k. By the definition of 1'0 and the fact that each component of f-I(D) is taken by f onto D, it follows that J" .. " J k are the only components of f-I(D). It also follows that I maps 1- 1(1'0) n J bijectively onto 1'0' for each component J= J;, 1 ~ i ~ k, ofl-I(D). We shall show thatfactually maps J bijectively onto D. If not, there exists ZI in D such that S(ZI) n J contains two distinct points, PI and P2' Thus there exist positive Borel measures /11 and /12 on a(J) such that
for all g in Band
~r.
Thus
2),
=
1(/11) and
2)2
= 1(/12) are positive measures On
~G(z)d2); = ~G(J(P»)d/1i(P) = G(J(p.»)
,
i = 1 or 2, for all polynomials G in one variable. Thus 2)1 - 2)2 is a real measure on E which annihilates all polynomials. Therefore JJ1 = 2)2' Since f maps/-I(r o) n J bijectively onto ro it follows that /11 and /12 are equal on subsets of I-I(ro) n J. Let g be any function in ~ with g(Pl) = g(P.). Let A,l and ~ be the measures l(g/11) and l(g/1.) on B. Thus we have
~dt-l = ~d(g/1J = ~gdf.ll
= g(p,) ,
i = 1 or 2. Therefore A,1 *- A,.. On the other hand, since /11 and p. are €qual on subsets of f- 1(r.) n J, the same is true of gf.-Ll and gp,. Therefore A.l and ~ are equal on subsets of roo and A,1 - ~ vanishes on subsets of 1'0' Since A,I - A., annihilates polynomials. by a theorem of F. and M. Riesz [14], it follows that A,1 - A,2 = O. This contradiction shows that 1 maps J bijectively, and therefore homeomorphically, onto D. Let a be the inverse homeomorphism. Let 9J be the algebra of all functi~ns on D Qf the form h 0 a, for h in ~(J). Thus 9J is naturally isomorphic to m(J), and the Silov boundary of 9J is a subset of B. It follows from a result of Wermer [14], or indeed from an earlier result of Rudin [11], that all functions in 9J are analytic interior to D. This completes the proof of
331
498
ERRETT BISHOP
the lemma. To finish off the proof of our theorem we shall need another lemma. which is a slight modification of a well-known result of Rado. Since the proof can be given by trivially modifying the proof of Heinz [5], for instance, we only state the lemma and refer to [51 for the proof. LEMMA. (Rado). Let f be a bounded function in D = {z: Iz 1~ I} which is analytic in the set Q = {z: 1 z I < 1, fez) O}. Let f be continuous on D except for a countable subset E of D - Q. Then f can be modified on E to become everywhere analytic interior to D. We return to the proof of our theorem. Let {gil be a sequence which is dense in 1lX. Let E consist of all z such that 8(z) is not totally disconnected. For each z in E, at least one of the functions gi maps 8(z) into a subset of C which is not totally disconnected. By the corollary of Lemma 12, however, for a fixed g, this can happen for at most a countable family of values of z. Thus E is countable. Assume for the moment that ~ is generated (as an algebra) by finitely many elements. Let F be the component of C - f(K) to which f(po) belongs. For each non-negative integer k let Fk consist of all z in F such that 8(z) has exactly k elements. Consider z in F - U Fl.' Then one of the finitely many generators of ~'( assumes infinitely many values on 8(z). By Corollary 2 to Theorem 1, it follows that F - U Fk has measure O. Let Gk consist of all z in Fk such that every neighborhood of z intersects Fk in a set of positive measure. Then G k and Fk have the same measure, so that F - U Gf.. has measure O. Thus for some value of k, which we fix, G k has positive measure. By Lemma 13, every z in Gk has a neighborhood U such that f-l( U) is a k-sheeted Riemann surface over U. In particular, G k is open. For each z in G'd let PI' "', Pk be the points of 8(z). Consider any g in ~r, and define the function ~ on G f. by
*'
~(z) =
IIt<J
(g(p;) - g(p;»)" •
We may choose g so that ~, which by Lemma 13 is an analytic function on G., does not vanish identically on any component of G k • Consider any point Zo in (bdry Gk - E) n F. If ~(z) does not converge to 0 as Z in Gk converges to Z(j' then there exist at least k points in 8(zo). By Lemma 13, if 8(zo) has more than Ie points then 8(z) has more than Ie points for all z sufficiently near to ZO, contradicting the assumption that Zo E bdryG k • Therefore 8(zo) has exactly k points, which means that Zo E G k • This is again a contradiction, from which it follows that ~(z)----->O as z----->zo, Z E G. Thus if we extend the definition of .6. by putting it equal to 0 on F - Gk
332
HOLOMORPHIC COMPLETIONS
it follows that ~ is defined everywhere on F, is bounded, is analytic where it does not vanish, and is continuous except possibly on E. By the Rado lemma it follows that bdry Gk is a countable set. Thus F - Gk is a closed countable subset of F. Because of this all of the symmetric functions of the quantities g(p,), ••. , g(Pk), which are bounded analytic functions of z in Gk , extend to be analytic functions in F. Thus there exist analytic functions h" ... , hk in F such that g(p)'
-t-
h,(J(P»)g(p)k-'
+ ... + hk(J(P»)
= 0
for all p in j-l(Gk). Now let R be the closure in j-'(F) of j-l(G,.). Thus (*) is valid for all pin R. Thus g assumes at most k distinct values on S(z) n R, for all Z in F. Moreover, since ~ can vanish at most an isolated set of points of F, g takes exactly k distinct values on S(z) n R, except for Z belonging to an isolated subset of F. Now this argument applies to any function g in '2t. Therefore F - G" is an isolated subset of F. We have seen, in Lemma 13, that j-'(G,J has a natural Riemann surface structure. Since F - Gh is an isolated subset of F, R has a natural Riemann surface structure, with possible branch points and identifications at the points of j-l(F - G,J n R. With this structure the functions in ~X are analytic on R. Since Pu Ej-J(F), to complete the proof of the theorem it is enough to prove R = j-'(F). Assume therefore that there exists z, in F with S(z,)ctR. Then there is g in ~l vanishing on S(Zj) n R which has the value 1 at some point of S(z,). There exists a closed disk D about Zl such that I g(p) I < 1 for all pin j-I(D) n R. We may in addition choose D so that bdry D is disjoint from the countable set F-G,. Thus bdry DcG k so that j -l(bdry D)cR, and I g(p) I < 1 for all p in j-l(bdry D). As above let ~(f-'(D» be the closure of ~ on j-'(D). Since the Silov boundary (J of ~(f-I(D» is a subset of j-l(bdry D)cR, we have Ig(p) 1 0 and representing measures JJ.s for x and JJ.u for Y such that CJJ.z ~ JJ.u and CJJ.II~JJ.Z.
\
PROOF. Let Cr(X) be the Banach space of all continuous realvalued functions on X. Let R be the set of all J in Cr(X) such that f+igE~ for some g in C.(X). Let c be a constant, 0 A as n ...... DO If An n X ---> A n X as n ---> co in the Hausdorff metric for compact subsets of X, for every compact subset X of U.
LEMMA 7. Let U be an open set in (In. Let {Il n } be a sequence ofnonnegative Baire measures on U, each having property M(b, c, k}. Let An = support Iln. Assume that Iln ...... Il and An ...... A, as n ...... DO, where Il is a positive Baire measure on U and A is a closed subset of U. Then support Il = A.
Proof. If x fore Iln(V}
£:
*
support Il and V is any neighborhood of x, then Il(V} O. Theren, so that An n V D. Therefore x £: A.
* 0 for all sufficiently large
*
If x £: A and V is a closed neighborhood of x, let W be a neighborhood of x with dist (W. (In - V) = d > O. For all sufflciently large n, we have An n W D. Take y In An n W, so that the closed sphere B of radius d about y is a subset of V. Since Il n has property M(b, c. k} it follows that
*
Il n(V} In the limit this gives Il (V)
L cdk
2. Iln(B} 2. cd k •
* O.
Therefore x
£:
support Il •
COROLLARY. If {An} is a sequence of closed subsets of an open set U C (In, and An ---> A as n ...... DO, then A has property S(b O, c, k) for some b o if An has property S(b, c, k} for each n.
Proof. Let Il n be a measure in M(b, c, k} supported by An. By passing to a subsequence, if necessary, we may assume that Iln converges to a measure Il on U as n ...... DO. By 'the lemma, Il = support A. Since Il has property M(b o , c, k} for some bo it follows that A has property S(b o , c, k}. LEMMA 8. Let A c (£n be a locally compact set of (2k + 1}-dimensional Hausdorff measure 0, for some k (0 < k < n - 1). Then the set F of all co mplex (n - k}-dimensional linear subspiiCes-P of G: n passing through 0 and intersecting A in a set that is not totally disconnected is of first category.
Proof. Without loss of generality we take A to be compact. Let 01 be any real-valued real-linear function on (In that is a linear combination with rational coefficients of the real and imaginary parts Xl, Y 1, ... , Xn, Yn of the coordinate functions z 1, ... , zn. Let c and d be rabonal numbers (0 < c < d). Let F(ClI, c, d} consist of all P for which [c, d] C ClI(A n p}. Clearly F(ClI, c, d} is closed and F c UF(ClI, c, d}. Thus it is sufficient to show that each F(ClI, c, d} is nowhere dense. Assume there exists an interior pomt Po of F(ClI, c, d}. Choose new coordinates z on (l; n so that 01 is the real part of Zk+ 1 and the equation of Po is zl ~ Z2 = ••• = zk = O. Then for all a = (al, ... , ak) sufficiently near to 0 (say for a i < 1:, 1 ~ i ~ k), the subspace P a whose equations are
I I
Zl + a 1 zk+l = •.• = zk + ~ zk+l = 0 belongs to F(ClI, c, d}, so that 01 assumes all values between c and d on ,A n P a . Let rr be the map of {Z £: G: n : ClI(Z} > ~} into (Ik X m defmed by rr(z) = (-Zl(Zk+l}-l, ... , -Zk(Zk+l}-l, ClI(Z».
339
ERRETT BISHOP
292
Then we see that 1T(A) contains the set
[~~, ~ ]
2k'>( [c, dl, and so has a nonvoid in-
terior. Hence the (2k + I)-dimensional Hausdorff measure of 1T(A) is not zero, contrary to the vanishing of the (2k + I)-dimensional Hausdorff measure of A. This proves that F(a, c, d) has void interior. as desired. THEOREM 1. Let {An} be a sequence of pure 2k-dimensional analytic sets in an open subset U of \j; n, converging to a set A CU. Let the 2k-dimensional volumes of the An be finite and bounded by some constant b. Then A it. an analytic subset of U. Proof. Define a measure !ln supported on An by taking !In(S) to be the 2kdimensional Hausdorff measure of An n S. Since !In(U) ~ b, we may assume, by passing to a subsequence if necessary, that JJ.n converges to some measure JJ. on U. By Lemma 3, JJ.n €M(b, c, 2k), so that An € S(b, c, 2k), By Lemma 7,
A (' S(b O, c, 2k). By Lemmas 5 and 6. the 2k-dimensional Hausdorff measure of A is finite. By Lemma 8, we may assume that the intersection of A WIth the set p
= {z: z 1 = Z l = .,. = Z k =- O}
is totally disconnected. Thus there exists an open neighborhood T of 0 in such that [k
and (G: k x such that
T)
I'
x bdry T ,[ (A
(I
p) =
\j;n-o
(2rr,-1 5 [ 5_t+ [8 [ y
2
1 - r 2 dq, 1 - 2r cos q, + r
t
345
]
dll(~)
298
ERRETT BISHOP
2
5 1+
r
rc
Yt
lTV
r
arctan
(t -
I e If./r dll(~) > - r -
1
5 1+
r
arctan
r=
YA
lTV
r
(t 1
Af./r - r
dll(~),
where A = A(t). From this lt follows that lim sup II(YA)m- 1 ~ 1. t->O
Hence lim sup II (Y A) A-1 ~ 1, t~O
or simply -1
lim sup II (Yt ) 7Tt
~ 1.
t->O
n S extending for length s on both sides of z 0, then O· at z , lt follows that
If we let As be the part of A
since
Idsd7T I = Icos RiJ;(zO) I
Therefore Il is absolutely continuous with respect to arc length, and
< I dill ds as desired. It remains to consider the case where zO E A n S - H and cos iJ;(zO) = O. Here let IIIi = 1I({~: I ~~ Ii }), for each Ii> O. The function
11
hm = m
is harmonic for ~
~
1 _ (1
"t"
x-{l+Ii) Ii) = (x _ (1 + Ii ))2 + y2
* 1 + Ii, and negative for
-1 -1 ~ ~ = h(O) =
I ~ I ::;: 1.
Thus
5
h(~)dll(~)
or
Combined with the above expression for
I~: I,
desired.
346
this gives
I=~ I
= 0 at zO, as
CONDITIONS FOR THE ANALYTICITY OF CERTAIN SETS COROLLARY 1. Under the hypothesis of Theorem 2, length (A n S)
~
299
21TH.
COROLLARY 2. There exists a constant c > 0, depending only on n, such that if B is an open ball of radius R about 0 in (l;n, and if P is an analytic subset of B, and A a pure k-dimensional analytic subset of B - P, with 0 £ A, then the 2kdimensional Hausdorff measure of A is at least cR2k.
Proof. This follows from Corollary 1 just as Lemma 3 followed from Lemma 1. THEOREM 3. Let U be an open subset of (l;n, and P an arUllytic subset of U. Let A be an analytic subset of U - P of pure dimension k and finite 2k-dimensional volume. Then A n U is an analytic subset of u.
Proof. Let V be any open subset of U containing P. Define a measure IlV by taking IlV (S) to be the 2k-dimensional volume of A n V n S. By Corollary 2 to Theorem 2, £ M( V c, 2k) on V. Thus support Ilv £ S( c, 2k). By Lemma 5,
Ilv
IIIl II,
Illlv II,
support Il V
n
£
N(K
IIIl V I c -I , 2k) . IIIl V I can be made arbitrarily small.
pcsupport Ilv for all V. Also, Now A Thus An P has 2k-dimensional Hausdorff measure O. By Lemma 9, analytiC set, as desired.
An
U is an
Stoll [9] has shown that Theorem 3 implies that If A is a pure k -dimens ional analytic set in (l; n, such that II r /2 is bounded as r -> 00, where IIr is the 2kdimensional volume of A n {z: I z ~ r}, then A is algebraic. Stoll [9J was not able to demonstrate Theorem 3 for general n and k, but got this condition for the algebraic character of A by other methods.
I
4. CAPACITY For an optimal generalization of the Remmert-Stein theorem, some notion of capacity in (l;n seems necessary. See Rothstem [5] for a Remmert-Stein theorem based on capacity in (l; I. After introducing the appropriate notions, we shall prove a Remmert-Stein type theorem (Theorem 4 below) which contains Rothstein's result and Lemma 9 above as special cases.
Definition 2. Let X be a compact Hausdorff space and .4 a subalgebra of C(X). A Baire subset B of X will be said to have capacity 0 for .4 If the set of all points z in X - B that admit no Jensen measure IJ. with IJ.(B) -t- 0 is dense in X-B. Definition 3. Let U be a bounded open set m (I n, and B a Baire subset of bdry U. Let.4 be the algebra of all bounded analytic functions on U, and a the spectrum of.4. Let a u be the closure of !! as a subset of a, 50 that au c a and au contams the ~ilov boundary of.4. Let B consist of all pomts x in a u that lie over some point p of B, 50 that f(x) = f(p) for all functions f analytic m a neighborhood of U. We say that B has capacity 0 relative to U If the set n consistmg of all pomts in U that have no Jensen measure IJ. on au with Il(B) t 0 is dense in U. LEMMA 10. Let U be a bounded open set in (I n, and let V be an open subset of U such that B = (bdry V) n U has capacity 0 relative to V. Then V is "de-lJ:..se in U, and every bounded analytic function on V extends to U.
Proof. Without loss of generality we take V to be connected. Consider first the case n = 1. If B is not totally disconnected, there eXIsts a disk
347
300
ERRETT BISHOP D = {z:
Iz
- Zo I < r}
about some point Zo of V such that the component C of V r int D containmg :l{) does not contain all of the set E = bdry D m its boundary. We may assume that Zo E Q. Let_F = bdry C, and let Ilo be a Jensen measurELfor_zo on F. Since IlO vanishes on B, we see that Ilo is actually a measure on E n F. Thus the projection of Ilo onto the complex plane is a Jensen measure !iO for Zo on E n F. Since E n F is a proper closed subset of E, this is impossible. Therefore B is totally disconnected. Hence V is dense in U. Since B is totally disconnected, if zO is any point in V there exists a simple closed curve y in V surrounding z o. Let JI 0 be the subalgebra of C(y) obtained by restricting the functions in JI to y, and let JlI be its closure in C(y). Now zO has a Jensen measure for JI on e, where e = y U (B n int y), and therefore on y = y. Thus evaluation at zO is a point of the spectrum of .A I. By Wermer's maximality theorem [10], it follows that all functions in .A I are boundary values of analytic functions on int y. Hence all functlOns in JI can be extended analytically to int y, as desired. Consider now the general case n 2. 1 Let Q be the dense Gil -subset of V consisting of the pomts having no Jensen measure Il on a v with Il (B) > O. Let P be any I-dimensional linear variety in ll: n, intersecting Q in a dense subset of V n P, and let Uo be any component of U n P containing a point of V. Let V n ~ = Va and Bo = (bdry Vol n Uo . Now Ii Bo were not of capacity zero relative to Vo (where V 0 is considered as an 0cPen subset of P and P is identifled with Cl: I ), there would exist a Jensen measure Il z for each z belonging to an open subset r of Vo ' relative to the algebra JI 0 of all bounded analytic functions on V o ' with Il~ (Bo) > O. Let w: JI ~ Jl o be the restriction map that takes bounded analytic functions on V into bounded analytic functions on Vo. The adj oint map w * takes a V0 into a V . Thus W*(Il~) = Ilz is a Jensen measure for z relative to the algebra .A, and 2. 1l~(Bo) > O. Hence reV - Q, contrary to the fact that Q n P is dense in V n P. Thus Bo is of capacity 0 relative to Va. By the case n = 1 already considered, Bo is totally disconnected, Vo is dense in U 0, and every bounded analytic function on Vo extends to Uo . Ilz(B)
Assume now that V is not dense in U. Then there exists a vanety P as above, with the additional property that Uo contains a point of int (U - V). This contradicts the fact that Vo is dense in U o. Hence V is dense in U. Consider a bounded analytic function f on V. Let z 0 be any point of B. Choose P as above, and so that in addition it passes through z 0 and intersects V in a point of the component Uo of U n P containing z o. Since B n U a is totally discOIUlected, there exists a simple closed curve y in Va surrounding z o. Take P to have equations Zz = ... = zn = O. There exists a neighborhood S of 0 in (5; n -I such that
is a subset of V. Thus f extends from V to W = {z: (z I' 0, •.. , 0) ( int y, (zz' ... , zn) ( S} to be an analytic function of the variable zl in Wand analytic in all variables in V n W. Thus f is analytic in W, as was to be proved.
348
CONDITIONS FOR THE ANALYTICITY OF CERTAIN SETS
301
THEOREM 4. Let U be a bounded open set in a;n, B a closed subset of U, A an analytic subset of U - B of pure dimension k such that B c A. Let B be of capacity 0 relative to the algebra .A of all continuous junctions on A that are analytic on A. Let there exist an analytic map 71 of U onto a connected open subset S of (l;k that is proper on B, with 71(B) S. Then An U is an analytic subset of U.
'*
Proof. We first reduce the problem to the case in WhICh 71 is proper on A and has countable level sets on A. To this end, replace B by the set of those pOints in A at which A is not analytic. Then we may assume that A is analytic at no point of B. If B is void, there is nothing to prove; we therefore assume B is not void, and consider zO € B with
Since 71- 1(7T(ZO» n B is compact, there exist a relatively compact open subset U 1 of U and a relatively compact open subset S 1 of S such that and By [1, Theorem 1J, we can find a mapping 711 of U into a;k, uniformly near to 71 on any compact subset of U, whose level sets on A are all countable. If S2 is any connected relatively compact subset of SI that contains 71(ZO), we may take 711 so near to 71 that
and 71 1(B) nS 2 *Sz· Let U z
=
U 1 n 71 11(Sz). Then 711 maps U Z onto Sz and 711(B n bdry U z ) n Sz =:::J,
so that 71 1 maps B n U z properly mto Sz. Take P
€
Sz r, bdry 711(B n U z ).
Now r = 71jl(p) n (B n U z ) IS compact, and 71jl(p) n (A C") U z ) is countable. There therefore exists a relatively compact open subset U 3 of U l such that r c U 3 and
There thus exists a connected open neighborhood S3 of p in a;k such that
Write
Then 7T 1 maps U 4 onto S 3 and maps (A U B) () U4 properly mto S3' Also, B n U4 * ["', 711 (B n U4 ) * S3' Therefore, if we can prove our theorem for the case
349
ERRETT BISHOP
302
where the map 7T is proper on A n will follow in the general case.
u
=
A U B and has countable level sets on A, it
Assume therefore that 7T is proper on A n U and has countable level sets on A. Let K be any component of S - 1T(B), and let L = S n bdry K. Then 1T maps An 1T- 1 (K) properly onto K. Associated with this map there is a multiplicity It, such that for each p in K there exist points PI' ••• , p,,- in A, counted with multiplicities, such that 7T(p t ) = p (1 s: i s: It). As in the proof of Lemma 9 above, we see that If a sequence {pn.r of pOints of K converges to a point p in L, then at least one of the paints p? converges to B. Since the set n of points in A - B not having Jensen measures J.L on A with > 0 is a dense G 15 -set in A - B, there exists a dense set of paints pO in K such that p ~, ... , p~ are distinct and such that P~ E n (1::;' i::;' It). For 0 < e < 1, for every compact subset D of B, and for 1::;' i::;' i\. there exists, by Lemma 3 of [1], !,. in A with J.L (B)
Choose D so that D = B n 1T- 1(1T(D». Let gi be any analytic function on ern with
There exist constants r
< 1,
c1, ... , ci\. with
Let E be any positive number. There exist a constant c and positive integers N 1 , ... , Ni\.' all depending on E, such that N.
Ilf.lll 1
f)
< c,
-
for 1 s: is: i\.. Define
N.
i\. g
and let y
= max { II gi II:
= i=2.;1 g.1 f.1
"
1 ~ i s: i\.}. Then 1
< i\.ye Also, g(p~)
=1
f)
-1 T:7! II II gD.s.i\.y(E) .
,
(1::;' i::;' i\.). Define the analytic function f on K by
i\. f(p)
=
II i=1
Then f is bounded, f(pO) = 1, (*)
350
g(Pi).
CONDITIONS FOR THE ANALYTICITY OF CERTAIN SETS and, since at least one of the points Pi converges to B, as p
---+
303
L,
where M = L n 7T(D) . Write make
1 - A(l - fi). Then 0' < 1, and by taking fi near enough to 1 we can arbitrarily near to 1. We compute, from (*) and (t):
0' = 0'
IlfllO'
Ilfll~O' ~ IlgIIO'A+(l-O')(A-I) Ilgll~-O' ~
where
T
(
-tJ)O'+A-I( -I l:fi)I-O' A'YC A'Y(r.C )
A Tf.
,
is a constant. Thus, for r. small enough, IlfllO' Ilfllk-O'
1 - 0'. Smce M can be an arbitrarily large compact subset of L, and since 0' canbe arbitrary, pO does not have a Jensen measure on L with Il(L) > O. Smce pO can belong to a dense subset of K, it follows that L is of capacity 0 relative to K. By Lemma 10, K is dense m S and every bounded analytic function in K extends to S. It follows as in the proof of the Remmert-Stein theorem that A n U is an analytic subset of U, as was to be proved. Actually, the hypothesis that B is of capacity 0 relative to .4 could be replaced by the following slightly weaker assumption, as the proof of Theorem 4 shows. Let .4 ° be the algebra of all bounded analytic fUllctions on A, and a A the closure of A as a subset of the spectrum a of .4 o. Let Bo consist of all points p in a A that lie over a pomt x in B, in the sense that some net (or filter) m A converges to x in the topology of A U B and converges to p in the topology of a A. The weaker assumptlOn is then that the set of pomts of A having no Jensen measure Il for the algebra .40 w1th Il (Bo) > 0 1S dense in A. Here is a jushfication of the definition of capacity given m Definitions 2 and 3 above. The proof is simple but tedious, and we omit it. Justification. Let r be a compact subset of the complex plane, and 'Y a Jordan curve surrounding r, so r c int 'Y. Let.4 be the algebra on X = 'Y U int'Y obtained by taking the closure of all polynomials. Let p be any point of int'Y - r. Then the following statements are equivalent. (a) r has capacity 0 in the usual sense, (b) if Il is any Jensen measure for p on X, relative to the algebra .4, then Il(r) = 0,
(c) r has capac1ty 0 relative to .4, m the sense of Definition 2, (d)
r has capacity 0 relahve to the open set mt'Y tion 3.
351
r, m the sense of Defini-
304
ERRETT BISHOP REFERENCES
1. E. Bishop, Mappings oj partially analytic spaces, Amer. J. Math. 83 (1961), 209-242. 2. - - - , Partially analytic spaces, Amer. J. Math. 83 (1961), 669-692. 3. - - - , Holomorphic completions, analytic continuation, and the interpolation of semi-norms, Ann. of Math. (2) 78 (1963), 468-500.
4. I. Gllcksberg, Maximal algebra" and a theorem of Radii, (to appear). 5. W. Rothstein, Zur Theorie der Singularitaten analytischer Funktionen und Flachen, Math. Ann. 126 (1953), 221-238. 6. W. Rudin, Analyticity, and the maximum modulus principle, Duke Math. J. 20 (1953), 449-457. 7. H. Rutishauser, Ube1" die Folgen und Scharen von analytischen und merorn01"phen Funktionen mehre."er Variablen, sowie von analytischen Abbildungen, Acta. Math. 83 (1950), 249-325. 8. W. Stoll, The growth of the area of a t."anscendental analytic set of dimension one, Math. Zeitschr. 81 (1963), 76-98. 9. - - - , The growth of the area of a transcendental analytic set (to appear). 10. J. Wermer, Banach algebras and analytic junctions, Academic Press, New York, 1961.
University of Callfornia Berkeley, California
352
.. , _ . ,
... -
.... _ -
-J
...
• ... -rr
.... ' -
- _ .. _ ...... _ .... -
..... .. ~~_r . . .
---rJ- ..o-....
~
..... - -
-r-
Uniform Algebras * By
E.
BISHOP
Recently concepts from the theory of Banach algebras have been increasingly useful in applications to complex analysis. Here we give a quick survey of some, but by no means all, of these applications. The algebras which occur are algebras of complex valued functions and they are given the uniform norm with respect to some compact set K. Since the usual terms for these algebras (sup. norm algebras or function algebras) are not euphonious, we shall call them uniform algebras. Definition 1. A uniform algebra III is a complex Banach algebra with unit, in which the norm equals the spectral norm. This condition means that lim I fn Illln = I f II, for all f in Ill. Here n-+oo
lim I fn Illln is the spectral norm of f, so called because it equals "->00
sup
{I q; (f) I:q; E spectrum Ill} ,
where spectrum III is defined to be the set of all non-trivial homomorphisms q; of III into the complex numbers C. Every such homomorphism q; is necessarily continuous. Following GELFAND, we topologize X = spectrum III by the weak topology induced by Ill, so that for each f in III the function q; --+ q; (f) on X is continuous. With this topology X is compact Hausdorff. For convenience we write x (instead of q;) for a generic element of X and write f(x) for x(f), so that we may consider III as an algebra of continuous functions on X. It is then a sub-algebra of the uniformlynormed Banach algebra C(X) of all continuous complex-valued functions onX. The category of all uniform algebras seems to be the natural setting for many of those problems in complex analysis which concern a cl~ of holomorphic functions (or holomorphic functions and their boundary values) known in advance to contain sufficiently many elements. The
* Received June 26, 1964.
353
273
Uniform Algebras
presently available methods of the theory do not seem to be suited to problems which require the construction of holomorphic functions from geometrical information, for example the E. E. Levi problem. The theory begins with the study of the Silov boundary r of m, which is defined to be the smallest closed subset of X such that
11I11 = 1!/llr=sup{lf(x)l·xEr} for all I in m. SILOV proved its existence. In case mis separable, there exists a smaller boundary, the minimal boundary M, defined as the smallest subset (not necessarily closed) of X on which all functions in attain their maximum absolute values. It turns out to consist exactly of the peak points of X, i e. those x in X for which there exists I in mfor which
m
If(x) I > II (y) I {x}. The closure of M is r.
for all y in X In case m= C (X) it is easy to show that r = X and, when M exists, M = X It is an open problem of whether conversely M = X implies m= C (X). In fact there are no general solutions to the problem of finding general conditions which imply that = C(X), although \VERMER [27] has recently gotten very pretty results in a special case. The importance of the Silov boundary lies in the fact that an element I of is completely determined by its values on r (i e. by its boundary values). In fact we have the possibility of representing the value of a function in ~ at a given point x of X as a boundary integral. More precisely: Theorem .. To each x in X there exists at least one finite positive Baire measure {tz on r such that
m
m
(*)
I (x) =
S fd{tz
lor all I in ~. In case M exists {tz can be taken to be a measure on M. Such a measure {tz is called a representing measure for the point x. Recently GLEASON [11], BUNGART [8], and others have proved in certain cases the possibility of taking the measure {tz to be an appropriately well-behaved function of x, thereby obtaining a kernel representation of functions in ~ in terms of their boundary values. A strengthened form of the last theorem which is sometimes needed is the following. Theorem. Let cp be any bounded linear functional on a uniform algebra ~. Then there exists a finite complex-valued Baire measure {t on such that
r
I {t I = Ii cp I lor all I in
~.
In case
~
and
cp (f)
=
Sf d{t
is separable, {t can be taken to be a measure on JJf. 18
354
274
E.
BISHOP
Looking at one of the standard examples, the uniform algebra of all continuous functions on the disk
~
D={z:lzl ~ 0, then there exists gl, ... , gn in Hoo with Lgdi = 1. Unfortunately CARLESON'S methods, which are classical and highly technical, do not give any insight into the corona conjecture for other algebras, for in-
357
Uniform Algebras
277
stance the algebra of all bounded analytic functions on the n-fold cartesian product of {z: IZ I < I}. It is not known how to give an abstract characterization of those uniform algebras W whose parts all have some one-dimensional analytic structure. Presumably some extension of the notion of a Dirichlet algebra is in order. Further information about applications of uniform algebras to one complex variable can be found in the book OfWERMER [25]. In a sense the theory of uniform algebras is a branch of the theory of several complex variables. To see how this comes about, call a uniform algebra Wfinitely generated if there are elements h, .. , In in Wsuch that the smallest closed subalgebra of W containing h, .. ,In is W itself. Associated with generators h, ... , In of Wis a map F : X -+ On defined by
F(x)
=
(h(x), ... , In(x)).
This map F is a homeomorphism, and the image K = F (X) is polynomially convex. This means that the polynomial convexification
K= {z= (Zl, ... ,zn)Ecn.l/(z)1
~sup{lf(w)1 :wEK}
for all polynomials t} of K is K itself The algebra W is isomorphic to the algebra ~K obtained by taking the uniform closure in 0 (K) of the set of all polynomial functions of Zl, ... , Zn. Conversely, if K is an arbitrary compact subset of Cn then the uniform algebra qJK is generated by the coorclmate functions Zl, .. , Zn, and the spectrum of qJK can be identified with K. Thus the study of finitely generated uniform algebras is eqUIvalent to the study of polynomially convex subsets of Cn. In fact however the study of an arbitrary uniform algebra can often be reduced to the firutely generated
case by studying appropriate finitely generated subalgebras. SILOV fin,t employed this technique to construct analytic functions of finitely many Banach algebra elements. His results were completed by \VAELBROECK [23] and ARENS-CALDERO~, giving the following theorem.
Theorem. Let gl, ... ,gle be elements of a Banach algebra W, and let G(X) be the image 01 the spectrum X 01 W under the map G defined by G (x) = (gdx), ... , gle (x)) of X into Cle. Let H be a lunction analytic in some neighborhood 01 G(X). Then there exists h in B such that h(x) = H(G(x)) lor all x in X. The proof of this theorcm makes non-trivial use of the theory of several complex variables, in particular of some integral formula of the type due to BERGMAN or "VEIL. Another result of fundamental importance which makes use of several complex variable theory applied to finitely generated subalgebras is the
358
278
E.
BISHOP
local maximum modulus principle, proved by ROSSI [19] and then by STOLZENBERG [22].
Theorem. Let K be a closed subset of the spectrum X of the uniform algebra ~. Then for every f in ~ [It [[K
=
[[
f [ bdry J( U (K n
r).
This result reinforces the belief that the spectrum of a uniform algebra should_ have some sort of analytic structure on the complement X of the Silo v boundary, but this belief was destroycd by STOLZENBERG [21], who gives an example of a polynomially convex subset Kin C2 for which this is not true. In STOLZENBERG'S example K is the limit of analytic sets, and this suggests the possibility of finding some such limit analytic structure on X in the general case. One of the most interesting problems in uniform algebras is to find the analytic structure on X when it exists. GLEASON [12] gives the following result.
r
r
r
Theorem. Let x be a point in the spectrum of a uniform algebra ~. Let the maximal ideal I of all functions in ~ vanishing at x be finitely generated, in the algebraic sense. Then there exists a neighbornhood U of x in X which can be given the structure of an analytic space in such a way that the func. tions in ~ are analytic on U. In the cases of most interest it is not possible to verify that I is finitely generated, and other methods are needed. Take the case of an algebra ~ of holomorphic functions on an n-dimensional complex manifold M. The natural topology for ~ is the topology of uniform convergence on compact sets, i. e. the topology determined by the family {[I [[K} of norms, where K is a compact subset of M. Let E denote the spectrum of ~, eonsisting of all continuous non-trivial homomorphisms of ~ into C. Then E = UK E K , where EK consists of all elements of E which are continuous with respect to [[ [[K. Thus to study the structure of E we may look at the structures of the E K . Now EK is the spectrum of the uniform algebra ~K obtained by completing ~ in the norm [[ [IK (i. e. by taking the closure of ~ in C (K)). In this general situation we have the following theorem [4]. Theorem. Let ~ be an algebra of analytic functions on an n-dimensional complex manifold M. Let K be a compact subset of M. Let ft, ... , fn be elements of ~. Let the map F· EK -+ Cn be defined by F(x)
= (fI(x),. ·,fn(x)).
Then for almost all z in Cn the set F-l (z) is finite. Thus EK is not too large. This fact can be applied to give strengthened versions of the result of OKA that the envelope of holomorphy of any
359
Uniform Algebras
279
domain M c en is a Stein manifold. The connection with envelopes of holomorphy comes from the fact that the envelope of holomorphy of a subset M of en can be identified with the spectrum 2: of the algebra 2l of all holomorphic functions on M. Thus for the case of an arbitrary algebra 2l of holomorphic functions on a complex manifold M it is natural to define the envelope ofholomorphy of M relative to 2{ to be the spectrum 2:. Much remains to be clarified about the structure of 2: in the general case-only the cases in which iII is a domain spread over en or in which M is one-dimensional have been thoroughly worked out GRAUERT [15J has shown that the structure of E may indeed be very complicated. However there is reason to hope that every point of E belongs to a subset of E which can be given the structure of a one-dimensional analytic space on which the functions in 2{ are analytic. The problem of analytic continuation falls naturally into this circle of ideas If 2{ is an algebra of analytic functions on a complex manifold M, if K is a compact subset of M, and if I III is some algebra semi-norm on K, one defines interpolated norms I IIIJ, by a process of geometric interpolation between I 11K and I I b for each () between 0 and 1. For each such () there is a spectrum E(IIIIIJ) defined, and it is suggestive to think of the functions in 2{ as having been continued analytically to E(lllilJ). In fact a necessary and sufficient condition for one function element to be the analytic continuation of another can be given by means of such in terpola ted semi-norms. Some recent applications [5J of the theory of uniform algebras involve the notion of capacity, which is the generalization to abstract complex analysis of the notion oflogarithmic capacity in Cl
Definition. Let Y be a Baire subset of the spectr'um X of a uniform algebra 2{. For each point Xo in X we define the capacity of Y with respect to Xo to be sup {,u (Y) ,u is a Jensen measure for Xo } Using this notion we can give a general criterion for the continuation of an analytic set through possible singularities.
Theorem. Let U be a bounded open set in Cn, B a closed subset of U, A an analytic subset of U - B of pure dimension k such that B c A. Let B be of capacity 0 for the algebra 2{ of all continuous functions on 2{ which are analytic on A, relative to every point Xo of a dense subset of A. Let there exist an analytic map n of U onto a connected open subset S of CTc which is proper on B, with n(B) '*' S. Then An U is an analytic subset of U. This generalizes now classical results on the removability of singularities due to THIMM, REMMERT-STEIN, ROTHSTEIN, etc. For many problems in complex analysis the category of uniform algebras seems too large, and it is worthwhile to examine the subcategory of differentiable uniform algebras. Such an algebra 2{ is a subalgebra of C (M), where M is a compact differentiable manifold, possibly with
360
280
E. BISHOP
boundary, which is generated by finitely many differentiable functions as a differentiable submanifold of en. In addition to the topological problem of the nature of the imbeddmg of Min R2n, there are equally deep and interesting problems concerning the relation of M to the complex structure of en. Many of these pertain to the structure of the family of all simple closed curves y in M which form the boundary of some Riemann surface Sy in en For instance, we may ask when the polynomially convex hull of M is the union of the Sy, or when there are no Sy, or how the family of all Sy can be paramaterized, or whether uSy contains an open set. We would also like conditions under which ~ (the closure in C(M) of the polynomials) coincides with the set of alII in C (M) which can be extended analytically to each Sy. The only deep fact about a general compact differentiable manifold Me en is due to A. BROWDER [7], who shows that if dim M ~ n then M is not polynomially convex. This implies, for instance, that the algebra C(S2) of all continuous complex-valued functions on a two-sphere cannot be generated by two elements. It would be of great interest to find a constructive explanation of Browder's result, for instance to show that under his hypotheses there necessarily exist Riemann surfaces Sy.
II, ... , In. These functions imbed M
References [1] ARENS, R., and 1. SINGER. Function values as boundary integrals. Proc. Amer. Math. Soc. 5, 735-745 (1954). [2] BISHOP, E.: Subalgebras of functions on a Riemann surface. Pac. J. Math. 8, 29-50 (1958). [3] Analyticity in certain Banach algebras. Trans. Amer. Math. Soc. ]02, 507-544 (1962). [4] Holomorphic completions, analytic continuation, and the interpolation of semi·norms. Ann. Math. 78,468-500 (1963). [5] Conditions for the analyticity of certain sets. Michigan J. Math. 11, 289-304 (1964). [6] Representing measures for points in a uniform algebra. Bull. Amer. Math. Soc. 70, 121-122 (1964). [7] BROWDER, A : Cohomology of maximal ideal spaces. Bull. Amer. Math. Soc. 76, 515-516 (1961). [8] BUNGART, L.· Holomorphic functions with values in locally convex spaces and applications to integral formulas. Trans. Amer. lvIath. Soc. 111, 317-344 (1964). [9] CARLESON, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76, 547-559 (1962). [10] GLEASON, A.: Function algebras. Seminars on Analytic Functions Vol. II, Inst. for Advanced Study, Princeton N.J., 1957. [11] The abstract theorem of Cauchy- Weil. Pac. J. Math. 12,511-525 (1962). [12] Finitely generated ideals in Banach algebras. J. Math. Mach. 13, 1~132 (1964). [13] GLICKSBERG, 1.: Maximal algebras and a theorem of ROOD. (To appear.) [14] -, and J. \VERMER: Measures orthogonal to a Dirichlet algebra. Duke Math. J. 30,661-666 (1963).
361
Uniform Algebras
281
[151 URAUI~ltT, H.: tu appear. [161 HELSON, H., and D. LOWDENSLAGER: Prediction theory and Fourier series in several variables. Acta Math. 99, 165-202 (1958). [17] HOFF'IAN, K.: Analytic functions and logmodular Banach algebras. Acta Math. 108, 271-317 (1962). [18] - Bmach sp'Lces of an'Llytic functions. Englewood Cliffs: Prentice Hall 1962. l19] ROSSI, H.: The lor-al m'Lximum modulus principle. Ann. Math. 72, 1-11 (1960). [20] RUDIN, W .. AWLlyticity and the maximum modulu8 principle. Duke Math. J. 20,449-458 (1953). [21] STOLZENBERG, G. A hull with no analytic structuTe. J. Math. Mech. 12, 103-111 (1963). [22] - Polynomially and rationally convex sets. Acta. Math. 109, 259-289 (1963). [2.1] WAELBROECK, L. Le calculsymbolique dans les algebre~ commutative8. J. Math. Pures App!. 33, 147-186 (1954). [24] WERMER, J : On algebras of continuous functions. Proc. Amer. Math. Soc. 4, 866-869 (1953). [25] B'mach algebras of analytic functions. New York: Academic Press 1961. [26] Function rings and Riemann surfaces. Ann. Math. 67, 45-71 (1958). [:nl PollfnomiaTlll COnl'ex disks. (To appear.) Department of Mathemati('.S University of California Berkeley, Calif.
362
DIFFERENTIABLE MANIFOLDS IN COMPLEX EUCLIDEAN SPACE By ERRETT BISHOP
We consider a k-dimensional differentiable submanifold Mk of n-dimensional complex coordinate space C", and interest ourselves in the relation of the submanifold M to the analytic subvarieties of C". Consider the tangent variety P to 1If at a point p in M. It is a k-dimensional real-linear variety and if k ~ n, it will contain a complex-linear variety of complex dimension Ie - n. The point p will be called exceptional if P contains a complex-linear variety of dimension k - n 1. We would like information about the structure of the set of exceptional points and about the structure of 1If in the neighborhood of an exceptional point. We \vould also like information about the submanifolds of M which bound analytic subvarieties of C". We would like to use this information to describe the polynomially convex hull and the hull of holomorphy of M, and perhaps eventually to give a satisfactory explanation of some of the known results [1] on the cohomology groups of polynomially convex sets in e". These problems seem to be very difficult At least it is hard to prove global results. Therefore in this paper we consider primarily the local situation. Our only global result, having to do with the exceptional points of a two sphere imbedded in e 2 , is a consequence of a theorem of Chern and Spanier [2]. In case M2 C e2 , as instances of our problems there arise questions in one complex variable which seem not to have been considered to any extent, although from our point of view they are very natural. Here is an example. Let f be a continuous complex-valued function on e'. What can be said about the family of all simple closed curves 'Y C e' such that t agrees on 'Y with the boundary values of some analytic function defined on the interior of 'Y? Although we get only the above-mentioned global result, we are able to obtain quite a bit of local information at a point p of M. We first investigate the case k > n, and put the equations of J.[ in a simple form, by means of a differentiable coordinate transformation on M at p and an analytic coordinate transformation on C" at p. The form of the simplified equations for M suggests the existence of certain families of simple closed curves on J.f each of which bounds an analytic disk in C". By an iteration argument we prove the conjectured families indeed exist. It is not clear whether locally the union of the corresponding disks gives the complete hull of holomorphy of 1l~ some appropriate sense. We next consider an exceptional point p of J.fk C e\ one which is not exceptionally exceptional. Such points are of two types, depending on whether
+
Received October 8, 1963. This research was partially supported by the National Science Foundation under grant GP2026. 1
363
2
ERRETT BISHOP
a certain quadratic form, analogous to the E. E. Levi form, is positive definite. For exceptional points of the first type, as we shall call them, the nature of the equations suggests the existence of a k - 1 parameter family of simple closed curves, filling up a neighborhood of p in M, each of which bounds an analytic disk in ek • Again an iteration argument, analogous to but more complicated than the previous one, proves such a family to exist. Given a k-dimensional differentiable manifold lJ,1 in en, and a compact subset K of M, we ask whether there is an open set U in en such that any function analytic in a neighborhood of K, no matter how small, extends to an analytic function in U. In case k = 2n - 1 and the E. E. Levi form is not degenerate at every point of K, the answer, as is well known, is affirmative. On the other hand, to have any hope for a good local result we must have k ~ n + 1. Thus k = n + 1 is the critical case. "\Ve study the case k = 4, n = 3, and show that if certain unavoidable non-degeneracy conditions are satisfied then the answer is affirmative. This is done by using the techniques here developed to construct a sufficiently large family of analytic disks bounded by lJf. An example of this general phenomenon was given by Lewy [3], who proved the additional fact, left open here, that any function on M satisfying the appropriate differential conditions is the boundary value function of an analytic function on U. It is likely that our techniques work for values of n larger than 3 as well as for n = 3. For previous work on problems of the type considered here, sec Lewy [3] and [4]. ' Thanks are due to M. Hirsch and P. Griffiths for pointing out the relevance of [2] and to G. Stolzenberg for correcting some mistakes. 1. Simplification of the local equations. Given lJ,[k C en, ",ith k > n, and a point p in M, we choose differentiable coordinates x, , ... , Xk for a neighborhood of p in M, vanishing at p, and analytic coordinates 2, , . . . , 2n for a neighborhood of p in en, vanishing at p. The equations of 111 in a neighborhood of p are then of the form (1)
Zn =
In(xl , ...
,Xk),
where I, , ... , In are complex-valued differentiable functions of Ie real variables, defined in a neighborhood of X, = 0, ... , Xk = O. The coordinate functions x, , ... ,Xk and 21 , ••. ,2k are to be chosen to give these equations a simplified form. To this end let P be the tangent space to M at p, which we identify with a real linear subspace of the tangent space T to en at p. Assume that p is not an exceptional point. Let p. denote the complex-linear subspace of T gener-
364
3
DIFFERENTIABLE MANIFOLDS
ated by P. Let Q denote the largest complex-linear subspace of P. Let compo dim. p. = t. Since real dim. P = k, we see that compo dim. Q = k - t. Since p is not an exceptional point, we therefore have t = n or Pc = T. By a linear coordinate change in en we may assume that p = (0, ' .. , 0) and that 1m ZI , ••• , 1m Z2.. - , vanish on P. Therefore Re ZI , ••• , Re z.. , 1m Z2 .. - H l , ••• , 1m z.. are coordinate functions for some neighborhood of p in M. Relative to these coordinates the equations (1) become
where
Zl , •.• ,
z. are the new coordinates at p in XI
I
•••
r
2"2,.-1
I
U1
I
.,.
en and
,Ul-n ,VI , . . .
,Vl- fl
are thc new coordinates at p in M. The differentiable functions hi , ... , hz.-, vanish to second order at XI = a, ... , V'_ = O. We write these equations in the form R
z = (x, w)
(3)
+ i(h(x, w), 0)
where Z =
(Zl , •••
,z.),
and hex, w)
=
(hl(x, w), ...
h2A _,(x,
I
w».
It would be possible to further simplify equations (3) by going into the structure of the functions h, but for our purposes this will not be necessary. Consider now an exceptional point p of M' C ct. Thus in a neighborhood of p the equations of M are of the fonn (4)
Zl =
fl(xl , •.•
I
x,)
Z, =
f,(x l , .•.
I
x.).
365
4
ERRETT BISHOP
That p is exceptional means that the Jacobian determinant
vanishes when x = (x, , ... , x k ) o. If J does not vanish to second order at x = 0, we say that p is not exceptionally exceptional. In this case an analysis similar to the one just given shows that we can choose analytic coordinates 2, , . . . , 2k for some neighborhood of p in C' and differentiable coordinates x, , ... , Xk-2 , U, v for some neighborhood of p in M such that equations (4) become (5)
2.-, = 2k =
U
+ iv
g(x, , ...
,X.- 2 ,U,
v)
where h, , ... , h'- 2 , g all vanish to second order at x, = Xk - 2 = U = v = o. Of course g is complex-valued and h, , ... , h'- 2 are real-valued. With conventions similar to those used previously, these equations become (6)
2 =
(x, w, g(x, w))
+ (ih(x, w), 0, 0),
+
where W = U iv. "Ve may thus write g(x, w)
=
L
aijXiX,
+L
bix,w
+ L C,XiW + Q + :\.(x, w),
where Q is a quadratic form in wand w, the coefficients ad , bi , Ci , are complex, and :\. vanibhcs to third order at x = W = O. Thus Q has the fonn
Q = aw'
+ (:lw + -yWU). 2
Assume that l{:ll ~ t 1"11 and "I ~ o. Replace w by w - L~:::~ lIi2, , where the II. are complex constants. This is a coordinate change on A-f. Replace 2 k _, by 2.-, L~:~ lIiZ, , a linear coordinate change on Ck • These coordinate transformations preserve the form of equations (6). We choose the constants II; so that in the new coordinates the constants c, will all vanish. This is possible because of the assumptions on {:l and "I, as can be easily checked. Now replace 0 is a I'ml parameter, then et is the boundary value of the mapping function of D onto the interior of another ellipse of the family. Defille (XO, WO) = (0, et) Define Xl = - Th(O, ct). Define WI = ct S til , \\ here til is the boundary value of an analytic function on D which vanishes at S = O. To define til we use the fact that z.(O, WI) should be almost the boundary value of an analytic function. Now we have in general, if ti is small,
+
(10)
z.(x, w
+ rti) ~ (3(w" + u/ + 2wrti + 2ws~) + W1[' + wrti + wrti + A(x, w),
370
DIFFERENTIABLE MANIFOLDS
where "~" means "is approximately equal to". (This notion is not precise; it is only for motivation.) We therefore define A to make the right side of (10) the boundary value of an analytic function. In our particular case this means we want (11)
to be the boundary value of an analytic function, since, by the choice of t, f3(t 2 + e) + tl is a constant. Since f3 < t, the winding number of
2f3tr
+ lr
is zero. Therefore there exists a differentiable function "I on DUE which is analytic on D, with T" 0 for all rED V E, such that
"1m
arg "I Let 0 = 'Y- 1 (2f3tr
+
=
arg (2f3tr
+ lr)·
oCr) > 0 for r
in, so that ech.1 1
E
E. Thus we want
+ eO'Y.:ll + X(O, et)
to be the boundary value of an analytic function. If we write X = Xl + iX 2 and subtract off - TX 2 (0, et) + iA2(0, et), we see that this is equivalent to the real function (12) being the boundary value of an analytic function. Therefore the function (12) is a constant C1 . Hence 2 Re 'Y.:lI
=
(eo)-I(C 1
-
XI(O, et) - TA2(O, et»
so that symbolically (13)
where C I is chosen to make the harmonic function whose boundary values are O-I(CI - AI(O, et) - TA2(O, et»
vanish at
r=
O. Thus since 0
>
0,
We now choose a sequence (XO, WO), (x\ WI), •.. , (Xi, Wi), ••• by induction, (XO, WO) and (x\ WI) already having been chosen. Assuming (Xi, Wi) to have been chosen we take ~ (14)
and take (15)
371
lU
ERRETT BISHOP
where 6 i +, is the boundary value of an analytic function whiC'h vanishes at O. We shall then have
where (16)
Define
By the previous motivation, we choose 6 i +, so as to make (17)
{Je(2tt6 i +'
+ 2tr6
i +')
+ ett6 i + + el(6 i + Q«(6 i ) + X(x', w') 1
+l
the boundary value of an analytic function. Using the equation (16) and remarking that (17) already holds with j replaced by i-I, we see that (18)
{Je(2tr6,+,
+ 2t(6 + + etr6,+, + elt6,+, + Q(t6 i ) - Q«(6 i - + X(x i , w') i
l)
l)
must be the boundary value of an analytic function. given previously shows that (19)
6i + = 1
+ iT)[r'(C i + Q«(6 i ) + Q(t6 i -
(2e1r'(l
+1
-
-
X(X'-I, Wi-I)
The same analysis as
+ XI(X i - Wi-I) TX.(x i , Wi) + TX2 (x Wi-I))].
XI(x i , Wi) l)
-
l ,
j-I ,
Again,
From the definitions and the above estimates we have
since t has been
fixed and therefore Iltll is a Ilxill
(21)
KII(xf-l -
constant. Also
~ KII(xi-l, wj-l)W
for
j ~ 1,
Ilxi - xi-III ~ Xi-I, Wi-I _ W i - 2) 11(II(xi-l, Wi-I) II + lI(x H , w i - 2)i/) IIAdl ~ 6- KIIA(O, 6t)11 ~ Kl, II(x', Wi) - (XO, wO)11 = Ilx'll + 116 11 ~ Ke 2• I
1
Also
372
for
j ~ 2,
DIFFERENTIABLE MANIFOLDS
if e :5 1, as we now assume. Also, by (19) and (20)
Ilw i +1 - will = II~i+111 :5 e-IK(II~(xi, U'i) - ~(Xi-l, wi-I)II + IIQ(t~i) - Q(t~i-I)II) :5 e-IJ( II(x i - Xi-I, W' - Wi-I)II (11(x i , wi)11 + II(xi-J, Wi- ')11)2 + e-IK Ilw i - wi-III (11~ill + II~i-IIi) for j ~ 1. Now fix K, not to be increased again. We may take K > 2. Consider only
(22)
values of e which are :5 (24K 3)-I. We shall show inductively that (23) and (24)
We have proved (23) and (24) for J = o. Assume them to have been proven for values less than a given j. Then by (21) and (22) and e ~ (24K 3)-1 we have, since K > 2 and, by (23), II(x i, wi)11 + II(X i- 1 , Wi- 1 )11 ~ Ke,
II(X i +l , WH1 ) ~ (K
+K
+ e-1K Ilw
2
-
(Xi,
)II(x i
i _
-
wi)11
Ilx i +1 - xiii
=
Xi-I, Wi
Wi-III
(11~ill
+ Ilwi +1 - will Wi-')II(II(x i , wi)11 + II(X i - wi-l)IJ) 1,
-
+
II~i-lll)
as desired. Also,
JI(X i +l , Wi+I)11 ~ II(x' , w')11 +
t
k-I
II(x·+
I ,
Wk+l) -
(x·,
wk)11
and ;+1
II~i+lll ~
L
Ilwk
-
w·-III
~ 2Ke 2 •
• -1
as desired. By (24), the sequence I (Xi, Wi) I converges in norm to a function (x·, w·) on E. By (23), II(x·, w·)11 ~ Ke/2. Since each Wi is the boundary value of an analytic function, w· is the boundary value of an analytic functio~on D, with w(O) = 0, w'(O) = e. Since (17) is the boundary value of an analytic function, for all j, and since this expression when added to the constant (je 2 (e 2
+ n + e el 2
373
ERRETT BISHOP
converges to (25) as j ~ co, we see that (25) is the boundary value of an analytic function z" on D. Similarly, for 1 ~ i ~ k - 2, x7
+ ihi(x"', w"')
is the boundary value of an analytic function Zi on D. Finally, we define the analytic function Zk-l on D to be w. Thus (Zl , .•. , Zk) is an analytic map from D to C\ whose boundary value is the map from E into M represented by (.1:=, w=) in our coordinates. lVe have Re Zi (0) = for ~ i ~ k - 2, since x~ = - Thi(x=, w=). Also
° °
Zk(O)
=
0, zt(O) = e,
and
/I (x'" , w"'). -
(0, et) II ~
'"
L ri Ki
S 2Ke2 •
Now to get other such systems of curves, do the same construction at the other exceptional points of M near p. These of course are found from the equation
J(Z' , ...
'Zk) =
0,
Xl , . , . ,Xk
and constitute a (k - 2)-dimensional regular submanifold of some neighborhood p in M. Thus in all we construct a system of simple closed curves in a neighborhood of pin M, each of which bounds an analytic disk, dependent on k - 2 parameters in addition to the parameter e. It is to be expected that these curves are disjoint, that they fill up some neighborhood of p, and that there are no other simple closed curves in this neighborhood which bound analytic disks. It would be of interest to simplify equations (3) and (8) to higher order, and to examine the terms up to that order, throwing the re&t away. If for these terms we could compute closely enough the structure of the constructed analytic disks, then we might hope to get a very good hold on the structure of the analytic disks for the original problem, perhaps enough to tell whether they fill up some open set for instance. We do this in a particular instance in §5 below 4. Exceptional points of a two sphere in C2 • Consider now a differentiable two sphere M C C2, with a given orientation. Let G be the Grassmann manifold of all oriented two-dimensional real-linear subspaces of C2 • We assume that the exceptional points of M are not exceptionally exceptional, so that near such an exceptional point the equations of M can be given the form (8), with {3 'iF- t. Of course there is no x-variable in equations (8) for k = 2. By mapping each point of M into its oriented tangent plane, we get a differentiable map t:M~G.
374
DIFFERENTIABLE MANIFOLDS
Let z, and z. be coordinates in C', and let e, , e. , ea , e4 be unit vectors along the Re z, ,1m 2, , Re 2. , 1m z. axes respectively. If P is any element of G, we choose 1', and 1'. to be orthogonal unit vectors in P such that 1', /\ 1'. determines the orientation of P. Then
The numbers aij are Plucker coordinates for P. They are independent of the choice of 1', and 1'. and satisfy the relations ~
ai/
=
l.
Conversely, any set of numbers aij satisfying these relations are Plucker coordinates of one and only one element of G. Thus G is homeomorphic to the subset of R6 consisting of all solutions of these equations. VIe introduce a linear coordinate change
(26)
after which G becomes homeomorphic to the subset of R6 consisting of all solutions to the equations x~
+ x~ + x~
= y~
+ y~ + y~
= 1,
the product of unit spheres 8, in x-space and 8. in y-space. We give 8, and 8 2 the orientations induced by the ambient space R a• This orients G also. We now make use of the complex structure of C2 , in particular of multiplication by i We have
If P t G is a complex-linear variety in C2 , whose orientation is induced by its complex structure, we say that P E H. Thus H C G, and a point p in M is exceptional if and only if t(p) E H or -t(p) E H, where -t(p) denotes t(p) with orientation reversed. In case P E H we may choose 1', and 1'2 with 1'2 = iI', , so that if
with complex ex and fl, then
From this we may compute the Plucker coordinates aij of P in terms of ex and o. Thus H is the subset of G
fl, and it turns out that x, = 1, x. = 0, Xa =
375
whose coordinates satisfy the additional relations
x, = I,
X2
= 0,
X3
o.
=
Now consider an exceptional point p of M, such that the orientation M induces on the tangent plane P is the orientation of P in its analytic structure, so that P E H. Without loss of generality we may assume that in some neighborhood of p the equations of 111 are of the form Z, = Z2
!,(u, v)
+ iv = W (3(w' + 1(/) + WW + A(W) ,
= u
= Mu, v) =
where u and v are oriented coordinates for 111 in a neighborhood of p and where A vanishes to third order at u = v = O. By the above, we take {3 ~ 0, {3 ~ ~. At a point q in M whose coordinates are (u, v) the vectors (f,., 12.) = (1, (2
+ 4(3)u + A,,)
and (f,,,
I •.) =
(i, (2 - 4(3)v
+ A.)
span the tangent plane p. to M at q. In terms of real coordinates, these vectors are
and e2
+ (2
- 4f3)ve3
+ A"e + Ao,e•. 3
Now since A, .. , A." , At. , Az. all vanish to second order at u within second order terms in u and v, these vectors equal ii,
= e,
il2
= e2
= 0 and v
=
0, to
+ (2 + 4(3)uea + (2 - 4(3)te
J •
Also, iI] and 112 are orthonormal to within second order terms, so that, to within second order terms, to find the coordinates of p. E G we look at ii,
/\ II.
=
e, /\ e2
+ (2
- 4(3)ve , /\ e3 -
(2
+ 4(3)ue2
/\ ea.
By (26), to within second order terms we have for the coordinates of p. 1
1
- (2
+ 4f3)u
- (2 - 4(3)v
(2
Y3
+ 4(3)u
(2 - 4(3)v.
We are interested in the intersection number of H = (1, 0, 0) X S2 with t(M). Now near the point (1, 0, 0) X (1,0,0) of G we can take X2 , Xa , Y2 , Y3 as oriented coordinates for G. To first order M maps into the subspace whose
376
15
DIFFERENTIABLE MANIFOLDS
equations are x. = Y. , X3 = Y3. Also x. , X3 are coordinates for the oriented image if fj < !, whereas X3 , x. are coordinates for the oriented image if fj > !. Thus if fj < !, the intersection number of Hand t(M) at (1, 0, 0) X (1, 0, 0) is +1, while if fj > ! this number is -1. By the theorem of Chern and Spanier [2], the sum of the intersection numbers of t(M) and H is + 1. Thus among those exceptional points of M which are analytically oriented there is one more point of the first kind than of the second. By applying the same argument to M in the reversed orientation, we see that among those exceptional points of M which are anti-analytically oriented there is one more point of the first kind than of the second. In particular, M has at least two exceptional points of the first kind.
5. The structure of a manifold M4 C C3 • It is thought that a manifold M",+l C C" has, in general, the property that holomorphic functions in a neighborhood of M extend to be holomorphic in some fixed open set. We show that this is indeed the case if n = 3. The techniques we use could probably be extended to treat the general case. For the case of M4 C C3 , equations (3) become
Z2 =
x.
Za =
U
+ ih.(x
l ,
X., U,
v)
+ iv.
The second order terms of hi are of the form
L: a,jX,Xj + L: aix,w + L: b,xiW + Q(w, w), is quadratic in wand w. By subtracting L: a"Z,Zj
where Q(w, w) we may assume a'i
=
from
Zl ,
0, and by adding
L: (b,
- a,)z,w
L: a,xiw + L: b,x,w is real. Xl + L: a,x,w + L: b,x,w,
to Zl ,we may assume that
we may assume that a, = bi = 0, as well as terms in hi are Q(w, w)
= aw'
aii
Thus if we replace Xl by
= 0, so that the second order
+ bww + ow'.
+
By adding (c - a)w' to Zl , we may make a = c. Thus aw 2 (Re b)w~+ ow 2 is real. If we replace Xl by the sum of Xl and this real quantity, we may assume that a = 0 = Re b = 0. We assume that M4 is not degenerate at 0, in the sense that 1m b ~ 0. Dividing Zl by 1m b, we get Q(w, w)
= iww,
377
hnn;.... .l.l
.tt.lt;HUJ:"
so that x,
+ iww + terms of higher order.
X2
+ iww + terms of higher order.
Z,
Similarly, =
Z2
Replacing
by
Z2
Z2
=
The third order terms in
L
aijkX,XjXk
where aijk
=
new, aij
+L
we may assume that
Z, ,
Z2 -
+
X2
terms of order 3 or higher.
wiII be of the form
Z2
+ L bijXiXiW + L O;XiWW + L
ai,XiX,W
dixjW 2
+L
ejxjw 2
+
U(w, w),
w) is a cubic form in wand W.
= bi; = d j =
If we replace
Z2
ej
=
As before, we may assume that \Ve may also take Cj = iCTj ,where CTj is real.
o.
by
and replace x. by
we obtain in addition
Cj
= O. Thus the cubic terms in = aw3 + (bw + cw)ww = d = 0, so that
Q(w, w)
As before, we may take a Q(w, w)
Z2
+ dw
are 3•
+ cw)ww = i(au + flv)w?iJ, We assume lal + Ifll ~ O. Then there exists a
= 1m (bw
where a and fl are real. plex constant jJ. such that if we replace w by Q(w, w)
jJ.W,
com-
we get
= ivww.
The equations of M in a neighborhood of 0 have now been given the form Z,
=
x,
+ i(ww) + ih,(x, w)
Z2
=
X2
+ iv(ww) + ih2(x, w)
Z3
=
W
=
U
+ iv,
where h, and h2 are real functions which vanish respectively to orders 3 and 4 at o. We know from the analysis of §2 that for all sufficiently small real parameters a, fl, b, r, with r > 0 and lal + 1,81 + Ibl < r, there will exist an analytic
378
DIFFERENTIABLE MANIFOLDS
disk D in Ca, which is bounded by a simple closed curve 'Y in M, and which is the image of Ir = s + it : Irl < 1} = E under an analytic map
r ---+ Cz,W, Z2W, zaW) of E into C with 3,
ZaCr) = rs
+ Crt -
Re (Zl(O» = a,
b)i,
Re (Z2(O» = fJ.
To study the dependence of this analytic disk on the parameters Q b r we _ a, fJ", first consider the case h1 = h2 = O. This defines a manifold M which approximates M, whose equations are
Z2 =
X2
Za =
U
+ ivww
+ iv.
Now we can compute Zl , Z2 , and Zs explicitly as functions of eters a, fJ, b, r. Indeed, on Irl = 1 we have u2
+ (v + b)2
r and
the param-
= r2
so that ww
=
u 2 + v2
=
r2 - 2vb - b2.
Thus ZlW
=
2ub
a -
+ i(r2
- 2vb - b2).
Also, vww
= v(r2 - 2vb - b2) = vCr2 - b2) - 2bv2 = v(r2 _ b2) _ 2b 2v 2 + r2 - u 2 - Cv
+W
2
= vCr2 - b') - b[v' - u' Thus Z2W
+ u(r2 - b 2buv + 2ub' + i[v(r2 - b') - b(v u 2 + r' - 2vb fJ + u(r2 + b2) - 2buv + i[b(u2 + v2) + (v - b)(r2 - 2vb - b
= fJ
2
)
-
2
=
-
2 )].
Of course, Z2
=
U
+ iv.
379
b2 )]
+ r2
- 2vb - b'].
.tuonur
.r...lt.lt.L~ ~
Thus the curve 'Y bounding the disk D is given by the equations Xl = a -
on
ItI =
1.
2ub,
x2
u
=
=
fj
+ u(r' + b
r8, V
=
rt -
2)
-
2uvb,
b
Defining norms as in §2, it follows that
II(x, w)11 :::;
KT,
because of the restrictions on the parameters imposed above. We now turn to our original problem of finding Zl , Z. , Z3 as functions of the parameters. Equivalently, we find Xl and x. as functions of t, on It I = 1. To do this we use the method of §2, modified in that we start with x~ = Xl , x~ = X. as our first approximation. As before, we define X i +l = X ThCx', w) for all positive integers j, and see that Xi converges in nonn to the desired function X of r as j ~ co. Also, from the fact that h vanishes to third order at 0, by an analysis similar to those in §2 we see that
Ilxi+lll:::; ilxll + KIICx i , w)W From this it easily follows that
Ilxill :::; KIICx, w)11 < To , where ro is a positive constant which is independent of j. Hence, K IICx, w)ll· Consider now a second set a', fj', b', r' of parameter values, subject also to the conditions la'i + 113'1 + WI < r', r' < To. Let fi, y;, and y be the values of the quantities x, x;, and x for these parameter values. We reserve then x, x;, and x as quantities corresponding to the original parameter values a, 13, b, r. We then have for all r
Ilxll :::;
llx;+' - yi+lll :::; Ilx - fill + Ilh(xi , w) - h(yi, w)11 < Ilx - fill + Kllxi - yillCllxil1 + Ily'll + Ilwli)" < Ilx - fill + Kllxi - yillCllxl1 + lifill + Ilwl/)' :::; Ilx - fill + K(r + r')'llx i - yill· From this it follows that
Ilxi
+!
-
yi+1/l:::; Kllx- fi/l,
independently of the value of j, if ro is sufficiently small. Also compute
/I(X;+l -
Xl) -
(y;+l - fi,)/I :::; Ilh1(:z/, w) - h1(y;, w)11 :::; K /Ix; - yi/l (/lx;1I + Ilyi/l + /lwlD 3 :::; K(r + T,)3 /Ix; - yj/l :::; K(T + r,)2 Ilx - fi/l.
380
DIFFERENTIABLE MANIFOLDS
Passing to the limit, we obtain II(x, -
(27)
:fl )
-
YI)II ::::; K(r
(YI -
+ r,)2Ii:f - yll ::::;
K(r
+ r,)2 X,
where
x
la - a'i + liS -
=
is' I + (r
+ r')lb
- b'l
+ (r + r')lr - r'l.
Similarly,
II(X2 -
(28)
(Y2 -
:£2) -
We shall now show that z = (ZI , eters a, f3, b, r, u, V on the region
Y2) II
Z2 , Z3)
:::;;
+ r,)3 X.
Ktr
is a one-to-one function of the param-
lal + 1f31 + Ibl < r < ro,u + 2
(v
+ b)2 < !l,
if ro is sufficiently small. To this end, let z correspond to the parameter values a, f3, b, r, u, v and Z to the values a', f3', b', r', u', v'. Assume that z = Z. Then, since Z3 = U + iv, Z3 = U' + iv', we have u = u', v = v'. From (27) and (28) we have I(z, - z,) - (Z, - 2,) I :::;; K(r + r,)2X, l(z2 - Z2) -
Since
Z
=
(Z2 -
2 2) I :::;;
K(r
+ r,)3X.
Z, this gives
(29)
Iz, - 2,1 :::;;
K(r
+ r,)2x,
221 :::;;
K(r
+ r,)3 X•
IZ2 -
Substituting into (29) the values for the real parts of z and Z computed above, we obtain (30)
l(r2 - 2vb -
+ (v
Ib(u 2 + v2)
(31)
b2) - (r,2 - 2vb' - b,2) I :::;; K(r
- b)(r2 - 2vb - b2)
- [b'(u 2 + v2)
From (30) and the inequality
I(v -
+ r,)2 X,
+ (v
Iv -
- b')(r,2 - 2vb' -
+ r')~X.
b'l :::;; Kr' we obtain
b)(r2 - 2vb - b2) - (v - b')(r,2 - 2vb' -
+ (b
b,2)11 :::;; K(r
- b')(r2 - 2vb -
b'~
b2)1
=
- Cr,2 - 2vb' - b,2) I :::;; K(r
Iv - b'll(r2
-
2vb - b2)
+ r,)3 X•
Combined with (31), this gives (I(b - b')(u
+
(v
+
b)' - r2)
I=
I(b - b')(u2 + v2 - (r2 - 2vb - b2)) I :::;; K(r
Since u
+
(v
+ b)2
:::;; !r2, this gives
Ib - b'lr2 5 K(r
381
+ r,)3 X•
+ r,)3 X•
ERRETT BISHOP
Similarly, Ib - b'lr"
~
K(r
+ r,)3 X.
Therefore (32)
Ib - b'l ~ K(r
+ r')X.
Combined with (30), this gives Ir" - r'"1 ~ K(r
+ r'lX,
or (33)
Ir - r'l ::; K(r
+ r')"A.
In addition to (30) and (31), we obtain the following equations by substituting the values of the imaginary parts of z and Z into (29).
I(a -
(34) (35)
1(,8
+ u(r" + b
2)
2ub) - (a' - 2ub') I ::; K(r
+ u(r'" + b'")
2uvb) - (,8'
-
+ r,)2X, - 2uvb') I ~ K(r
+ r,)3"A.
Because of (32), (34) gives (36)
Because of (32) and (33), (35) gives 1,8 - ,8' I ~ K(r
+ r,)3X.
Thus, from the definition of A, we have
x=
la -
a'i +
1,8 - ,8'1
+ (r + r')lb
- b'l
+ (r + r') 1("" -
r') I ::; K(r
+ r')"X.
If we take To so small that 4r~K < 1, this gives a contradiction. Hence we see that Z is a one-one continuous function of the parameters on the region R:
lal
+
1,81
+
Ibl
< T < ro, u" + (v
+ W < !r2.
By the theorem on invariance of domain, R is mapped homeomorphically into an open set U C C3 • It is clear that the image point converges to 0 as r --? o. We now show that for each neighborhood W of 0 in M there exists a neighborhood V of 0 in C3 with the following property. Let f be an analytic function in an open subset Wo of with W C Wo. Let (; be that component of U (\ Wo (\ V whose closure contains o. Then f admits an analytic extension from (; to U (\ V. To define V, let U o consist of all points in U which lie on a disk D whose parameters a, ,8, b, r lie in R and which has the property that the boundary curve "I' of the disk D' with parameter values ai, ,8', b', r' lies in W whenever r' ::; r. Thus U o is an open subset of U. Let U 1 be that component of U o
Ca,
382
DIFFERENTIABLE MANIFOLDS
whose closure contains o. Since U 1 is unique, and is open in U, there exists a neighborhood V of 0 in Cl such that un V = U 1 • We show next that if D is one of our disks, whose parameters ex, (j, b, T have the property used in defining Uo , f can be extended to an analytic function in some neighborhood of D. To see this, let n be the envelope of holomorphy of W o , so that Wo C nand n is a domain spread over Cl • By the so-called Kontinuitatssatz, the boundary curve 'Y of D, which lies in Wo , must bound an analytic disk jj in n. Of course jj projects into D under the natural projection of n into C3 • Therefore the analytic extension of f to n gives rise to an analytic extension f n of f to a neighborhood of D in Cl . Define the function g on U I by taking the value of g at a point z of such a disk D to be fnCz), i.e. to be the analytic extension to z over D of the values of t on the boundary curve 'Y of D. It is clear then that gCz') = fnCz) for all z' in U 1 which are sufficiently near to z. Hence g is analytic in U 1 = U n V. It only remains to show that g = f on V, or, equivalently, on some open subset of V. To this end, choose the disk D discussed above to have the additional property D C Wo , and take the domain r of fn to be a subset of Wo. Then
so that
fir n V = fn/V = g/V, as was to be proved. REFERENCES 1. A. BROWDER, Cohomology of maximal ideal spaces, Bulletin of the American Mathematical Society, vol. 67(1961), pp. 515-516. 2. S. S. CHERN AND E. SPANIER, A theorem on orientable surfaces in four-dimensional space, Commentarii Mathematici Helvetici, vol. 25(1951), pp. 205-209. 3. H. LEWY, On hulls of holomorphy, CommunicatioIlll on Pure and Applied Mathematics, vol. 13(1960), pp. 587-591. 4. H. LEWY, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Annals of Mathematics, vol. 64(1956), pp. 514-522.
UNIVERSITY OF CALIFORNIA, BERKELEY
383
CONSTRUCTIVE METHODS IN THE THEORY OF BANACH ALGEBRAS ERRETT BISHOP University of Californw, Berkeley The method of Banach algebras, like so many methods of functional analysis, gives with one hand and takes away with the other. It gives a powerful technique for existence proofs in certain parts of analysis, but it takes away the possibility of constructing the objects asserted to exist. To explain adequately the precise places at which this nonconstructivity intervenes would require a long journey into the philosophy of intuitionist or constructive mathematics. However, the point can be put across by an example. In the proof of the HahnBanach theorem one considers the distance d of an element x of a Banach space B to a closed subspace F' of B, d
=
inf{y
E
F'. p(x,y)1.
In general there is no way to program a computer to calculate d, in other words there is no finite routine method to compute a rational approximation to d to within a prescribed limit of accur-
acy. The assertion that d exists lacks empirical, or constructive, or numerical, content. The Hahn- Banach theorem, which relies on this assertion, therefore also lacks constructive content, and it becomes necessary, if one believes it important that mathematics have empirical Significance, to look for a satisfactory constructive version of the Hahn-Banach theorem. The following result is one possibility. THEOREM 1. Let B be a separable Banach space, F' a closed subspace of B, and ¢ a continuous linear functional on F'. If the distance from an arbitrary point of B to F' is computable then for every f > 0 there exists a continuous linear functiorw.l rp on B with rp/F' = ¢ and I rp II S II ¢ II + f. In view of this result it is natural to ask when the distance from a point x of a Banach space B to a closed subspace F' of B is computable. For instance, consider a compact subset K of the complex plane C. Let B be the Banach space C(K) of all continuous complexvalued functions on K. Let F' be the closed subspace of B consisting 343
384
344
E.
BISHOP
of all functions in C (K) which are uniformly approximable on K by polynomials in the complex variable z. Is the distance from an arbitrary element f of B to F' computable? It is easy to see that the distance d n from f to the set of polynomials of degree n is computable, so that an equivalent question is whether the limit of the monotone decreasing sequence Idnl of non-negative real numbers is computable. To compute this limit we must have some method of knowing how large to take n in order to make all of the subsequent d m lie within a prescribed distance of dna To find such a method is an important mathematical problem which does not even exist within the framework of formal mathematics. We now turn our attention to a separable commutative Banach algebra a with unit 1. The spectrum 1. of a consists of all continuous multiplicative linear functionals ¢ on with ¢m = I, and is a closed subset of the unit sphere S of B*. Since S is compact in the weak 4< topology, it follows that 1. is compact. This argument is not valid in the constructive framework, and indeed it is possible to give examples to show that 1. need not be compact in the constructive meaning of the word. Before giving such an example we remark that the term compact will only be applied to metric spaces (it is easy to put a metric on S), and it will mean complete and totally bounded. For an example in which 1. is not totally bounded, and therefore not compact, let
a,
X={z·lzl=I!,
a be the sub algebra of C (X) generated by the two functions .... z and z .... 1: 8n 2- n Z-, where 18711 is a sequence of integers, such
and let z
<Xl
n =1
that 871 = 0 or 871 = I for each n and such that it is not known whether there exists a value of n for which 871 = 1. Now in case 871 = 0 for all n then is the subalgebra of C(X) generated by z .... z, and therefore consists of all functions on X which can be extended to be continuous on {z : I z I ~ I} and analytic on {z . \ z I < I}. In this case l is homeomorphic to {z.1 z I 5 I}. In case 871 = 1 for some n then a = C (X) and 1. is homeomorphic to X. Since we can't preclude either of these possibilities the space 1. is not totally bounded. In general we do have the result that 1. is the intersection of a decreasing sequence of compact subsets of S. Gelfand defines the spectral norm of an element x of a by
a
II x liE
=
max {I ¢(x) I : ¢
E
1.}.
Since 1. is not in general compact this maximum need not Gelfand shows that 1
IlxilE
=
lim 71 .... ""
385
Ilxnll" ,
e~.
345
CONSTRUCTIVE METHODS
but in general this limit will not exist. The equation however remains valid in that one can prove: 1
THEOREM 2. II x Ill: exists if and only if lim I xn lin exists, and if n->oo
they exist they are equal.
Perhaps the core of the theory of Banach algebras lies in the techniques for "constructing" maximal ideals, specifically in the result that every ideal is included in a maximal ideal. Involving as it does a heavy reliance on the principle of the excluded middle, the proof of this result and indeed the result itself are not constructively valid. Fortunately there exists a good constructive substitute. THEOREM 3. Let xl' ... 'Xk' Y be elements of a Banach algebra let G be a finite-dimensional compact convex subset of a, and let rand s be real numbers with 0 < r < 1, 0 < S < 1. Then there exists a finite-dimensional compact convex subset H of a such that whenever al' ... ,ak are complex numbers with
a,
then there e:dsts a complex number b with
Theorem 3 can be applied in the usual way to give the result that are elements of a such that there exists f > 0 with I q,{x 1) I + ••• + I q,(Xn ) I ~ f for all ¢ in I then there exist Yl' ... 'Y n in with Y1 xI + '" + Yn xn = 1. Since we are speaking in a constructive framework, this means that the proof affords a method of computing the elements Y I, ...• Yll. This is the advantage of Theorem 3 over Gelfand's theorem that every ideal is contained in a maximal ideal. The disadvantages are obvious. if
xl ••..• Xn
a.
386
AN UPCROSSING INEQUALITY WITH APPLICATIONS
Errett Bishop The theorem of Lebesgue that a function of bounded variation has a derivative almost everywhere, the ergodic theorem of l3irkhoff, and the martingale theorem have the common property of bemg nonconstructive; they are not true in mtuitionist mathematics. (See Brouwer [1] for an example of a function of bounded variation that does not have a derivative almost everywhere.) This paper developed from an attempt to save the phenomena. The martingale theorem has in fact already been saved: the upcrossing inequality of Doob [2, p. 314l, which almost trivially implies the martingale theorem, is (with slight modifications) constructively valid. Moreover, Doob's inequality contains important information that can't be gotten from the martingale theorem itself. Thus to a formalist Doob's me quality is interestmg because it improves the martingale theorem, and to a constructivist it has the additional ment of making empirical sense. In this paper we prove an upcrossing mequality (Theorem 1) that stands in the same relation to Lebesgue's theorem as Doob's inequality stands to the martingale theorem: it is a constructively valid inequality which in the formal system constitutes a generalization of Lebesgue's theorem. By a very simple argument, Theorem 1 leads to another constructively valid inequality (Theorem 2), which in turn constitutes in the formal system a generali:lation of Birkhoff's ergodic theorem. Fmally, by an argument unfortunately not so simple, we derive Doob's upcrossing mequality from Theorem 1.
The theorem of Lebesgue, Birkhoff's ergodic theorem, and the martingale theorem are thus consequences of a smgle inequality (Theorem 1); regarded in this way, these theorems gam both depth and empirical valiruty. Although, as we have indicated, it is possible to develop our results within a strictly constructive framework, at present this would lead too far afield; therefore We stay within the formal system. Definition 1. For each x in R 20 we let Xl be the abscissa and x2. the ordinate of x, so that x = (XI, xz). For each Ct in R and x in R Z, we define
Definition 2. If x € RZ, Ct, f3 € R, and RZ are said to have property P(x, Ct, (3) lf
Ct
< f3, then pOints u I, vi, .•. , uk, v k in
I I Z 20 k k xI a OI(x I ), or
xz - {3xI
< yzI
1
Xz - OIxl •
Similarly I
- {3YI •
Subtraction gives the mequality ({3 - OI)xI
>
xi - yi + ({3 - a)xI.
Therefore
o
of3(ym+l) will be called case 2, and the case oa(x m ) < U a(x m + ), 0 f3(ym) ~ 0 f3(ym+l) will be called case 3. Consider next case 2. We defme 0 = r - {x m + 1 , ym+l}. Since r is elementary, to verify that 0 is elementary we need only note that
The power of 0 is n - 1. By the inductive hypothesis,
5wet, ~, 0,
O)d! '" {Jl - 0)-1
l;~ (yi - xi) - (y~H - X~H)j
w.. e
Wish to compare w(x, a, f3, r) with w(x, a, f3, 0). To this end, choose elements u . v , "', uk, v k in r having property P(x, a, (3) with w(x, a, f3, r) = k. (a) If u i '" x m +1 and v j '" ym+l for all i, then w(x, a, f3, 0) - k. (b) If u i '" x m +! for all i but v j = ym+l for some value of i, replace ~ by VJ = ym. Then " and
389
Thus the pOints u l w(x, a, fl, 0) = k.
iii
(c) If vi
= xm+Z.
,
vI, ••• , vi, ..• , v k in
*
have property P(x, a,
0
= xm+1
ym+l for all j, but u i Then
/H. Therefore
for some value of j, replace ui by
and Thus the pOints u I, vi, .•. , iii, ... , v k in 0 have property P(x, a, fJ). Therefore w(x, a, fJ, 0) = k. (d) If u i = x m + l , vi = ym+l for some j, then w(x, a, fJ, 0) ~ k - I, w(x, a, fJ, r m-t-I) ~ 1. Therefore w(x, a, fJ,
r) = k
~ w(x, a,
fJ, 0) + w(x, a, fJ, r m+I)'
This last inequality therefore holds in all cases. Takmg maxima, we see that
wet,
a, fl,
r)
~ wet, a,
fJ, 0) +
wet,
a, fl,
r m+l)'
Integration now gives the inequality (.):
5wet, ., ~, r)dt ~ (~- .)-, t~ (yl- xl) - (y;:,H - x;:'H)! n
m+l) -_ + (~f.l - a )-1 (m+1 Yl - Xz
(f.l ~
-
01)-1" (i i) L.J Yl - Xl . i=l
In the final case, case 3, we obtain 0 from r by leaving out the pomts xm and ym and replacing x m + 1 by the point ;em+l ,,(xf'+l, x~ + a (x?'-t-1 - x?,». Smce r is elementary, to verIfy that 0 is elementary we need only note that ym+1 _ im+1 Z
l
= ym+1
_ xm _
0I(X m + 1
Z
1
= YZ+1
- xZ'+l +
l
U
- x;n) 1
a (xm+l ) -
Ua
(xm)
>
yZ'-t-1 - Xz + 1
>
0
and (-m+l) .2:uaX ( m) -uOIx (-m+1) ( m-I) -uax uaY (m+1 -Xlm) -OIxlm+1} Xlm -aXlm - {m Xl +OIxl
o.
By the inducbve hypothesis,
5
wet, a, fJ, O)dt ~ (fJ - a)
= (fJ
- 01)-1
-11::-
i
i
m
m
m+1
i~ (Yl - XZ) - (Yl - Xl) + Xl
_m+l! - Xl
l.~ (y~ - X~) - (Yz - xl' - xl'+! + xl' + a(xl'+! - Xl'»! >=1
390
AN UPCROSSING INEQUALITY WITH APPLICATIONS n
= ({3
- a)-l L; (y~ - X~) - ({3 - a)-l (aa(yrn) - aa(Xm.+l». i=1
Consider any x in R 2. To compare w(x, a, (3, r) with w(x, a, (3, 0), choose pOints u 1 • vI, ... , uk, v k m r with property p(x, a, {3) such that w(x, a, (3, r) = k. Define "
iF
!
whenever u i
ui
=
'"
xtn and u i '" xm.+l ,
xrn+l whenever u i = xm or u i = xm+l.
Dehne vi = vi whenever vi", ym., and vi = yrn+l whenever vi = ym. The points iii are all in 0 and satisfy the inequalities
vi
and
" Now tii.::;; vi for all i, and vi < tii+l for 1'::;; i ~ n - 1, except when vi = ym and = xm+l. In this exceptional case, Xl < xl and
U 1 +1
a a(x)
Xz - aXl
>
a a(x rn + l ) = xz+ l - axr+l,
Subtraction gives the inequality «(3 -a)xl
> flyr
-axr+l+xz+l- yZ
({3 - a)x?, + a(x?, - x?,+l) + x~+l - y~ ({3 - a) xl" + a O'(xm+l) - a O'(ym.).
Therefore
Thus, unless Xl satisfies this inequality, the points iII, vI, ... , iIk, v k have property P{x, 0', {3), so that w(x, a, (3, 0) L. w(x, O!, f3, r). Therefore
wet,
0',
(3, O)?
wet,
O!, (3,
r),
except possibly when 0< x l - t < «(3 - O!)-I(aO!(yrn) - aa(x m + l », and in any case wet, 0, (3, 0)';:: wet, a, (3, r) - 1. Integration gives
n
~ «(3 - 0')-1
6
(Y~ - x~).
i =1
The inequality (*) has been shown to hold in all cases, and the lemma is Definition 5. If f is of bounded variation on [a, b}, we de"fme
391
pr~d.
ERRETT BISHOP
6
= sup
V+(f)
~.~ (f(Yi) - f(xi »: a .::; Xl < Y1 < Xz < yz < ... < xn < Yn ~ b I·
( 1=1
The following upcrossing inequality is our main result.
wet,
THEOREM 1. If f is of bounded variation and a a, (3, f) of t is integrable, and
Swet, where
Id
< {3,
then the function
a, {3, f)dt .::; ({3 - a)-l V+(f - aid),
is the identity function
X ~ X
from R to R.
Prooj. First consider the case where f is a continuous and piecewise linear function of bounded variation on a fmite interval [a, bJ. Here there exist real numbers a ~ r 1 < sl ~ r Z < Sz ~ ... ~ rn < sn ~ b such that f - a id is linear and increasmg on each of the intervals [ri' si] and is nonincreasing on each of the intervals [si' ri+1], [a, rd, [sn' b]. For 1 ~ i ~ n we define
Write
r= {x 1 ,y 1 , · •• ,x n,y n }. To see that r is elementary, note that
and
Lemma 1 imphes that
Swet,
n
O!, {3, r)dt .::; ({3 - O!r 1 L; i=l
(Y~
-
x~)
n
(f3 - O!r 1 L; (f(Si) - f(ri) - O!(Si - ri» = ({3 - O!r 1 V+(f - aid). i=1 To compare w(x, O!, (3, r) with w(x, O!, f3, f), choose points u l , vI, .. , uk, v k in the graph of f having property P(x, O!, (3) and such that w(x, O!, (3, f) = k. The set of points zi in the graph of f for which and is closed. Therefore there exists such a point z i for which zi is a maximum, and whenev:er z is on the graph of f and zi < zl ~ If we replace. each u ' by the point Zl, we reduce the argument to the case where aO!(z) > aO!(u ' )
aO!(z) :::- aO!(u i)
vt.
392
AN UPCROSSING INEQUALITY WITH APPLICATIONS
whenever z is on the graph of f and ui ment further to the case where
< Z I < vi.
Similarly, we reduce the argu-
and whenever Z is.on the graph of f and ul < zi < vi. Thus the derivative fro~ the right of f at u~ is greater than 01, and the derivative from the left of f at vi is greater than (3. It follows that there exist integers j = j(i) and m = m(i) with
v~
€
(r m' sm] .
Clearly, Sj .:::;; sm. Write iP '" yi '" ym. Then all t € (vi, Sm), we have the relations xj ,
>
a c/Z) - a OI(X)
a Oi(Vi }
-
< u II .:::;; u-II'
Since fl(t)
> (3 > 01
(] (3(v i ) - a (3(X) + «(3 - 01) (vi - XI)
a OI(X)
whene\ er z is in the graph of f and zi that
Ui .: :; VI'
€
(vi, Sm(i})' Therefore ui+l
for
>0
> Srn(i} ,
so
so that XI
_I < u_Il _< VI
Moreover,
and (] j3(vi )
'" (] 13 (yk)
> (] (3 (v i) > (] j3(x) •
Therefore the pOints iii, vI, ... , iik, yk of r have property P(x, 01, M. Thus Oi, 13, r} 2 k = w(x, 01, 13, f). Since this is true for all x,
w{x,
Swet,
Oi,
(3, f}dt
~
Swet,
Oi,
(3, r)dt
~
«(3 - 01)-1 v+(f - 01 id).
Next we consider the general case. Let {t n } be a dense sequence of points in the domain of f that includes all points of discontinuity of f. For each positive mteger n, let fn be the function whose domain is the smallest closed interval that contains the paints tl, "', tn' agrees with f at these points, and is linear on each complimentary interval. By the case already considered,
For each x in R2,
and lim w(x, 01, (3, f n )
n __
DO
393
= w(x,
01,
(3, f).
ERRETT BISHOP
Taking suprema, we see that the same inequalities hold for each real number t. Lebesgue's monotone convergence theorem now gives the desired inequality:
5wet,
=
a, {3, f)dt
lim n-"O
5wet,
Ct, {3,
r.,)dt .:::; ({3 -
Ct)-l
Y+(f - aid).
COROLLARY l. Let f be a function of bounded variation on R, and let a and {3 be real numbers (a < 13). For each t in R, let vet, a, (3, f) be the supremum of all positive integers k such that there exist points vi, u l , v l , ... , uk, v k in the graph off with
and
Then
5
vex, a, {3, f)dx
~
({3 - ar I Y+(f - aid).
Proof. It is clear that if Il is a suffiCiently small positive constant, then
vet,
Ct, (3,
f) ~ w((t - Ill, f(t) + Il), a, (3, f) ~ wet - Il Z , a, (3, f).
The inequality now follows from Theorem 1. COROLLARY 2. Let f be a function of bounded variation, and let a and {3
< (3)
(a
be real numbers. Then
5w((t,
f(t», a, {3, f)dt
~
({3 - a)-I Y(f),
where Y(f) is the total variation of f. Proof. If a L 0, Theorem 1 implies that
5
wet,
a,
If (3 ~ 0, then a ~ fore, by Theorem 1,
{3, f)dt
~
({3 - a)-I Y+(f - aid)
° also, and w((t, f(t», a,
(3, f) =
~
({3 - a)-IY+(f).
vet, -
(3, - a, - f), and there-
5
w((t, f(t», a, {3, f)dt ~ ({3 - a)-I y+(- f + a id) ~ ({3 - a)-l y+(- f) ~ ({3 - a)-l Y(f). There remains the case a
wet, a,
(3, f) ~
°
Clearly,
(3, f)
and
< < {3.
wet, 0,
wet,
Ct, (3,
f) ~ wet, a, 0, f).
From the above cases we also see that {3
5wet,
0, (3, f) dt
~
Y+(f),
-a
394
5
w((t, f(t», a, 0, f)dt ~ Y+(-f).
9
AN UPCROSSING INEQUALITY WITH APPLICATIONS Therefore (,3 - 0')
S
wrIt, fIt)), 0', {3, f) dt
~
{3
S
S
wet, 0, {3, f) dt - 0'
~ V+(f) + v+(- f)
wert, f(t», a, 0, f) dt
= V(f).
As our first application of Theorem 1, we show that a function of bounded variation has a derivative almost everywhere. Indeed, Theorem 1 and its corollaries can be regarded as a strong generalization of this fact. They assert tha.t the difference quotient of a function of bounded variation cannot oscillate too much. THEOREM (Lebesgue). A junction f of bounded variation on a closed interval [a. bJ Izas afinite derivative at almost all points x of [a, b].
?Yoof. Let S be the subset of [a, b] on which the derivative of f from the right does not exist. Then
U
S
{x
[a, b]: w(x, 0', {3, f)
f
""} ,
0',{3
where the union is taken over all pairs a, {3 of rational numbers with a < {3. By Theorem 1, w(x, 0', {3, f) < "" almost everywhere, so that S has measure O. Thus the derivative g of f from the right exists almost everywhere. Let T be the set of all x for which g(x) = +"". Then vex, 0, n, f) 2 1 for all x in T and all positive integers n. Thus the measure of T is at most
s
v (x, 0, n, f) dx
~
n -1 V+(f) ,
and this tends to 0 as n ---> "". Replacing f by - f, we see that the set of all x for which g(x) = -"" also has measure O. It follows that g is finite almost everywhere. Similarly, f has a finite derivative h from the left almost everywhere. Thus to each I: > 0 there corresponds a Ii > 0 and a measurable set U c [a, b] of measure at most I: with the property that when x f [a, b] - U, then
I
fey;
=~x)
- g(x)
Ifey; = ~(x) - hex) Thus, if x
Ig(x)
E
I< I
a at most N times if N is an upper bound for all integers k such that there exists a finite subsequence of length 2k whose terms are alternately a at most finitely many times, whereas constructively convergence is much the stronger statement. Thus it is not surprising that many classical convergence results that fail constructively find constructive substitutes in terms of upcrossing inequalities; in fact these inequalities usually constitute considerable extensions of the classical results, from the classical point of view For instance, one can give an upcrossing inequality for ergodic theory that not only implies (within the classical system) the Chacon-Ornstein and DunfordSchwartz ergodic theorems and various generalizations thereof, but has the merit of a relatively simple proof From the ergodic upcrossing inequality one can derive an upcrossing inequality that affords a constructive substitute for Lebesgue's theorem, and an upcrossing inequality that generalizes Doob's upcrossing inequality for martingales. The theory of separable normed linear spaces can be given a constructive basis. In particular such theorems as the Hahn-Banach
412
312
nOJlYYACOBbIE ,nOK.'1A.D:bI
HALF HOUR REPORTS
theorem, the separation theorem, the spectral theorem, the KreinMilman theorem, and various theorems on the forms of linear fundionals all have satisfactorv constructive versions. For instance, a bounded linear functional on Lp is induced by an element of Lq (where p-l + q-l = 1) if and only if it is normable This implies that Lq is not the complete dual of Lp because (except in trivial situations) there always exist non-normable linear fundionals. In addition to constructive versions of well-known classical results, there are results that, while trivial classically, have considerable constructive content. An instance is the theorem that the normable linear functionals are den~e in the dual space relative to a certain norm (which corresponds to the weak* topology). The theory of locally compact abelian groups (provided the underlying space is metric) can be constructivized. A constructive proof of the existence of Haar measure can be extracted from work of H. Cartan. The theory of the Fourier transform and Pontryagin duality theory are developed along classical Ijnes, the difference being that certain least upper bounds, which trivially exist classically, must be proved to exist. For instance it is not trivial to give a constructive proof that an L j function acting by convolution induces a normable linear operator on L 2 • The theory of a commutative Banach algebra ill is a rare instance of a theory whose constructive version is substantially more complicated than its c1assial counterpart. The reason is that the classical techniques for building maximal ideals are not constructively valid. It is therefore necessary, instead of considering an ideal {xlal Xnan : ai, ... , an E ill} generated by elements Xj, ... , Xn , to consider a partial ideal {xjaj Xn~: ai, ... , an E A}, where A is some compact subset of 12I. This makes the statements and proofs more awkward, but for the applications to anah sis the constructive results seem to have the same force as their classical prototypes. It is interesting to speculate whether we can relax the separability and metrizability assumptions that occur throughout most of the above development. Surprisingly, the \vay to do this does not seem to be to introduce larger spaces, but rather to examine functorial properties of mappings between the spaces already introduced. An example will suffice. Let I2I be a commutative uniformly-closed algebra of normable Hermitian operators on a Hilbert space H. In case 121 is separable the spectrum ~ of I2I is compact, and the spectral theorem asserts that I2I is isomorphic to C (:L) In case I2I is non·separable, on the other hand, there seems to be no way of constructing even one element of L, so that the spectral theorem is apparently not constructively valid unless I2I is se~able. Nevertheless, the folowing theorem shows the situation in its true
+ ... +
+ ... +
413
-.
,~~-
--- ..~.- -- ••••
F. W. OEHlfmO
313
light: If \!I and \!z are separable commutative uniformly-closed algebras of normable Hermitian operators, with ml c: m2 , and f.. is the inclusion map, then the adjoint map f..* from the spectrum ~ 2' of m2 to the spectrum ~I of ml has dense range (classically, the range therefore equals ~ I)' Thus the partially ordered family of all uniformly closed separable subalgebras of a given (not necessarily separable m), together with the inclusion maps f.., gives rise to a dual family /F* of compact sets, with corresponding maps f..*. Classically it would be a simple matter to use !F* and the maps f..* to extend the spectral theorem to the algebra m, but constructively we can go no further.
s:
University of California, San Diego, USA
414
