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Y is an isomorphism of topological vector spaces (Exercise!), Y being another tvs. We consider that the empty set is convex, ideally convex, bcs-complete, cs-complete and cs-closed. It is worth to point out that when X is a locally convex space, every b-convex series with elements of X is Cauchy (Exercise!). The class of cs-closed sets (and consequently that of ideally convex sets) is larger than the class of closed convex sets, as the next result shows. Proposition 1.2.1
Let A C X be a nonempty convex set.
(i) / / A is closed or open then A is cs-closed. (ii) If X is separated and dim A < oo then A is cs-closed. Proof, (i) Let £ n >! A n i n be a convergent convex series with elements of A; denote by x its sum. Suppose that A is closed and fix a £ A. Then, for every n G N we have that X)fc=i ^kXk + (l — Yl'kLn+i ^*) a S A. Taking the limit for n -> oo, we obtain that x G cl A = A. Suppose now that A is open. Assume that x $. A. By Theorem 1.1.3, there exists x* G X* such that (a - x, x") > 0 for every a G A. In particular (xn — x,x*) > 0 for every n G N. Multiplying by \ n > 0 and adding for n G N we get (since A„ > 0 for some n) the contradiction
0 < ^ n > 1 A„ (xn - x, x*) = (X]„>i XnXn' x*)~ \52n>i
An
) ^' x ")
=
°'
Therefore x G A. So in both cases A is cs-closed. (ii) We prove the statement by mathematical induction on n := dim A. If n = 0 A reduces to a point; it is obvious that A is cs-closed in this case. Suppose that the statement is true if dim A < n G N U {0} and show it for dim A = n + 1. Without any loss of generality we suppose that 0 G A; then Xo := aff A is a linear subspace with dimX 0 = n + 1. Because on a finite dimensional linear space there exists a unique separated linear topology and in such spaces the interior and the algebraic interior coincide for convex sets, we have that iA — intx 0 A ^ 0. Let J2n>i ^n%n be a convergent convex series with elements of A and sum x. Assume that x fi A. Because A is convex, the set P :— {n G N | A„ > 0} is infinite, and so we may assume that P — N. Applying now Theorem 1.1.3 in XQ, there exists XQ G XQ \ {0} such that (x — x, XQ) > 0 for every x G A. But X)n>i An (xn —X,XQ) = 0. Since {xn —X,XQ) > 0 and Xn > 0 for every n, we obtain that (x n ) C AQ := A(~)Hx*t\, where A :— (X,XQ). Since
Closedness and inferiority
notions
11
dimAo < dimHx*t\ = n, from the induction hypothesis we obtain the contradiction x e A0 c A. Therefore x € A. The proof is complete. • Other properties of cs-closed and ideally convex sets are given in the following result. Proposition 1.2.2 (i) If Ai C X is cs-closed (resp. ideally convex) for every i £ I then P | i e / A{ is cs-closed (resp. ideally convex). (ii) / / Xi is a topological vector space and Ai C Xi is cs-closed (resp. ideally convex) for every i 6 I, then Ylizi -^-i is cs-closed (resp. ideally convex) in Yliei Xi (which is endowed with the product topology). Proof. The proof of (i) is immediate, while for (ii) one must take into account that a sequence (x n ) n € N C X := YlieI Xi converges t o x G X (resp. is bounded) if and only if (xln) converges to xl in Xi (resp. is bounded) for every i £ I. • We say that the subset C of Y is lower cs-closed (Ics-closed for short) if there exist a Frechet space X and a cs-closed subset B of X x Y such that C = Pry (B). Similarly, the subset C of Y is lower ideally convex (li-convex for short) if there exist a Frechet space X and an ideally convex subset B of X xY such that C = Pry (B). It is obvious that any cs-closed (resp. ideally convex) set is Ics-closed (resp. li-convex), any Ics-closed set is li-convex and any li-convex set is convex, but the converse implications are not true, generally. Note also that T(A) is Ics-closed (resp. li-convex) if A C X is Ics-closed (resp. li-convex) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!). The classes of Ics-closed and li-convex sets have very good stability properties as the following results show. We give only the proofs for the "li-convex" case, that for the "Ics-closed" case being similar. Proposition 1.2.3 Suppose that Y is a Frechet space and C C F x Z is a li-convex (Ics-closed) set. Then Prz(C) is a li-convex (Ics-closed) set. Proof. By hypothesis, there exists a Frechet space X and an ideally convex subset B C X x (Y x Z) such that C = P r y x z ( B ) - Since I x 7 i s a Frechet space and Prz(C) = Prz(B), we have that Prz(C) is a li-convex subset of Z. • Proposition 1.2.4
Let I be an at most countable nonempty set.
(i) If Ci C Y is li-convex (Ics-closed) for every i £ I then (~)ieI Ci is li-convex (Ics-closed).
12
Preliminary
results on functional
analysis
(ii) / / Y{ is a topological vector space and Ci C Yi is li-convex (Icsclosed) for every i 6 I, then Yli&iCi 8S li-convex (Ics-closed) in W^jYi. Proof, (i) For each i £ I there exist X» a Frechet space and an ideally convex set Bt C X» xY such that d = PrY(Bi). The space X := fliei -%i is a Frechet space as the product of an at most countable family of Frechet spaces. Let Bi •= {{(xj)jei,y)
e X x Y | (xi,y) £ Bi} .
Then Bi is an ideally convex set by Proposition 1.2.2(h). It follows that B := f]ieI Bi is ideally convex by Proposition 1.2.2(i). Since PrY(B) = Hie/ Ci, ^ f ° u o w s that flie/ Ci is li-convex. (ii) For each i £ I there exist Xj a Frechet space and an ideally convex set Bi cXiX Yi such that d = Pry ; (Bi). By Proposition 1.2.2(h), ]JieI Bt is an ideally convex subset of r i i e / ( ^ j x ^ ) - The space X := riig/ ^i 1S a Frechet space; let Y := FJie/ ^i- Consider the set B:= {((xi)ieI,(yi)ieI)
£XxY\
(xi,yi)
£ Bi Vt 6 / } .
Since T : I l i e / ( X * x y4) - • X x y , T ( ( x i , y i ) i 6 / ) := ((xi)ieI,(yi)iei) is an isomorphism of topological vector spaces, B — T (Y\ieI Bi) is ideally convex. As C := riig/ C* = ^VY(B), C is li-convex. D Before stating other properties of li-convex and lcs-closed sets, let us define some notions and notations related to multifunctions. Let E,F be two nonempty sets; a mapping ft : E —• 2F is called a multifunction, and it will be denoted by ft : E =X F. The set domft := {x £ E \ 5l(x) -£ 0} is called the d o m a i n of the multifunction 51; the image of 51 is Imft := UigB^( a ; )i the g r a p h of 51 is the set grft :— {(x,y) \ y £ 5l(x)} C E x F; the inverse of the multifunction 51 is the multifunction ft"1 : F =} E defined by ft_1(?/) := {x £ E | y £ %(x)}. Therefore domft- 1 = Imft, Imft- 1 = domft and grft- 1 = {(y,x) | (x,y) £ grft}. Frequently we shall identify a multifunction with its graph. For A C E and B C F one defines 51(A) := \JxeA5l(x) and ft"1^) := Uj, e B# - 1 (2/); i n particular Imft = ft(.E) and domft = ft_1(F). If 8 : F =4 G is another multifunction, then the composition of the multifunctions S and 51 is the multifunction Soft :E=lG, (Soft) (a) := \Jy€Ci{x) S(y). If ft, S : E =} F and F is a linear space, the sum of ft and S is the multifunction ft + S : E =t F, (ft + S)(x) :=5i(x) + S(x).
Closedness and interiority notions
13
Let now K : X r{ 7 ; we say that 51 is convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) if its graph is a convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) subset of X x Y. Note that 31 is convex if and only if V i , a ' eX,
VAe [0,1] : \3l(x) + (l-\)3l(x')c3l(\x
Proposition 1.2.5
+
Let A, B C X, 31, S : X =4 Y and7:Y
{l-\)x'). =} Z.
(i) If X is a Frechet space and A, 31 are li-convex (resp. lcs-closed), then 01(A) is li-convex (resp. lcs-closed). (ii) If X is a Frechet space and A, B are li-convex (resp. lcs-closed), then A + B is li-convex (resp. lcs-closed). (hi) If Y is a Frechet space and 31, T are li-convex (resp. lcs-closed), then T o 31 is li-convex (resp. lcs-closed). (iv) IfY is a Frechet space and 31,$ are li-convex (resp. lcs-closed), then 31 + § is li-convex (resp. lcs-closed). Proof,
(i) We have that %{A)
=PrY((AxY)ngrJl).
Using successively Propositions 1.2.4(h), 1.2.4(i) and 1.2.3, it follows that 31(A) is li-convex. (ii) Let T : X xX —> X, T(x,y) := x + y. Since T is a continuous linear operator, grT is a closed linear subspace; in particular T is a li-convex multifunction. Since Ax B is li-convex, by (i) A + B = T(A x B) is a li-convex set. (hi) We have that gr(To3?) = P r X x 4 ( g r : R x Z) n (X x grT)). The conclusion follows from Propositions 1.2.4(h), 1.2.4(i) and 1.2.3. (iv) The sets T~{(x,z,y,y')\x€X,
y,y' eY,
z = y + y'} C (X x Y) x (Y x Y),
A:= {(x,z,y,y') \ (x,y) G g r # , y',z € Y} and B := {(x,z,y,y') \ (x,y') G grS, y,z € Y} are li-convex sets (the first being a closed linear subspace). Therefore gr(# + S) = PrXxY (THAHB) is li-convex. •
14
Preliminary
results on functional
analysis
Let Y be another topological vector space and A C X xY; we introduce the conditions (Ha;) and (Hwa;) below, where x refers to the component xeX: (Ha:) If the sequences ((xn, yn)) C A and (A„)„>! C ffi+ are such that E „ > i K = 1, E n > i ^nVn^as sum y and X)„>i A«a;n is Cauchy, then the series X^n>i ^nxn *s convergent and its sum x G X verifies {x,y)EA. (Hwi) If the sequences ({xn,yn))n>1 C A and (A n ) n >i C K+ are such that ((xn,yn)) is bounded, E n > x An = 1, £ „ > i A„2/n has sum y and ^ r a > 1 A n x n is Cauchy, then the series X^ n>1 ^« x « ^s convergent and its sum x € X verifies (x, y) 6 A. Of course, when X is a locally convex space, deleting "^2n>1 Ana;n is Cauchy" in condition (Hwa;) one obtains an equivalent statement. In the next result we mention the relationships among conditions (Ha;), (Hwi), ideal convexity, cs-closedness, cs-completeness and convexity. The proof being very easy we omit it. Proposition 1.2.6 sets.
Let A C X xY
and BcXxYxZbe
nonempty
(i) Assume that Y is complete. Then B satisfies (H.(x,y)) if and only if B satisfies (Ha;); B satisfies (Hwa;) if and only if B satisfies (Hw(x,j/)). (ii) Assume that X is complete. Then A satisfies (Ha;) if and only if A is cs-closed; A satisfies (Hwa;) if and only if A is ideally convex. (iii) Assume that Y is complete. Then A satisfies (Ha;) if and only if A is cs-complete; A satisfies (Hwa;) if and only if A is bcs-complete. (iv) If A satisfies (Ha;) then A satisfies (Hwx); if A satisfies (Hwa;) then A is convex. (v) Assume that X is a locally convex space and Prx(A) is bounded. If A satisfies (Ha;) then Pry(^4) is cs-closed; if A satisfies (Hwa;) then Pry (A) is ideally convex. We define now several interiority notions. Let 0 / A c X. We denote by rint A the interior of A with respect to aff A, i.e. rint A := intafj A A. Of
Closedness and inferiority
notions
15
course, rint A C %A. Consider also the sets c
. lA 1
.
riA:= | i6
.
-{
if aff A is a closed set, otherwise,
rint A 0
| lA
if aff A is a closed set, otherwise,
if XQ is a barreled linear subspace, otherwise,
where X0 = \in(A — a) for some (every) a G A; XQ is the linear subspace, parallel to aff A. In the sequel, in this section, A C X is a nonempty convex set. Taking into account the characterization (1.1) of *A, we obtain that x € tcA cone(A — x) is a closed linear subspace of X & M
X(A — x) is a closed linear subspace of X,
and x £
A -£> cone(^4 — a;) is a barreled linear subspace of X M
n(A — x) is a barreled linear subspace of X.
If X is a Frechet space and aff A is closed then %CA = lbA, but it is possible to have ibA ^ 0 and icA = 0 (if aff A is not closed). The quasi relative interior of A is the set qri A := {x € A | cone(yl — z) is a linear subspace of X}. Taking into account that in a finite dimensional separated topological vector space the closure of a convex cone C is a linear subspace if and only if C is a linear subspace (Exercise!), it follows that in this case qri A = %A = ic A = ibA Below we collect several properties of the quasi relative interior. Proposition 1.2.7 L{X,Y). Then:
Let A C X be a nonempty convex set and T £
a £ qri A •& a G A and cone(A — a) = cone(A - A) O a E i and a - A C cone(yi - a), i
A C qriA = AnqriA,
V a G q r i A , Vx £ A : [a,z[CqriA,
(1.4) (1.5)
Preliminary
16
results on functional
analysis
and T(qriA) C qriT(A). In particular qriA is a convex set. Assume that qriA ^ 0; then qriA = A and *(T(A)) C T(qriA) C qriT(A) C T(A) c T(qriA).
(1.6)
Moreover, if Y is separated and finite dimensional then T(qiiA)=i(T(A)).
(1.7)
Proof. The first equivalence in Eq. (1.4) is immediate from the definition of qriA. Of course, cone (A — a) = cone (A — A) implies that a — A C cone(A - a). Conversely, if a — A C cone(A - a) then -cone(A - a) = cone(a — A) C cone (A — a), which shows that cone (A — a) is a linear subspace. Therefore Eq. (1.4) holds. If a £ lA, from Eq. (1.1) we have that a £ A and cone(A — a) is a linear subspace, and so cone(A — a) is a linear subspace, too. Hence a € qriA. The equality qriA = A n qriA follows immediately from the relation coneC = cone C, valid for every nonempty subset C of X. Let a € qri A, x e A, A € [0,1[ and a\ :— (1 — X)a + Xx. Then A-ADA-a
A
= ( l - A)(A - a) + A(A - z) D (1 - A)(A - a),
and so, taking into account Eq. (1.4), we have cone(A - A) D cone(A - a\) D cone ((1 - A)(A - a)) = cone (A — a) = cone (A — A). Therefore cone(A — A) = cone(A — a^)- Since a\ € A, from Eq. (1.4) we obtain that a\ G qriA. The proof of Eq. (1.5) is complete. Let a £ qriA; then, by Eq. (1.4), a - A C cone(A — a), and so Ta - T{A) = T(a-A)cT = cone (T(A) -
(cone(A - a)) C T (cone(A - a)) Ta),
which shows that Ta € qriT(A). Assume now that qriA ^ 0 and fix a £ qriA. It is sufficient to show Eq. (1.6); then the equality qriA = A follows immediately from the last inclusion in Eq. (1.6) for T = Idx and from Eq. (1.5). Let y e i (T'(A)); using Eq. (1.1), there exists A > 0 such that (1 + X)y XT a € T(A). So, (1 + X)y - XTa = Tx for some x £ A. It follows that y = Txx, where xx ~ (1 + A) _1 (Aa + x). But, using Eq. (1.4), x\ 6 qriA, and so y 6 T(qriA).
Closedness and inferiority
notions
17
The second inclusion in Eq. (1.6) was already proved, while the third is obvious. So, let y G T(A); there exists x G A such that y = Tx. By Eq. (1.5), (1-X)a + Xx G qriAfor A G]0,1[, and so (l-X)Ta + Xy G T(qriA) for A G ]0,1[. Taking the limit for A —> 1, we obtain that y G T(qri A). When Y is separated and finite dimensional we have (as already observed) that i(T(A)) = qriT(A); then Eq. (1.7) follows immediately from Eq. (1.6). • The notion of quasi relative interior is related to that of united sets. Let X be a locally convex space and A,B C X he nonempty convex sets; we say that A and B are united if they cannot be properly separated, i.e. if every closed hyperplane which separates A and B contains both of them. The next result is related to the above notions. Proposition 1.2.8 Let X be a locally convex space, A,BcX empty convex sets and x G X. (i) A and B are united O- cone(A-B) is a linear subspace.
be non-
is a linear subspace •& (A — B)~
(ii) Assume that cone (A — x) is a linear subspace. Then x G c\A. Moreover, i/aff A is closed and rint A ^ 0 then ~x G rint A. Proof, (i) Assume that A and B are united but C := cone(A — B) is not a linear subspace. Then there exists XQ G (—C) \ C. By Theorem 1.1.5 there exists x* G X* such that {x0, x*) < 0 < (z, x*) for every z G C. Then 0 < (x - y,x*) for all x G A, y G B, and so (x,x*) < X < (y,x*) for all x G A, y G B, for some A G M. Therefore Hx* t\ separates A and B. It follows that (x,x*) = X = (y,x*) for all x G A, y G B, and so 0 < (z,x*) for every z G C. Thus we have the contradiction (x0,x*) = 0. Therefore cone (A — B) is a linear subspace. Let C := cone (A — B) be a linear subspace and (x,x*) < X < (y, x") for all x G A, y G B, for some x* G X* and A e i . Then 0 < (z,x*) for every z G A — B, and so 0 < (z, x*) for z € C. Then, since C is a linear subspace, 0 = (z,x*) for every z G C which implies immediately that Hx*,\ contains A and B. Therefore A and B are united. The other equivalence is an immediate consequence of the bipolar theorem (Theorem 1.1.9). (ii) By (i) we have that {x} and A are united. Assuming that x $ cl A, using again Theorem 1.1.5, we obtain that {x} and A can be properly separated. This contradiction proves that x E. cl A.
18
Preliminary
results on functional
analysis
Suppose now that aSA is closed and rint A / 0. Without loss of generality we suppose that x = 0. By what was shown above we have that x = 0 € clA C cl(affA) = aff A. Thus X0 := aS A is a linear space. Assuming now that 0 £ intx 0 A ^ 0, we obtain that {x} and A can be properly separated (in XQ, and therefore in X) using Theorem 1.1.3. This contradiction proves that x € rint A. • From the preceding proposition we obtain that when X is a locally convex space and A C X is a nonempty convex set, the quasi relative interior of A is given by the formula
qri A = A (~1 {x € X \ {x} and A are united}. The next result shows that the class of convex sets with nonempty quasi relative interior is large enough. Proposition 1.2.9 Let X be a first countable separable locally convex space and A C X be a nonempty cs-complete set. Then qri A ^ 0. Proof. Since X is first countable, by Proposition 1.1.11, the topology of X is determined by a countable family 7 = {pn \ n 6 N} of semi-norms. Without loss of generality we suppose that pn < pn+i for every n € N. Since Ty is semi-metrizable, the set A is separable, too. Let A0 = {xn | n € N} C A be such that A C cl A0. Consider j3n € ]0,2~n] such that /3npn(xn) < 2 _ n . The series ^ n > 1 Pnxn is Cauchy (since for m > n and p e N w e have that PniT^Z^Xk)
< ET=mPkPn(xk)
< ET=Z^Pk(xk)
< 2"™+!). Taking
^n '•= (Z)n>i Pn) 1Pn, ]C n >i ^ n 1 " *s a Cauchy convex series with elements of A. Because A is cs-complete, the series ^ n > 1 Xnxn is convergent and its sum x € A. Suppose that x ^ qri A. Then there exists XQ € (—C)\C, where C := cb~m(A — x). Using Theorem 1.1.5, there exists x* € X* such that (xo,x*) < 0 < {z,x*) for all z € C. In particular ( *) > 0 for — = every x £ A. But E n > i ^» ( ^ ^^o) 0- Since {xn — X,XQ) > 0 and A„ > 0 for every n, we obtain that (x - x, x*) = 0 for every x € AQ. Since AQ is dense in A and x* is continuous, we have that {x — x, x*) = 0 for every x € A, and so (z,a;*) = 0 for every z E C. Thus we get the contradiction (x0,x*) = 0. Therefore qri A ^ 0. D
Open mapping
1.3
theorems
19
Open Mapping Theorems
Throughout this section the spaces X, Y are topological vector spaces if not stated otherwise. We begin with some auxiliary results. Lemma 1.3.1 Let X,Y be first countable topological vector spaces, 31 : X =$ Y be a multifunction and XQ G X. Suppose that grIR satisfies condition (Hwa;). Then
where Nxl^o) ** the class of all neighborhoods of XQ. Proof. Let y0 G C\u£7fx(x0)int (clft(t/)). Replacing grft by gv% {x0,y0) if necessary, we may assume that (xo,yo) = (0,0). Let U G NxSince X is first countable, there exists a base (Un)n>i C N x of neighborhoods of 0 such that Un + Un C Un-i for every n > 1, where [7o := ?/• Because 0 G f " ) ^ ^ int (cl#([/)), there exists (V n ) n >i C Ny such that V„ C int (cl3i([7n)) for every n > 1. Since Y is first countable, we may suppose that (Vn)n>i is a base of neighborhoods of 0 G Y and, moreover, y n + i + Vn+i C KJ for every n > 1. Consider j / ' G int (clCR.(C/i)); there exists fi G]0,1[ such that y := (1 — lJ)~1y' £ cl3?([/i). There exists (£1,2/1) e g r $ such that x\ G C/i and 2/ - 2/i G M^2- It follows that /x -1 (y - j/i) G V2 C cl!R(t/2)- There exists (^2,2/2) G grR such that xi G L/2 and /x_1(2/ _ 2/i) — 2/2 G M^3- It follows that /U~2y — fJ-~2yi — H~lyi 6 V3 C c\"R{Uz). Continuing in this way we find ({xm,ym))m>1 C grR such that xm e Um and n~m+xy - n~m+1yi -m+2 M 2/2 - ••• - ym G fiVm+i for every m > 1. Therefore u m := y 2/1-/^2/2 At",_12/m G ^ m K i + i C V m + i for every m > 1. Moreover, m_1 M 2/m = vm-i -vm e \im~xVm - fimVm+1, and so y m £ Vro - fiVm+1 C Vm + Vm C V m _i for m > 2. It follows that (x m ) m >i -> 0, (y m ) m >i -> 0 and (u m ) m >i -> 0, whence J2m>i nm~lym has sum y. Taking Am := ( l - / i ) ^ m _ 1 , we have that (A m ) m >i C R)., E m > i A™ = *> E m > i Am2/m h a s sum (1 - ju)y = y', the sequence ((x TO ,y m )) is bounded (being convergent) and, because £mtf„ + i Ama;TO € C/„+i + Un+2 + ••• + Un+P C f7„ +1 + Un+1 C f/n for every n > 1, the series 5Z m > 1 A m x m is Cauchy. By hypothesis there exists x' £ X, sum of the series ]Cm>i Am#m> such that (x',y') £ g r X Let x := (1 - n)~lx'- We have that YZi=i Hm~lxm G t/i + U2 + • • • + Un C t/i -I- Ui C t/. Since U is closed we have that x E.U, and so a;' G (1 — n)U C
20
Preliminary
results on functional
analysis
U. Thus y' £ "R{U). Therefore int (cl3l(l7i)) C R(U). 0 € int%(U). The proof is complete.
In particular •
Note that we didn't use the fact that X or Y is separated. Observe also that it is no need that x0 £ dom"R when y0 £ f\ue^x(xo) * nt ( c ^ ( ^ 0 ) ' but, necessarily, xo £ cl(domD?) and yo £ int(ImlR) in our conditions. Note also that condition (Hwa;) may be weakened by asking that the sequence ((xm,ym)) C A be convergent instead of being bounded. In the case of normed spaces one has the following variant of the result in Lemma 1.3.1. Having the normed vector space (nvs) (X, ||-||), we denote by Ux the closed unit ball {x £ X \ \\x\\ < 1}, by Bx the open unit ball {x € X | ||a;|| < 1} and by Sx the unit sphere {x 6 X | ||a;|| = 1}. Lemma 1.3.2 Let (X, || • ||), (Y, || • ||) be two normed linear spaces and let 31: X =4 Y be a multifunction. Suppose that condition (Hwa;) holds and (x0,y0) &X xY. If yo + nUy C cl (^(XQ +
pUx)),
where rj,p > 0, then yo + nBy C%(x0 + pUx). Proof. We may take (a;o,2/o) = (0,0). One follows the same argument as in the proof of the preceding lemma, but with Un := pUx and Vn := nBy for n > 1. Consider y' 6 nBy and take p, £]0,1[ such that y :— (1 — p) y' £ c\"R(pUx)- We find the sequence {{xn,yn))n>l C grft such that (xn) C pUx and v„ := y - 2/1 - py2 ^ " _ 1 2 / n £ pnr)BY for n > 1. _1 Hence (u n ) -> 0 and /x" 2/„ = u„_i - vn, whence \\yn\\ < 77(1 + p) for n > 1. Taking An := (1 — p)pn^1 > 0 for n > 1, X) n >i ^" = 1> * n e s e r i e s S « > i ^n%n is Cauchy and the series X3n>i ^n2/n is convergent with sum y'. Since 51 satisfies the condition (Hwi), we obtain that the series J2n>i ^nXn is convergent with sum x' and (x',y') £ gr!R. Of course, x' £ pUx- Hence
nBy C npUx)-
•
Lemma 1.3.3 Let X,Y be topological vector spaces, 3? : X =4 Y be a closed convex multifunction and XQ € X. Suppose that X is complete and first countable. Then condition (1.8) holds.
Open mapping
theorems
21
Proof. Let y0 G f)u£Nx(x0)int ( cl %( U ))- Replacing gr51 by g r R - { x 0 , y0) if necessary, we may assume that (xo,yo) = (0,0). Let us first show that V i £ ] 0 , l [ , VU,U' £Kx
• tc\3l(U)c
31 {t{U + U')).
(1.9)
Fix t G ]0,1[ and take tn := t2 for n > 1. Of course, lim£„ • • • t\ = t. Consider U' G Kx- There exists a base of neighborhoods (t/„) n eN C K x such that U\ +U\ C U' and Un+i + Un+i C Un for every n € N. Let Un := (1 - i „ ) - 1 i „ • --tyUn- For every n G N there exists Vj( 6 Ky such that V^ C cl#([/'„). Consider V„ := ^ ( l - t„)V n . Let £/ G K x and j/ G cl %{U); we intend to show that ty G 31 (*(£/ + £/'))• We construct a sequence x = XQ G U, X\ G t/i, . . . , xn G J7 n ,... with the property: V n G N : t n .. .hy G clK (i„ .. .tx (x + ••• + xn-i
+Un)).
(1.10)
Since (t/ - Vi) n 3?([7) ^ 0, there exist yx G Vi and x G J7 such that y — yi G 3£(a:); hence tij/ = *i(y - yi) + (i - *0 ((i - hyhm) C ti3i(a;) + (1 - t i ) c l K ( ^ ) C c\5t{h(x
G t&ix)
+ (l -
W
+ Ui)).
Suppose that we already have x £ U, xi £ Ui,... ,xn-i desired property. Since
G £/n-i with the
(i„ . . . tiy - Vn+i) n 31 (i„ . . . h(x + • • • + x„_i + Un)) + 0, there exist yn+i G Vn+i
an
d xn G C/n such that
tn---hy-yn+i
G3i(tn...ti(H
F x „ - i + xn)).
Therefore tn+l •••hy
= tn+l(tn-
--hy
— 2/n+l) + ( 1 - < n + l ) ( ( 1
£ tn+i$.(tn...ti(x-{
ha;„_! +xn))
_
tn+1)~~
+ (1
tn+1yn+i)
-tn+1)V„+1
C i„+i^(*n .. . t i ( i + • • • + a; n -i + a;„)) + (1 - t„+i) d 3 l ( t / ; + 1 ) C cl3t(t n +i ...h(x-\
hx„_i +xn +
Un+i)).
So the desired sequence is obtained. Since xn+\ H h x „ + p G Un+i + • • • + Un+P C Un for all n,p € N, the series 2 n > i x™ ' s convergent. Denote by x1 its sum. Since i i + - + i » E i i + f/i and Ui is closed we have that x1 Gxi+Ux C U'.
22
Preliminary
results on functional
analysis
Let now U" E Nx and V" E Ny be arbitrary. There exists n E N such that tn ...h(x-\ \-xn-i+Un) C t(x+x')+U" andtn .. .hy G ty+V". By Eq. (1.10) we obtain that tn ... txy E cl!R(t(x + x') + U"). It follows that (ty + V")nJi(t{x + x') + U") ^ Hi, and so there exist x" E U" and y" G V" such that ty + y" E $.(t(x + x')+x"), i.e. (t(x + x'),ty) + (x",y") € g r # . It follows that (t{x + x'), ty) E cl (gr 3V) = grft. Therefore ty E ft (t(U + U')). To complete the proof, let U E Nx(0). Then there exists U' E Nx such that [/' + [/' C 1U. From Eq. (1.9) we obtain that | c l # ( E / ' ) C 3? ( | ( [ / ' + [/')) C #(*/). Since 0 E C\ue^x{xo) int (cl3i(U)), we have that 0G int£([/). ° D Note that Lemma 1.3.3 is a particular case of Lemma 1.3.1 when Y is first countable; otherwise these results are independent. Corollary 1.3.4 Let Y be a first countable topological vector space and C CY be an ideally convex set. Then intC = int(clC). Proof. Consider X an arbitrary Frechet space (for example X = R) and ft : X =t Y denned by IR(0) = C, R(x) = 0 for x ^ 0. Of course condition (Hwir) is satisfied. Taking y0 E int(clC) and xo = 0 we have that 2/o G f)ue^x(xo)int ( c l 3 i ( C 7 ))' a n d s o 2/o € int C. O Theorem 1.3.5 (Simons) Let X and Y be first countable. Assume that X is a locally convex space, "R : X =$ Y satisfies condition (Hwa;), yo E lb (ImJi) and x0 E 3l_1(j/o)- Then yQ E int af f(i m s) R(U) for every U E N x ( z 0 ) . In particular i6(ImIR) = rint(Imft) if i6 (ImK) ^ 0. Proof. Once again we may consider that (xo,yo) = (0,0); so Y0 := aff(ImCR) — lin(Im3i). Replacing, if necessary, Y by YQ, we may suppose that Y is barreled and 0 G (ImIR)\ Let U E Ncx; since grft is a convex set and R(U) = Pry (gr ft n U x Y), %(U) is convex, too. "R{U) is also absorbing. Indeed, let y E Y. Because Imft is absorbing, there exists A > 0 such that Xy E Imft. Therefore there exists x E X such that (x,Xy) E grft. Since U is absorbing, there exists fi G]0,1[ such that fix E U. As grft is convex we have that (fj,x,fj,Xy) = fx(x,Xy) + (1 — /x)(0,0) G grft, whence /xAj/ G R(U). Therefore $.(U) is also absorbing. It follows that cl (£([/)) is an absorbing, closed and convex subset of the barreled space Y. Therefore 0 G int (clft(C/)). Using Lemma 1.3.1 we obtain that 0 G intft(Z7) for every neighborhood U of 0 G X. The last part is an immediate consequence of what was obtained above. •
Open mapping
theorems
23
Corollary 1.3.6 Let X be a Frechet space, Y be first countable and 51 : X =$ Y be li-convex. Assume that yo € t6(Im!R) and XQ 6 5i~1(yo)Then y0 £ intaff(im3j) 5l(U) for all U E Nx(zo)- In particular *b(Im5l) = rint(Im^) provided i6 (ImK) ^ 0. Proof. There exist a Frechet space Z and an ideally convex multifunction S : Z x X =4 Y such that grIR = P r x x y ( g r S ) - Then § verifies condition (Hw(z,i)) by Proposition 1.2.6 (ii). Of course, there exists ZQ 6 Z such that yo £ S(zo,zo)- Since ImS = Imft, by the preceding theorem, j / 0 € intaff(imX) #(E0 = intaff(imK) HZ x ^ ) for every U G Kx(^o)• Theorem 1.3.7 (Ursescu) Let X be a complete semi-metrizable locally convex space and 51 : X =£ Y be a closed convex multifunction. Assume that yo £ t6(ImCR) and xo € 5l~1(yo)- Then yo € int a ff( Im ^) 5l(U) for every U £ Xj(xo). In particular i 6 (Im^) = rint(ImK) if ib{Im 51) ^ 0. Proof. If Y is first countable it is obvious that the conclusion follows from Simons' theorem. Otherwise the proof is exactly the same as that of Simons' theorem but using Lemma 1.3.3 instead of Lemma 1.3.1. • An immediate consequence of Theorem 1.3.5 is the following corollary. Corollary 1.3.8 Let Y be a first countable barreled space and C CY be a lower ideally convex set. Then Cl = intC. Proof. Let yo G Cl. There exist a Frechet space X and an ideally convex multifunction 51: X =3 Y such that C = Im 51. The conclusion follows from Theorem 1.3.5 taking again U — X. O In fact the conclusion of the above corollary holds if C is the projection on Y of a subset A of X x Y with (a') X is a first countable locally convex space and A satisfies condition (Hwx) or (b') X is a semi-metrizable complete locally convex space and A is closed and convex. Putting together Corollaries 1.3.4 and 1.3.8 we get the next result. Corollary 1.3.9 Let Y be a first countable barreled space and C C Y be an ideally convex set. Then Cl = intC = int(clC) = ( c l C ) \ • In normed spaces the following inversion mapping theorem holds. T h e o r e m 1.3.10 (Robinson) Let (X, ||-||) and (Y, ||-||) be normed vector spaces, 51 : X =S Y be a convex multifunction and (xo,yo) € g r ^ - Assume
Preliminary
24
results on functional
analysis
that y0 + rjUy C 3\{XQ + pUx) for some n,p > 0. Then d f c . K - 1 ^ ) ) < P+\\x-x0\\ v-\\y-yo\\
,d(
^ ( a .))
VarGX, Vy£y0
+ r,BY.
Proof. Replacing g r # by g r # — (xo,yo), we may assume that (xo, yo) = (0,0). Let x E X and y G nBy. The conclusion is obvious if x £ domIR or y € &(z), so suppose that neither is true. Choose 6 > 0 and find ye € 3£(:c) such that 0 < \\ye - y\\ < d[y, R(x)) + 6; define a := n - \\y\\ > 0 and take e G ]0, a[. Consider ye :=y + {a-e)\\y-ye\\~1
(y - ye);
thus ||y £ || < \\y\\ + (a — s) =n — e, and so ye G nBy- Therefore there exists xe G pUx with y£ G R(x€). Define A := \\y -yg\\(ae + \\y - y ^ l ) - 1 G ]0,1[. Then 2/ = (1 - X)ye + Xys G (1 - A)tt(a:) + A#(x £ ) C E((l - A)z + Aa;£). Thus (1 - A)z + Xx£ G 3£_1(j/), whence d ( x , ^ - 1 ^ ) ) < A | | x - x e | | . As ll^ — ^ell < \\x\\ + ll^ell < P + INI a n d A < (a — e ) _ 1 ||y — yg\\, we obtain that d (x^iy))
0, VA G [0,1] : 2/0 + XnUy C IR(a;o + At/ X ); (iv) 37,77 > 0, Vz G z 0 + r?t/ X; Vy G y0 + nUy : d ( ^ S T ^ y ) ) < 7-d(y,K(a;)); (v)
3 r ? > 0 , V y G y o + r ? [/x, V x G X
d(y,3i(i)).
: d(x,-R-\y))
< ^Efff •
Open mapping
theorems
25
Proof. The implications (hi) =>• (ii) =>• (i) are obvious. The implication (i) =$> (ii) follows immediately from Simons' theorem, while the implication (ii) =$> (v) is given by the preceding theorem. (iv) => (iii) Let 7' > 7; we may assume that 777' < 1, because, in the contrary case, we replace 77 by 77' := I / 7 ' < 77. Let y £ yo + A?7?7y, y 7^ j/o, w i t h A e ] 0 , l ] . Thend(a;o,^- 1 (y)) < 7 • d{y,3L(x0)) < 7' ||y - yQ\\. Hence there exists x € 5l~1(y) such that ||x —io|| < 7'|l2/~2/o|| < 7'Ar/ < 77. Therefore y0 + A^C/y C R(x0 + \UX)(v) => (iv) Taking a; € XQ + §t/x, 2/ G j/o + f^y' w e obtain that d (x, ft"1 (y)) Y be a linear operator. Then T is continuous if and only if gr T is closed in X x Y. Proof. It is obvious that gr T is closed if T is continuous (even without being linear). Suppose that g r T is closed and consider the multifunction 51 := T _ 1 : Y =$ X. It is obvious that grlR is closed and convex (even linear subspace). Moreover ImO? = X. So we can apply Theorem 1.3.5 for (Tx0,x0) G Y x X. Therefore
VVeNHTio) :
R(V)=T-1(V)£Nx(x0),
which means that T is continuous at XQ.
•
Corollary 1.3.13 (Banach-Steinhaus) Let X, Y be Frechet spaces and T : X —> Y be a bijective linear operator. Then T is continuous if and only if T _ 1 is continuous; in particular, if T is continuous then T is an isomorphism of Frechet spaces. Proof.
Apply the closed graph theorem for T and T - 1 , respectively. D
Theorem 1.3.14 (open mapping theorem) Let X, Y be Frechet spaces and T G L(X,Y) be onto. Then T is open. Proof. Of course, T is a closed convex relation and Im T = Y. Let D C X be open and take 7/0 G T{D)\ there exists XQ G D such that 7/0 = Tx0. Applying the Ursescu theorem for this point, since D is a neighborhood of XQ, we have that T(D) is a neighborhood of T/0. Therefore T(D) is open.D
Preliminary
26
results on functional
analysis
An interesting and useful result is the following. Corollary 1.3.15 Let X, Y be Frechet spaces, A C X and T : X ->• Y be a continuous linear operator. Suppose that I m T is closed. Then T(A) is closed if and only if A + ker T is closed. Proof. Replacing, if necessary, F by I m T and T by T" : X -> I m T , T'{x) := T(x) for x G X, we may suppose that T is onto. Consider T : Xj kerT —> Y, T(x) := T(x), where x is the class of x. It is easy to verify that T is well defined, linear and bijective. Let q : X —> M. be a continuous semi-norm; since T is continuous, p := qoT is a continuous semi-norm, too. For all x € X and u € ker T we have that (