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Existence of Best Approximants in Banach Spaces of Cross-Sections
J.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)
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EXISTENCE OF BEST APPROXIMANTS IN BANACH SPACES OF CROSS-SECTIONS Joao B. PROLLA Departamento de Matemtitica, Universidade Estadual de Campinas, Campinas, SP, Brazil Department of Mathematics, Center for Approximation Theory, Texas A & M University, College Station, Texas, U.S.A. To Professor Leopolda Nachbin on the occasion of his sixtieth birthday
O. Introduction
In this paper we exploit the idea of using a selection theorem similar to Michael's selection theorem to solve the problem of finding best approximants. We consider Banach spaces (and Banach algebras) L of cross-sections over some base space X, i.e. vector subspaces (or subalgebras) of the Cartesian product of families of normed spaces (or normed algebras) {Ex I x E X} endowed with the sup-norm /~sup{II/(x)lIlx E X} together with an upper semicontinuity hypothesis on the function x ~ 11/(x)ll. In such spaces we consider closed vector subspaces We L such that their cross-sections W(x) = {g(x)1 g E W} are proximinal in L(x), which we assume to fill up the fiber Ex, i.e. L(x) = Ex' The following problem was then posed by the late S. Machado (personal communication with the author): find conditions on X, Land W such that proximinality of W in L will accrue. In Section 1 we prove a selection theorem, which is then applied in Sections 2 and 3 to show that certain C({b(X, ~)-modules are proximinal in L. The author acknowledges his debt to the late Professor S. Machado with whom he had planned to write a joint paper on this subject as a continuation of their paper [9]. The author is also grateful to Dr. Jaroslav Mach for many conversations on the subject, while the latter was visiting the Department of Mathematics of the Universidade Estadual de Campinas during August 1982. Finally the author expresses his warm appreciation to all the members of the Center for Approximation Theory at the Texas A & M University, where this work was partly performed in the period January-May 1983.
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1. A Selection Theorem
Throughout this paper X will denote a non-empty completely regular Hausdorff space. Further properties of X will be stated when needed. For each point x in X, let us be given a normed space Ex whose norm we denote by v ~ /lv/l. The elements of the Cartesian product of all spaces Ex will be called cross-sections over X, i.e. cross-sections are functions defined on X and such that their values at a given point x lie in the corresponding space Ex. The Cartesian product of all the spaces Ex is made a vector space under coordinatewise operations, since we assume that all the vector spaces Ex are over the same field of scalars ~, where ~ denotes either the real or the complex field. By ce(X, ~) we denote the algebra over ~ of all continuous functions defined on X and taking values in ~. The subalgebra of <€(X,~) consisting of all elements of <€(X,~) which are bounded on X is denoted by ceb(X, ~). When X is locally compact, two more subalgebras will be considered. The first one is ceo(X, ~), the subalgebra of ce(X,~) consisting of all elements of ce(X,~) which vanish at infinity, i.e. those IE ce(X,~) such that, given e >0 there is a compact subset KCX such that I/(x)1 < e for all x in X not in K. It follows that <€o(X,~) is contained in ceb(X, IK). The second sub algebra of ce(X, IK) that will be considered, when X is locally compact, is <€c(X, IK), the subalgebra of ce(X, IK) consisting of all elements of <€(X, IK) which have compact support, i.e. those IE ce(X, IK) such that the closure in X of {x E XI I(x) oj. O} is a compact set. It follows that cec(X, IK) is contained in ceo(X, ~). Now returning to the general situation, the Cartesian product of all the spaces Ex is also made a module over the algebra ce(X, ~), also called a ce(X, ~)-module, by means of pointwise operation, i.e. given a function a E ce(X,~) and a cross-section I, then af is defined to be the crosssection whose value at x is the vector a(x )/(x).
Definition (1.1). A vector subspace L of the Cartesian product of all the spaces Ex is called a normed space 01 cross-sections over X, if the following conditions are satisfied: (i). For each IE L, the non-negative function x ~ II/(x )11 is upper semicontinuous and bounded on X. (ii). For each x EX, L(x) = {f(x)/ I E L} = Ex. It follows from (i) that the function I ~ sup{/l/(x )/11 x E X} is a norm on L. It is understood that L will always be equipped with this norm. In
J.B. Prolla / Existence of Best Approximants
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particular, if L is complete under this norm, we say that L is a Banach space 01 cross-sections over X. Historically, the notion of spaces of cross-sections goes back to von Neumann, whose paper [11] was written in 1937/38, Godement [4] and Shilov [10]. Kaplansky [5] treated the case in which each Ex is a normed algebra. For a survey of the subject and its relation to Banach bundles, see [3]. We shall be interested in the following problem. Let L be a normed space of cross-sections over X, and let We L be a closed non-empty subset. Under what circumstances, assuming that for each x E X the set W(x) = {f(x)1 lEW} is proximinal in Ex, will W be proximinal in L? This problem arises in analogy with the Stone-Weierstrass theorem which implies that, when X is compact and both Wand L are Y6'(X'; ~)-modules, then W is dense in L if, and only if, W(x) is dense in Ex for each x EX. This density result is an easy corollary of the 'strong' version of the Stone-Weierstrass theorem for Y6'(X, ~)-modules which states that, when X is compact, L is a normed space of cross-sections which is a Y6'(X, ~) module, and We L is a vector subspace which is a Y6'(X, ~)-submodule, then for any IE L, dist(f, W) = sup{dist(f(x), W(x))1 x E X}. For a proof of this 'strong' version of the Stone-Weierstrass theorem see [8, Th. 1.30]. (See also [2, Lemma 4] and [1].) Proposition (1.2). Let L be a normed space 01 cross-sections over X. Let we L and I E L be given. Then, an element g E W is a best approximant to I in' W if, and only if, g(x) E ip(x) lor each x E X, where ip(x) = {h(x)1 hEW; Ilh(x) - I(x)/I ~ d} and d = dist(f, W).
Proof. Suppose g E W is a best approximant to I in W Then, for each x E X'; IIg(x) - I(x )11 ~ Ilg - III ~ d. Hence g(x) E tp (x), for each x E X. Conversely, assume that g E W satisfies the condition g(x) E ip(x) for each x E X. Then Ilg - III = sup{llg(x) - l(x)II I x E X} ~ d,
and so g is a best approximant to
I in W 0
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To deal with the situation presented by Proposition (1.2) we will introduce some terminology.
Definition (1.3). Let L be a normed space of cross-sections over X. A carrier 01 X into L is set-valued mapping lp defined on X and such that lp(x) C Ex for each x E X. If W C L, and lp is a carrier of X into L, we shall say that lp is a W -carrier if
a
lp(x) C W(x)
= {/(x)I/E
W}
for every x E X. If tp is a carrier of X into L, a selection lor lp is a cross-section IE L such that I(x) E lp(x) for each x E X. If a selection for rp belongs to some set l¥, we may say that it is a W-selection for lp.
Remark (1.4). By Proposition (1.2) the problem of finding best Wapproximants to a given IE L is reduced to the problem of finding W-selections for the corresponding carrier ip, Obviously, to find such selections it is necessary that lp(x) ¥- 0 for each x E X. Now, this is certainly the case, if W(x) is proximinal in L(x) = Ex for each x E X. Indeed, if ux'E W(x) is a best W(x)-approximant to I(x), then
lIux -
l(x)11 ~ dist(f(x), W(x».
However, if h E l¥, then
Ilh(x) - l(x)11 ~ IIh -
III.
Hence dist(f(x), W(x» ~ d. Therefore lIux - l(x)1I ~ d, and so u, E lp(x), i.e. lp(x) ¥- 0. However, we cannot assume that for some gEl¥, g(x) = u, for all x E X. Even gEL with this property may not exist. Hence the following problem arises:
Problem (1.5). Let L be a normed space of cross-sections over X. Let we L be a closed and non-empty subset such that, for every x E X, the set W(x) is proximinal in Ex' Find sufficient conditions for W to be proximinal in L.
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We will direct our efforts in the direction pointed out by Proposition (1.2), i.e. to search for sufficient conditions on W so that, for any given f E L, the corresponding carrier cp admits a selection. Now Michael's selection theory suggests that existence of a W-selection for tp may be obtained by appropriate lower semicontinuity with respect to W Such a concept was introduced by Prolla and Machado [9, Dei. 3.1] and will be used here. Definition (1.6). Let cp be a carrier of X into L, and let W C L. The carrier cp is said to be lower semicontinuous (abbreviated l.s.c.) with respect to W at a point x E X if, for each g E Wand r> 0 such that cp(x) n B,(g(x» :r! 0, there exists a neighbourhood V of x in X such that cp(t) n B,(g(t» = 0 for all t E V. (We used the following notation: B,(g(x» ={v E Exlliv - g(x)1I < r}.)
The carrier cp is said to be lower semicontinuous with respect to W if it is l.s.c. with respect to W at all points of X. This is equivalent to saying that, for each g E Wand r > 0, the set
is open. Definition (1.7). Let L be a normed space of cross-sections over X, and let MeL. Let A C C€(X'; IK). We say that M is a locally finite A-module if, for every collection {(fa' wa)1 a E I}, with wa EM and fa E A such that every point x of X has a neighbourhood in which all but finitely many of the fa's vanish, the cross-section x ~ La fa(x)wa(x) belongs to M. Clearly any vector subspace which is a locally finite A-module is an A-module. (Recall that an A-module, say W; is a vector subspace of L such that af E W for every pair a E A and fEW.) We can now state our main selection result, analogous to [7, Th. 3.2", part (a)~ (b)], whose proof we adapt. Theorem (1.8). Let X be a paracompact Hausdorff space, and let L be a Banach space of cross-sections over X'; which is a locally finite C€b(X, IK)-
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J.B. Prolla / Existence of Best Approximants
module. Let we L be a closed non-empty subset which is a locally finite [0, 1])-module. Let rp be a W-carrier which is lower semicontinuous with respect to Wand such that, for each x E X, the set rp(x) is a closed and convex non-empty subset of E". Then, there exists a W-selection for rp.
~(X,
For the proof we shall need some lemmas. Lemma (1.9). Let L be a normed space of cross-sections over X and let We L. Let rp be a W-carrier of X into L which is lower semicontinuous with respect to W Let g E Wand r > O. Define a carrier l/J of X into L by setting l/J(x) = rp(x)
n B,(g(x))
for each x E X. Then I/J is lower semicontinuous with respect to W
Proof. Let w E W, e > 0 and x in the set
T
= {t E Xil/J(t) n BE(w(t))
~
0}
be given. There is some element f in W such that f(x) E rp (x), Ilf(x)g(x)/I < rand Ilf(x) - w(x)11 < E. Choose sand" such that Ilf(x)- g(x)11 < s 0 such that 'Y < min{r - s, E - S}. Now, f(x) E rp(x) n By(f(x)) and since rp is lower semicontinuous with respect to W, there exists some neighbourhood N of x, contained in V; such that tEN implies rp(t) n By(f(t)) ~ 0. Let u E rp(t) n By(f(t)), with tEN. Since t E V; Ilu - f(t)11 < 'Y implies Ilu - g(t)11 < rand Ilu - w(t)11 < e. Therefore u belongs to l/J(t) n BE(w(t)) for all tEN, and the set T is open. This proves that I/J is I.s.c. with respect to
WD
Remark (1.10). For each vEE" we have been using the notation B,(v) to denote the set {u E E"lllu - vii < r}. Let us denote by B,(S) the set S + B,(O) C E" for every non-empty subset SeE". Lemma (1.11). Let X, L, Wand q; be as in Theorem (1.8). Then for each e > 0, there is an element g E W which is an e-approximate selection for tp, i.e. g(x)E BE(rp (x)) for each x in X.
J.B. Prolla / Existence of Best Approximants
Proof. Let e > 0 be given. For each
U(W) = {x E
W
625
E W consider the set
XI w(x) E B.«(,O(x»}.
By lower semicontinuity of (,0 with respect to W the set U(w) is open. Now, for each x E X, the set (,O(x) is non-empty and contained in W(x). Thus {U(w)1 wE W} is an open covering of X. Since X is a paracompact Hausdorff space, there exists a collection {fal a E I} of non-negative real-valued continuous functions fa : X ~ [0,1] such that (1). La fa(x) = 1, for every x E X. (2). Every point x of X has a neighbourhood in which all but finitely many of the fa's vanish. (3). For every a E I, there is wa E W such that fa(x) = 0 for all x ft U(wa ) . Define a cross-section g by setting
a
for all x E X. Then gEl¥, because W is a locally finite ee(x, [0, 1])module. Let x E X. Let I(x) be the finite subset of I such that a E I(x) if, and only if,fa(x) ¥- 0. Then
g(x) =
2:
fa(x)wa(x).
aEI(x)
For each a E I(x), we have x E U(wa). Hence wa(x) E B.«(,O(x». Since B.«(,O(x» is convex, and La E I(x) fa(x) = 1, g(x) E B. «(,0 (x». 0 Proof of Theorem (1.8). We will construct a sequence gn (n = 1,2,3, ...) of elements of W satisfying for all x E X the following two conditions: (1).
gn(x)EB p+2(gn_!(x»
(n=2,3, ...).
For n = 1, the existence of a cross-section g, satisfying (2) follows from Lemma (1.11). Let us assume that the cross-sections g!, B» ... , gn belonging to W have been defined satisfying (1) and (2) above.
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J.B. Prolla I Existence of Best Approximants
For each x E X, define
By Lemma (1.9), l/Jn+1 is a lower semicontinuous carrier with respect to W. By (2), l/Jn+I(X) is non-empty for every x E X. By Lemma (1.11) applied once again, there is gn+1 in W such that gn+I(X) E B 2-n-l(l/Jn+I(X)) for all x E X. But then gn+i(x) E B 2-n+l(gn(X)) for all x E X, which is (1) for n + 1. On the other hand, gn+I(X) E B 2-n-l (cp (x)) for all x E X, which is (2) for n + 1. This proves that the sequence {gn} satisfying (1) and (2) can be obtained by induction. From (2) it follows that
for all n = 2,3, .... Since L is a Banach space, and W is closed, there is some element g E W such that {gn} converges to g. We clain that g is a W-selection for tp, Assume that g(x) g cp(x) for some point x E X. Since cp(x) is closed, there is some integer k such that cp(x) n B 2-k(g(X)) is empty. Since gn(x)~ g(x), there is some n > k + 1 such that gn(x) E BrH(g(X)). On the other hand, from (2) it follows that
Hence
But since cp(x)n Brk(g(X)) = 0, it follows that
a contradiction. Therefore g(x) E cp (x) for all x E X. 0
Remark (1.12). An analysis of the proof of Lemma (1.11) reveals that Theorem (1.8) remains true with the following modifications: (a). Assume X is compact. Then we may assume that both Land W are only convex ee(X, t\)-modules, i.e, given any finite number of functions ai' a2 , ••• , an in ee(x, 1<) and cross-sections 11'/2' .. ''/n in L (resp. W) such that 0,,;;; a, ,,;;; 1 and L a, = 1, then L af, belongs to L (resp. W). This
J.B. Prolla I Existence of Best Approximants
627
is certainly the case if W c L is a vector subspace and both Wand L are I'€(X; IK)-modules. (b). Assume that the carrier q; is bounded on X, i.e. there is some constant k > 0 such that sup{sup{lltlll t E q;(x)}1 x E X} ~ k ; and that, for any x E X and t E q;(X) there is some g E ~ with Ilgll ~ M, such that t = g(x). In this case, Theorem (1.8) remains true even if we relax the condition defining a locally finite module M by requiring that x ~ ~ fa (x) Wa(x) necessarily belongs to M only when the family {w a I a E I} C M is uniformly bounded. In this case we shall say that M is a bounded locally finite A-module. For example, M = I'€b(X, E) is a bounded locally finite r5b(X , ~) module, but it is not a locally finite I'€b(X, ~)-module. Moreover, in our applications of Theorem (1.8) the carrier will be tp as defined in Proposition (1.2), and q; is certainly bounded on X, with constant k = Ilfll + dist(f, W).
2. Proximinality of Certain I'€b (X; K)-ModuJes
Let L be a normed space of cross-sections over X. Let we L be a closed non-empty subset such that, for each x E X; the set W(x) = {g(x)/ g E W} is a closed non-empty subset of Ex = L(x). For each fE L, f ~ ~ let q; be the corresponding W-carrier: q;(x) = {h(x)/ h E ~ Ilh(x)- f(x)1I ~ d},
where d = dist(f, W) > O. If we define d(x) = dist(f(x), W(x», for each xEX; then d(x)~d. Hence D = sup{d(x)1 x E X}:s;; d.
Under the hypothesis above, let us denote by X(f, W) the possibly empty set of all points x E X such that d(x) = d, i.e. X(f, W) = {x E
XI dist(f(x), W(x» = dist(f,
W)}.
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I.B. Prolla I Existence of Best Approximants
One instance in which X(f, W) ¥
0 is the following: X is compact and
we L is a vector subspace such that Awe W, where A = C€(X';
~). See Machado and Prolla [6], where we show that x ~ d(x) attains its supremum d = dist(f, W).
Proposition (2.1). Let we L be a closed non-empty subset such that for each x E X, the set W(x) is proximinal in Ex = L(x). Then, for each f E L, f ti. W, the W-carrier ({) is lower semicontinuous with respect to W at each point x ~ X (f, W). Proof. Let x ~ X(f, W) be given, i.e. d(x) = distU(x), W(x) < d. Let g E Wand r> 0 be such that
tp(x)nB,(g(x» ¥ 0. There is some wE W with w(x) E tp(x), so that lIw(x) - f(x)lI::E; d and /Iw(x) - g(x)/1 < r. Firstly, suppose /Iw(x) - f(x)/1 < d. Then, since w - f and w - g belong to L, by upper semicontinuity there is a neighbourhood Vof x such that
Ilw(t)- f(t)11 < d
and
lIw(t) - g(t)/1 < r,
for all t E V. Hence (()(t) n B,(g(t» is non-empty for all t E V. Suppose now that /Iw(x) - f(x)/1 = d. We know that d(x) < d, because x ~ XU, W). Since we have assumed that W(x) is proximinal in Ex there exists some hEW such that /If(x)- h(x)11 = d(x) < d. Now for each O::E; A ::E; 1 let gA E W be the element gA = (1- A)w + Xh. We claim that gA (x) E (() (x). Indeed, gA E Wand IlgA (x) - f(x )'I::E; (1- A)llw(x) - f(x)11 + Allh(x) - f(x)/I ::E; (1 - A)d + Ad (x) < (1 - A)d + Ad = d .
».
On the other hand, for A sufficiently small gA (x) E B,(g(x Indeed, IlgA (x) - w(x )/I::E; A/lh (x) - w(x )/1. Now if we take A > 0 so small that AlIh(x) - w(x)11 < e, where e = r -llw(x)- g(x)/1 > 0, then IlgA (x) - g(x)/I::E; Ilw(x) - g(x)11 + IlgA (x) - w(x)l/
< /lw(x)- g(x)/I+ e < r.
J.B. Prolla / Existence of Best Approximants
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Therefore, we can take A > 0 so that IlgA (x) - f(x )11 < d and IlgA (x) g(x)11 < r and argue as in the first case: for some neighbourhood V of x, it is true that IIgA (t) - f(t)1I < d and IIgA (t) - g(t)1I < r for all t E V. Hence lp(t) n B,(g(t» is non-empty for all t E V. 0 As a consequence of Proposition (2.1), we have the following result: Theorem (2.2). Let X, Land W be as in Theorem (1.8). If W(x) every x E X, then W is proximinal in L.
=
Ex for
Proof. For any f E L, f g ~ the set X (f, W) is empty, since d (x) = 0 < d for every x E X. Now lp is a W-carrier such that lp(x) is a closed and convex non-empty subset of Ex. By Theorem (1.8), lp admits a Wselection, which by Proposition (1.2) is a best approximant to f in W. 0 Corollary (2.3). Let X be a paracompact Hausdorff space and let E be a Banach space. Let We C(fb (X, E) be a closed vector subspace which is a locally finite C(f(X, [0, 1])-module. If W(x) = E for each x E X, then W is proxim inal in (X, E).
e,
Proof. For each x E X, take Ex = E and let L = C(fb (X, E). Now C(fb (X, E) is a locally finite C(fb(X, ~)-module and it remains to apply Theorem (2.2), and Remark (1.12). 0 Proposition (2.4). Under the hypothesis of Proposition (2.1), for each fE L, f g ~ the W-carrier tp is lower semicontinuous with respect to W at all points x E X(f, W) such that for some neighbourhood V of x and some constant c > 0, d - d(t);;?: c, for all t E V, t:;e x. Proof. Let g E Wand r > 0 be given such that Ip(x)
n B,(g(x» :;e 0.
Choose 8> 0 such that 8 < d and 8 < 3- 1d-1rc. By upper semicontinuity there is a neighbourhood N C V of x such that IIf(t) - g(t)11 < d + 0 for all t E V. Now for any tEN with IIf(t) - g(t)11 ~ d, clearly lp(t) n B,(g(t» :;e 0. Fix now tEN with Ilf(t) - g(t)1I > d. Since N C V, d - d(t);;?: c. Let y(t) > 0 be defined by y(t) = I/f(t) - g(t)l/- d. Clearly, y(t) < o. Define 0 < A < 1 by
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J.B. Prolla I Existence of Best Approximants
A = (d - d(t) + y(t»-l(d - d(t»
=(lIf(t) -
g(t)/I- d(t)r1(d - d(t» .
Since W(t) is proximinal, choose hEW with Ilf(t) - h (t)1I = d(t). Let gA E W be defined by gA = (1- A)h + Ago We claim that s. (t) E rp(t) n B,(g(t». Firstly: IlgA (t) - g(t)1I = (1- A )llg(t)- h (t)11 :so; (1- A) (lig(t):so; (1-
f(t)lI + lIf(t) - h (t)IO
A)(d + y(t) + d(t»
= y(t)(d - d(t) + y(t»-l(d + y(t) + d(t» < 8(d - d(t) + y(t)r1(d + y(t) + d(t»
< 8(d - d(t) + y(t»- 13d < rc(d - d(t) + y(t)r 1< r. On the other hand: lIgA (t) - f(t)II:so; Allf(t) - g(t)1I + (1- A)/If(t) - h(t)1I = A (d
+ y(t» + 1(1- A)d(t)
0
From the definition of A it follows that A(d + y(t» + (1- A)d(t) = do Hence l/gA (t) - f(t)II:so; d
and therefore gA (r) E
tp (r)
n B,(g(t» as claimed. 0
Theorem (2.5). Let X be a paracompact Hausdorff space, and let L be a Banach space of cross-sections over X which is a locally finite 'f5b (X , K)module. Let W be a closed vector subspace of the form W
=
{f ELI f(t)
=
O,[or all t E T},
I.B. Prolla I Existence of Best Approximants
631
where T C X is some closed subset. Then W is proximinal in L. Moreover, lor each IE L, I fit l¥, the subset X (f, W) is contained in T and, if X is compact, then d
= dist(f, W) = sup{lI/(t)111 t E T}.
Proof. Let x E X; x g T, be given. Let v E Ex be also given. There is some gEL such that g(x) = v. Since X is completely regular, there is some a E cgb(X, &(), 0::::; a::::; 1, such that a(x) = 1 and a (t) == 0 for all t E T. Then w = ag belongs to Wand w(x) = v. Thus W(x) = Ex for all x E X, x fit T. Therefore, one has d(x) = 0 < d = dist(f, W), and x fit X(f, W). When X is compact, we know that d = sup{d(x)1 x E X}. Now, when t E T, W(t) = {O} and therefore d(t) = 11/(t)ll. Hence
d = sup{d(t)1 t E T} = sup{lI/(t)II I t E T} . To prove the proximinality of l¥, by Theorem (1.8) and Proposition (2.1), only the lower semicontinuity of cp at x E XU, W) needs proof. Let IE L, I fit l¥, x E XU, W) be given. Notice that, for any given x E X we have
cp(x)={O}
if xE T,
cp(x) = {v E Exlliv - l(x)II::::; d}
if x g T.
Let g E Wand r> 0 be given such that cp (x) n BAg(x » ~ 0. Notice that, by the first part of the proof, x E T and therefore g(x) = O. Moreover, since x E XU, W) C T, d = d(x) = II/(x)II, i.e. 11/(x) - g(x)11 = d < d + r. By upper semicontinuity of 1- gEL, there is some neighbourhood V of x in X such that
II/(t)- g(t)11 < d + r
for all t E V.
(i). t E V n T. Then cp(t) = {O} and g(t) = O. Hence, cp(t) n B,(O) ~ 0. (ii). t E V and t fit T. Then W(t) = E, and cp(t) = {u E E,lllu - 1(t)11 : : ; d}. Choose wE W such that I(t) = w(t), and let gA = Ag + (1- ..\)w, where ..\ = d(d + y(t))-l, and y(t) = 1I/(t) - g(t)ll- d. Firstly: l!gA (t) - 1(t)II = ..\llg(t) - 1(t)II = ..\(d + y(t» = d.
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J.B. Prolla / Existence of Best Approximants
Secondly: /IgA (t) - g(t)II = (1- A )IIg(t)- /(t)11
= r(t) = II/(t) Hence
g(t)ll- d < r.
e, (t) E fP(t) n B,(g(t». 0
Theorem (2.6). Let X be a paracompact Hausdorff space and let E be a Banach space. For every closed subset T C X, the closed vector subspace W
= {IE "€b(X, E)I/(t) = 0, t E T}
is proximinal in cgb (X, E). Proof. For each x E X and let Ex = E. For each / E "€b(X, E), / g W; fP is a bounded carrier and W satisfies the condition that given x E X and t E rp(x) there exists agE W with g(x) = t and Ilgll.,; Ilfll + d. By the same argument as in Theorem (2.5), fP is lower semicontinuous and then by Remark (1.12)(b), the W-carrier fP has a selection in w: By Proposition (1.2) this selection is a best approximant to / in w: 0
3. Proximinality of Closed Two-Sided Ideals In this section we shall assume that each normed space Ex is in fact a normed algebra and that the space of all cross-sections over X has been made a linear algebra by defining the product of two cross-sections / and g to be x~ /(x)g(x). If L is now a sub algebra of the Cartesian product of all the spaces Ex which is a normed space of cross-sections over X, then L is also a normed algebra. Indeed
II/gil = sup{II/(x)g(x)II' x E X} .,; sup{II/(x)11·llg(x)111 x E X} .,; 1If11·llg II , for all / and g in L. Thus, if L is both a subalgebra and a normed space of cross-sections over X, according to Definition (1.1), then we shall say that
J.B. Prolla / Existence of Best Approximants
633
L is a normed algebra 01cross-sections over X. If, moreover L is complete, then we say that L is a Banach algebra 01 cross-sections over X. Let us now consider the following condition, where L itself is assumed to be a '?6b(X, ~)-module: (3.1)
Every closed right (resp. left) ideal in L is a '?6b(X,
~)-module.
With regard to (3.1), let us point out that it is implied by the following condition (see Kaplansky [5]): (3.2)
For every 1 E L,
1 lies
in the closure of IL (resp. LI) .
When X is compact, and L is a '?6b(X; ~)-module, using the StoneWeierstrass theorem for '?6(X, ~)-modules of cross-sections (see [8, Th. 1.30]), (3.2) follows from (3.3)
For every x E X, and every v E Ex, v lies in the closure of vEx (resp. Exv) .
Clearly, (3.3) is satisfied if Ex has an identity, or an approximate right (resp. left) identity. Theorem (3.1). Let X be a compact Hausdorff space, and let L be a Banach
algebra 01 cross-sections over X which is a '?6(X; ~)-module. Let us assume that each normed algebra Ex is topologically simple (i.e. has no proper closed two-sided ideal). Then every proper closed two-sided ideal I in L, which is a '?6(X, K)-module, is proximinal in L. Proof. By Theorem (2.7) it is sufficient to prove that I has the form 1= {f ELI I(t) = 0, t E T} for some closed subset. Let
T = {x E
xl I(x) = 0, for
all 1 E I} .
By condition (i) of Definition (1.1), the set T is closed. Let
W
= {fE
LI/(t) = 0, tE T}.
Clearly, lew, let now 1 E W, and assume, by contradiction that 1 Ii 1. Since I is a '?6(X; IK)-module, and X is compact, it follows from [8, Th.
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J.B. Prolla / Existence of Best Approximants
1.30], that for some point x EX; f(x) does not belong to the closure of I(x) in Ex' Now the closure of I(x) is a closed two-sided ideal in Ex, and therefore reduces to {O}. Hence f(x) -::I- O. Since fEW; x g T. However, we have just seen that I(x) = {O}. Hence x E T. This contradiction shows that in fact W C 1. 0 Corollary (3.2). Let X, L and each Ex be as in Theorem (3.1), and assume
that (3.3) is true. Then every proper closed two-sided ideal in L is proximinal. Proof. We have seen that, since X is compact, (3.3)=? (3.2)=? (3.1). Corollary (3.3). Let X be a compact Hausdorff space. Every proper closed
ideal in C6?(X; K) is proximinal.
References [1] R.c. Buck, Approximation properties of vector-valued functions, Pacific J. Math. 53 (1974) 85-94. [2] F. Cunningham jr. and N.M. Roy, Extreme functionals on an upper semicontinuous function space, Proc. Amer. Math. Soc. 42 (1974) 461-465. [3] G. Gierz, Bundles of Topological Vector Spaces and Their Duality, Lecture Notes in Math. 955 (Springer, Berlin, 1982). [4] R. Godement, Theorie generate des sommes continues d'espaces de Banach, c.R. Acad. Sci. Paris 228 (1949) 1321-1323. [5] 1. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951) 219-255. [6] S. Machado and J.B. Prolla, An introduction to Nachbin spaces, Rend. Circ. Mat. Palermo 21 (1972) 119-139. £7] E. Michael, Continuous selections I, Ann. of Math. 63 (1956) 361-382. [8] J.B. Prolla, Approximation of Vector Valued Functions (North-Holland, Amsterdam, 1977). [9] J.B. Prolla and S. Machado, Weierstrass-Stone theorems for set-valued mappings, J. Approx. Theory 36 (1982) 1-15. [10] G. Shilov, On continuous sums of finite-dimensional rings, Sbornik N.S. 27 (1950) 471-484. [11] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401-485.
619
EXISTENCE OF BEST APPROXIMANTS IN BANACH SPACES OF CROSS-SECTIONS Joao B. PROLLA Departamento de Matemtitica, Universidade Estadual de Campinas, Campinas, SP, Brazil Department of Mathematics, Center for Approximation Theory, Texas A & M University, College Station, Texas, U.S.A. To Professor Leopolda Nachbin on the occasion of his sixtieth birthday
O. Introduction
In this paper we exploit the idea of using a selection theorem similar to Michael's selection theorem to solve the problem of finding best approximants. We consider Banach spaces (and Banach algebras) L of cross-sections over some base space X, i.e. vector subspaces (or subalgebras) of the Cartesian product of families of normed spaces (or normed algebras) {Ex I x E X} endowed with the sup-norm /~sup{II/(x)lIlx E X} together with an upper semicontinuity hypothesis on the function x ~ 11/(x)ll. In such spaces we consider closed vector subspaces We L such that their cross-sections W(x) = {g(x)1 g E W} are proximinal in L(x), which we assume to fill up the fiber Ex, i.e. L(x) = Ex' The following problem was then posed by the late S. Machado (personal communication with the author): find conditions on X, Land W such that proximinality of W in L will accrue. In Section 1 we prove a selection theorem, which is then applied in Sections 2 and 3 to show that certain C({b(X, ~)-modules are proximinal in L. The author acknowledges his debt to the late Professor S. Machado with whom he had planned to write a joint paper on this subject as a continuation of their paper [9]. The author is also grateful to Dr. Jaroslav Mach for many conversations on the subject, while the latter was visiting the Department of Mathematics of the Universidade Estadual de Campinas during August 1982. Finally the author expresses his warm appreciation to all the members of the Center for Approximation Theory at the Texas A & M University, where this work was partly performed in the period January-May 1983.
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1. A Selection Theorem
Throughout this paper X will denote a non-empty completely regular Hausdorff space. Further properties of X will be stated when needed. For each point x in X, let us be given a normed space Ex whose norm we denote by v ~ /lv/l. The elements of the Cartesian product of all spaces Ex will be called cross-sections over X, i.e. cross-sections are functions defined on X and such that their values at a given point x lie in the corresponding space Ex. The Cartesian product of all the spaces Ex is made a vector space under coordinatewise operations, since we assume that all the vector spaces Ex are over the same field of scalars ~, where ~ denotes either the real or the complex field. By ce(X, ~) we denote the algebra over ~ of all continuous functions defined on X and taking values in ~. The subalgebra of <€(X,~) consisting of all elements of <€(X,~) which are bounded on X is denoted by ceb(X, ~). When X is locally compact, two more subalgebras will be considered. The first one is ceo(X, ~), the subalgebra of ce(X,~) consisting of all elements of ce(X,~) which vanish at infinity, i.e. those IE ce(X,~) such that, given e >0 there is a compact subset KCX such that I/(x)1 < e for all x in X not in K. It follows that <€o(X,~) is contained in ceb(X, IK). The second sub algebra of ce(X, IK) that will be considered, when X is locally compact, is <€c(X, IK), the subalgebra of ce(X, IK) consisting of all elements of <€(X, IK) which have compact support, i.e. those IE ce(X, IK) such that the closure in X of {x E XI I(x) oj. O} is a compact set. It follows that cec(X, IK) is contained in ceo(X, ~). Now returning to the general situation, the Cartesian product of all the spaces Ex is also made a module over the algebra ce(X, ~), also called a ce(X, ~)-module, by means of pointwise operation, i.e. given a function a E ce(X,~) and a cross-section I, then af is defined to be the crosssection whose value at x is the vector a(x )/(x).
Definition (1.1). A vector subspace L of the Cartesian product of all the spaces Ex is called a normed space 01 cross-sections over X, if the following conditions are satisfied: (i). For each IE L, the non-negative function x ~ II/(x )11 is upper semicontinuous and bounded on X. (ii). For each x EX, L(x) = {f(x)/ I E L} = Ex. It follows from (i) that the function I ~ sup{/l/(x )/11 x E X} is a norm on L. It is understood that L will always be equipped with this norm. In
J.B. Prolla / Existence of Best Approximants
621
particular, if L is complete under this norm, we say that L is a Banach space 01 cross-sections over X. Historically, the notion of spaces of cross-sections goes back to von Neumann, whose paper [11] was written in 1937/38, Godement [4] and Shilov [10]. Kaplansky [5] treated the case in which each Ex is a normed algebra. For a survey of the subject and its relation to Banach bundles, see [3]. We shall be interested in the following problem. Let L be a normed space of cross-sections over X, and let We L be a closed non-empty subset. Under what circumstances, assuming that for each x E X the set W(x) = {f(x)1 lEW} is proximinal in Ex, will W be proximinal in L? This problem arises in analogy with the Stone-Weierstrass theorem which implies that, when X is compact and both Wand L are Y6'(X'; ~)-modules, then W is dense in L if, and only if, W(x) is dense in Ex for each x EX. This density result is an easy corollary of the 'strong' version of the Stone-Weierstrass theorem for Y6'(X, ~)-modules which states that, when X is compact, L is a normed space of cross-sections which is a Y6'(X, ~) module, and We L is a vector subspace which is a Y6'(X, ~)-submodule, then for any IE L, dist(f, W) = sup{dist(f(x), W(x))1 x E X}. For a proof of this 'strong' version of the Stone-Weierstrass theorem see [8, Th. 1.30]. (See also [2, Lemma 4] and [1].) Proposition (1.2). Let L be a normed space 01 cross-sections over X. Let we L and I E L be given. Then, an element g E W is a best approximant to I in' W if, and only if, g(x) E ip(x) lor each x E X, where ip(x) = {h(x)1 hEW; Ilh(x) - I(x)/I ~ d} and d = dist(f, W).
Proof. Suppose g E W is a best approximant to I in W Then, for each x E X'; IIg(x) - I(x )11 ~ Ilg - III ~ d. Hence g(x) E tp (x), for each x E X. Conversely, assume that g E W satisfies the condition g(x) E ip(x) for each x E X. Then Ilg - III = sup{llg(x) - l(x)II I x E X} ~ d,
and so g is a best approximant to
I in W 0
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J.B. Prolla / Existence of Best Approximants
To deal with the situation presented by Proposition (1.2) we will introduce some terminology.
Definition (1.3). Let L be a normed space of cross-sections over X. A carrier 01 X into L is set-valued mapping lp defined on X and such that lp(x) C Ex for each x E X. If W C L, and lp is a carrier of X into L, we shall say that lp is a W -carrier if
a
lp(x) C W(x)
= {/(x)I/E
W}
for every x E X. If tp is a carrier of X into L, a selection lor lp is a cross-section IE L such that I(x) E lp(x) for each x E X. If a selection for rp belongs to some set l¥, we may say that it is a W-selection for lp.
Remark (1.4). By Proposition (1.2) the problem of finding best Wapproximants to a given IE L is reduced to the problem of finding W-selections for the corresponding carrier ip, Obviously, to find such selections it is necessary that lp(x) ¥- 0 for each x E X. Now, this is certainly the case, if W(x) is proximinal in L(x) = Ex for each x E X. Indeed, if ux'E W(x) is a best W(x)-approximant to I(x), then
lIux -
l(x)11 ~ dist(f(x), W(x».
However, if h E l¥, then
Ilh(x) - l(x)11 ~ IIh -
III.
Hence dist(f(x), W(x» ~ d. Therefore lIux - l(x)1I ~ d, and so u, E lp(x), i.e. lp(x) ¥- 0. However, we cannot assume that for some gEl¥, g(x) = u, for all x E X. Even gEL with this property may not exist. Hence the following problem arises:
Problem (1.5). Let L be a normed space of cross-sections over X. Let we L be a closed and non-empty subset such that, for every x E X, the set W(x) is proximinal in Ex' Find sufficient conditions for W to be proximinal in L.
J.B. Prolla / Existence of Best Approximants
623
We will direct our efforts in the direction pointed out by Proposition (1.2), i.e. to search for sufficient conditions on W so that, for any given f E L, the corresponding carrier cp admits a selection. Now Michael's selection theory suggests that existence of a W-selection for tp may be obtained by appropriate lower semicontinuity with respect to W Such a concept was introduced by Prolla and Machado [9, Dei. 3.1] and will be used here. Definition (1.6). Let cp be a carrier of X into L, and let W C L. The carrier cp is said to be lower semicontinuous (abbreviated l.s.c.) with respect to W at a point x E X if, for each g E Wand r> 0 such that cp(x) n B,(g(x» :r! 0, there exists a neighbourhood V of x in X such that cp(t) n B,(g(t» = 0 for all t E V. (We used the following notation: B,(g(x» ={v E Exlliv - g(x)1I < r}.)
The carrier cp is said to be lower semicontinuous with respect to W if it is l.s.c. with respect to W at all points of X. This is equivalent to saying that, for each g E Wand r > 0, the set
is open. Definition (1.7). Let L be a normed space of cross-sections over X, and let MeL. Let A C C€(X'; IK). We say that M is a locally finite A-module if, for every collection {(fa' wa)1 a E I}, with wa EM and fa E A such that every point x of X has a neighbourhood in which all but finitely many of the fa's vanish, the cross-section x ~ La fa(x)wa(x) belongs to M. Clearly any vector subspace which is a locally finite A-module is an A-module. (Recall that an A-module, say W; is a vector subspace of L such that af E W for every pair a E A and fEW.) We can now state our main selection result, analogous to [7, Th. 3.2", part (a)~ (b)], whose proof we adapt. Theorem (1.8). Let X be a paracompact Hausdorff space, and let L be a Banach space of cross-sections over X'; which is a locally finite C€b(X, IK)-
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J.B. Prolla / Existence of Best Approximants
module. Let we L be a closed non-empty subset which is a locally finite [0, 1])-module. Let rp be a W-carrier which is lower semicontinuous with respect to Wand such that, for each x E X, the set rp(x) is a closed and convex non-empty subset of E". Then, there exists a W-selection for rp.
~(X,
For the proof we shall need some lemmas. Lemma (1.9). Let L be a normed space of cross-sections over X and let We L. Let rp be a W-carrier of X into L which is lower semicontinuous with respect to W Let g E Wand r > O. Define a carrier l/J of X into L by setting l/J(x) = rp(x)
n B,(g(x))
for each x E X. Then I/J is lower semicontinuous with respect to W
Proof. Let w E W, e > 0 and x in the set
T
= {t E Xil/J(t) n BE(w(t))
~
0}
be given. There is some element f in W such that f(x) E rp (x), Ilf(x)g(x)/I < rand Ilf(x) - w(x)11 < E. Choose sand" such that Ilf(x)- g(x)11 < s
WD
Remark (1.10). For each vEE" we have been using the notation B,(v) to denote the set {u E E"lllu - vii < r}. Let us denote by B,(S) the set S + B,(O) C E" for every non-empty subset SeE". Lemma (1.11). Let X, L, Wand q; be as in Theorem (1.8). Then for each e > 0, there is an element g E W which is an e-approximate selection for tp, i.e. g(x)E BE(rp (x)) for each x in X.
J.B. Prolla / Existence of Best Approximants
Proof. Let e > 0 be given. For each
U(W) = {x E
W
625
E W consider the set
XI w(x) E B.«(,O(x»}.
By lower semicontinuity of (,0 with respect to W the set U(w) is open. Now, for each x E X, the set (,O(x) is non-empty and contained in W(x). Thus {U(w)1 wE W} is an open covering of X. Since X is a paracompact Hausdorff space, there exists a collection {fal a E I} of non-negative real-valued continuous functions fa : X ~ [0,1] such that (1). La fa(x) = 1, for every x E X. (2). Every point x of X has a neighbourhood in which all but finitely many of the fa's vanish. (3). For every a E I, there is wa E W such that fa(x) = 0 for all x ft U(wa ) . Define a cross-section g by setting
a
for all x E X. Then gEl¥, because W is a locally finite ee(x, [0, 1])module. Let x E X. Let I(x) be the finite subset of I such that a E I(x) if, and only if,fa(x) ¥- 0. Then
g(x) =
2:
fa(x)wa(x).
aEI(x)
For each a E I(x), we have x E U(wa). Hence wa(x) E B.«(,O(x». Since B.«(,O(x» is convex, and La E I(x) fa(x) = 1, g(x) E B. «(,0 (x». 0 Proof of Theorem (1.8). We will construct a sequence gn (n = 1,2,3, ...) of elements of W satisfying for all x E X the following two conditions: (1).
gn(x)EB p+2(gn_!(x»
(n=2,3, ...).
For n = 1, the existence of a cross-section g, satisfying (2) follows from Lemma (1.11). Let us assume that the cross-sections g!, B» ... , gn belonging to W have been defined satisfying (1) and (2) above.
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J.B. Prolla I Existence of Best Approximants
For each x E X, define
By Lemma (1.9), l/Jn+1 is a lower semicontinuous carrier with respect to W. By (2), l/Jn+I(X) is non-empty for every x E X. By Lemma (1.11) applied once again, there is gn+1 in W such that gn+I(X) E B 2-n-l(l/Jn+I(X)) for all x E X. But then gn+i(x) E B 2-n+l(gn(X)) for all x E X, which is (1) for n + 1. On the other hand, gn+I(X) E B 2-n-l (cp (x)) for all x E X, which is (2) for n + 1. This proves that the sequence {gn} satisfying (1) and (2) can be obtained by induction. From (2) it follows that
for all n = 2,3, .... Since L is a Banach space, and W is closed, there is some element g E W such that {gn} converges to g. We clain that g is a W-selection for tp, Assume that g(x) g cp(x) for some point x E X. Since cp(x) is closed, there is some integer k such that cp(x) n B 2-k(g(X)) is empty. Since gn(x)~ g(x), there is some n > k + 1 such that gn(x) E BrH(g(X)). On the other hand, from (2) it follows that
Hence
But since cp(x)n Brk(g(X)) = 0, it follows that
a contradiction. Therefore g(x) E cp (x) for all x E X. 0
Remark (1.12). An analysis of the proof of Lemma (1.11) reveals that Theorem (1.8) remains true with the following modifications: (a). Assume X is compact. Then we may assume that both Land W are only convex ee(X, t\)-modules, i.e, given any finite number of functions ai' a2 , ••• , an in ee(x, 1<) and cross-sections 11'/2' .. ''/n in L (resp. W) such that 0,,;;; a, ,,;;; 1 and L a, = 1, then L af, belongs to L (resp. W). This
J.B. Prolla I Existence of Best Approximants
627
is certainly the case if W c L is a vector subspace and both Wand L are I'€(X; IK)-modules. (b). Assume that the carrier q; is bounded on X, i.e. there is some constant k > 0 such that sup{sup{lltlll t E q;(x)}1 x E X} ~ k ; and that, for any x E X and t E q;(X) there is some g E ~ with Ilgll ~ M, such that t = g(x). In this case, Theorem (1.8) remains true even if we relax the condition defining a locally finite module M by requiring that x ~ ~ fa (x) Wa(x) necessarily belongs to M only when the family {w a I a E I} C M is uniformly bounded. In this case we shall say that M is a bounded locally finite A-module. For example, M = I'€b(X, E) is a bounded locally finite r5b(X , ~) module, but it is not a locally finite I'€b(X, ~)-module. Moreover, in our applications of Theorem (1.8) the carrier will be tp as defined in Proposition (1.2), and q; is certainly bounded on X, with constant k = Ilfll + dist(f, W).
2. Proximinality of Certain I'€b (X; K)-ModuJes
Let L be a normed space of cross-sections over X. Let we L be a closed non-empty subset such that, for each x E X; the set W(x) = {g(x)/ g E W} is a closed non-empty subset of Ex = L(x). For each fE L, f ~ ~ let q; be the corresponding W-carrier: q;(x) = {h(x)/ h E ~ Ilh(x)- f(x)1I ~ d},
where d = dist(f, W) > O. If we define d(x) = dist(f(x), W(x», for each xEX; then d(x)~d. Hence D = sup{d(x)1 x E X}:s;; d.
Under the hypothesis above, let us denote by X(f, W) the possibly empty set of all points x E X such that d(x) = d, i.e. X(f, W) = {x E
XI dist(f(x), W(x» = dist(f,
W)}.
628
I.B. Prolla I Existence of Best Approximants
One instance in which X(f, W) ¥
0 is the following: X is compact and
we L is a vector subspace such that Awe W, where A = C€(X';
~). See Machado and Prolla [6], where we show that x ~ d(x) attains its supremum d = dist(f, W).
Proposition (2.1). Let we L be a closed non-empty subset such that for each x E X, the set W(x) is proximinal in Ex = L(x). Then, for each f E L, f ti. W, the W-carrier ({) is lower semicontinuous with respect to W at each point x ~ X (f, W). Proof. Let x ~ X(f, W) be given, i.e. d(x) = distU(x), W(x) < d. Let g E Wand r> 0 be such that
tp(x)nB,(g(x» ¥ 0. There is some wE W with w(x) E tp(x), so that lIw(x) - f(x)lI::E; d and /Iw(x) - g(x)/1 < r. Firstly, suppose /Iw(x) - f(x)/1 < d. Then, since w - f and w - g belong to L, by upper semicontinuity there is a neighbourhood Vof x such that
Ilw(t)- f(t)11 < d
and
lIw(t) - g(t)/1 < r,
for all t E V. Hence (()(t) n B,(g(t» is non-empty for all t E V. Suppose now that /Iw(x) - f(x)/1 = d. We know that d(x) < d, because x ~ XU, W). Since we have assumed that W(x) is proximinal in Ex there exists some hEW such that /If(x)- h(x)11 = d(x) < d. Now for each O::E; A ::E; 1 let gA E W be the element gA = (1- A)w + Xh. We claim that gA (x) E (() (x). Indeed, gA E Wand IlgA (x) - f(x )'I::E; (1- A)llw(x) - f(x)11 + Allh(x) - f(x)/I ::E; (1 - A)d + Ad (x) < (1 - A)d + Ad = d .
».
On the other hand, for A sufficiently small gA (x) E B,(g(x Indeed, IlgA (x) - w(x )/I::E; A/lh (x) - w(x )/1. Now if we take A > 0 so small that AlIh(x) - w(x)11 < e, where e = r -llw(x)- g(x)/1 > 0, then IlgA (x) - g(x)/I::E; Ilw(x) - g(x)11 + IlgA (x) - w(x)l/
< /lw(x)- g(x)/I+ e < r.
J.B. Prolla / Existence of Best Approximants
629
Therefore, we can take A > 0 so that IlgA (x) - f(x )11 < d and IlgA (x) g(x)11 < r and argue as in the first case: for some neighbourhood V of x, it is true that IIgA (t) - f(t)1I < d and IIgA (t) - g(t)1I < r for all t E V. Hence lp(t) n B,(g(t» is non-empty for all t E V. 0 As a consequence of Proposition (2.1), we have the following result: Theorem (2.2). Let X, Land W be as in Theorem (1.8). If W(x) every x E X, then W is proximinal in L.
=
Ex for
Proof. For any f E L, f g ~ the set X (f, W) is empty, since d (x) = 0 < d for every x E X. Now lp is a W-carrier such that lp(x) is a closed and convex non-empty subset of Ex. By Theorem (1.8), lp admits a Wselection, which by Proposition (1.2) is a best approximant to f in W. 0 Corollary (2.3). Let X be a paracompact Hausdorff space and let E be a Banach space. Let We C(fb (X, E) be a closed vector subspace which is a locally finite C(f(X, [0, 1])-module. If W(x) = E for each x E X, then W is proxim inal in (X, E).
e,
Proof. For each x E X, take Ex = E and let L = C(fb (X, E). Now C(fb (X, E) is a locally finite C(fb(X, ~)-module and it remains to apply Theorem (2.2), and Remark (1.12). 0 Proposition (2.4). Under the hypothesis of Proposition (2.1), for each fE L, f g ~ the W-carrier tp is lower semicontinuous with respect to W at all points x E X(f, W) such that for some neighbourhood V of x and some constant c > 0, d - d(t);;?: c, for all t E V, t:;e x. Proof. Let g E Wand r > 0 be given such that Ip(x)
n B,(g(x» :;e 0.
Choose 8> 0 such that 8 < d and 8 < 3- 1d-1rc. By upper semicontinuity there is a neighbourhood N C V of x such that IIf(t) - g(t)11 < d + 0 for all t E V. Now for any tEN with IIf(t) - g(t)11 ~ d, clearly lp(t) n B,(g(t» :;e 0. Fix now tEN with Ilf(t) - g(t)1I > d. Since N C V, d - d(t);;?: c. Let y(t) > 0 be defined by y(t) = I/f(t) - g(t)l/- d. Clearly, y(t) < o. Define 0 < A < 1 by
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J.B. Prolla I Existence of Best Approximants
A = (d - d(t) + y(t»-l(d - d(t»
=(lIf(t) -
g(t)/I- d(t)r1(d - d(t» .
Since W(t) is proximinal, choose hEW with Ilf(t) - h (t)1I = d(t). Let gA E W be defined by gA = (1- A)h + Ago We claim that s. (t) E rp(t) n B,(g(t». Firstly: IlgA (t) - g(t)1I = (1- A )llg(t)- h (t)11 :so; (1- A) (lig(t):so; (1-
f(t)lI + lIf(t) - h (t)IO
A)(d + y(t) + d(t»
= y(t)(d - d(t) + y(t»-l(d + y(t) + d(t» < 8(d - d(t) + y(t)r1(d + y(t) + d(t»
< 8(d - d(t) + y(t»- 13d < rc(d - d(t) + y(t)r 1< r. On the other hand: lIgA (t) - f(t)II:so; Allf(t) - g(t)1I + (1- A)/If(t) - h(t)1I = A (d
+ y(t» + 1(1- A)d(t)
0
From the definition of A it follows that A(d + y(t» + (1- A)d(t) = do Hence l/gA (t) - f(t)II:so; d
and therefore gA (r) E
tp (r)
n B,(g(t» as claimed. 0
Theorem (2.5). Let X be a paracompact Hausdorff space, and let L be a Banach space of cross-sections over X which is a locally finite 'f5b (X , K)module. Let W be a closed vector subspace of the form W
=
{f ELI f(t)
=
O,[or all t E T},
I.B. Prolla I Existence of Best Approximants
631
where T C X is some closed subset. Then W is proximinal in L. Moreover, lor each IE L, I fit l¥, the subset X (f, W) is contained in T and, if X is compact, then d
= dist(f, W) = sup{lI/(t)111 t E T}.
Proof. Let x E X; x g T, be given. Let v E Ex be also given. There is some gEL such that g(x) = v. Since X is completely regular, there is some a E cgb(X, &(), 0::::; a::::; 1, such that a(x) = 1 and a (t) == 0 for all t E T. Then w = ag belongs to Wand w(x) = v. Thus W(x) = Ex for all x E X, x fit T. Therefore, one has d(x) = 0 < d = dist(f, W), and x fit X(f, W). When X is compact, we know that d = sup{d(x)1 x E X}. Now, when t E T, W(t) = {O} and therefore d(t) = 11/(t)ll. Hence
d = sup{d(t)1 t E T} = sup{lI/(t)II I t E T} . To prove the proximinality of l¥, by Theorem (1.8) and Proposition (2.1), only the lower semicontinuity of cp at x E XU, W) needs proof. Let IE L, I fit l¥, x E XU, W) be given. Notice that, for any given x E X we have
cp(x)={O}
if xE T,
cp(x) = {v E Exlliv - l(x)II::::; d}
if x g T.
Let g E Wand r> 0 be given such that cp (x) n BAg(x » ~ 0. Notice that, by the first part of the proof, x E T and therefore g(x) = O. Moreover, since x E XU, W) C T, d = d(x) = II/(x)II, i.e. 11/(x) - g(x)11 = d < d + r. By upper semicontinuity of 1- gEL, there is some neighbourhood V of x in X such that
II/(t)- g(t)11 < d + r
for all t E V.
(i). t E V n T. Then cp(t) = {O} and g(t) = O. Hence, cp(t) n B,(O) ~ 0. (ii). t E V and t fit T. Then W(t) = E, and cp(t) = {u E E,lllu - 1(t)11 : : ; d}. Choose wE W such that I(t) = w(t), and let gA = Ag + (1- ..\)w, where ..\ = d(d + y(t))-l, and y(t) = 1I/(t) - g(t)ll- d. Firstly: l!gA (t) - 1(t)II = ..\llg(t) - 1(t)II = ..\(d + y(t» = d.
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J.B. Prolla / Existence of Best Approximants
Secondly: /IgA (t) - g(t)II = (1- A )IIg(t)- /(t)11
= r(t) = II/(t) Hence
g(t)ll- d < r.
e, (t) E fP(t) n B,(g(t». 0
Theorem (2.6). Let X be a paracompact Hausdorff space and let E be a Banach space. For every closed subset T C X, the closed vector subspace W
= {IE "€b(X, E)I/(t) = 0, t E T}
is proximinal in cgb (X, E). Proof. For each x E X and let Ex = E. For each / E "€b(X, E), / g W; fP is a bounded carrier and W satisfies the condition that given x E X and t E rp(x) there exists agE W with g(x) = t and Ilgll.,; Ilfll + d. By the same argument as in Theorem (2.5), fP is lower semicontinuous and then by Remark (1.12)(b), the W-carrier fP has a selection in w: By Proposition (1.2) this selection is a best approximant to / in w: 0
3. Proximinality of Closed Two-Sided Ideals In this section we shall assume that each normed space Ex is in fact a normed algebra and that the space of all cross-sections over X has been made a linear algebra by defining the product of two cross-sections / and g to be x~ /(x)g(x). If L is now a sub algebra of the Cartesian product of all the spaces Ex which is a normed space of cross-sections over X, then L is also a normed algebra. Indeed
II/gil = sup{II/(x)g(x)II' x E X} .,; sup{II/(x)11·llg(x)111 x E X} .,; 1If11·llg II , for all / and g in L. Thus, if L is both a subalgebra and a normed space of cross-sections over X, according to Definition (1.1), then we shall say that
J.B. Prolla / Existence of Best Approximants
633
L is a normed algebra 01cross-sections over X. If, moreover L is complete, then we say that L is a Banach algebra 01 cross-sections over X. Let us now consider the following condition, where L itself is assumed to be a '?6b(X, ~)-module: (3.1)
Every closed right (resp. left) ideal in L is a '?6b(X,
~)-module.
With regard to (3.1), let us point out that it is implied by the following condition (see Kaplansky [5]): (3.2)
For every 1 E L,
1 lies
in the closure of IL (resp. LI) .
When X is compact, and L is a '?6b(X; ~)-module, using the StoneWeierstrass theorem for '?6(X, ~)-modules of cross-sections (see [8, Th. 1.30]), (3.2) follows from (3.3)
For every x E X, and every v E Ex, v lies in the closure of vEx (resp. Exv) .
Clearly, (3.3) is satisfied if Ex has an identity, or an approximate right (resp. left) identity. Theorem (3.1). Let X be a compact Hausdorff space, and let L be a Banach
algebra 01 cross-sections over X which is a '?6(X; ~)-module. Let us assume that each normed algebra Ex is topologically simple (i.e. has no proper closed two-sided ideal). Then every proper closed two-sided ideal I in L, which is a '?6(X, K)-module, is proximinal in L. Proof. By Theorem (2.7) it is sufficient to prove that I has the form 1= {f ELI I(t) = 0, t E T} for some closed subset. Let
T = {x E
xl I(x) = 0, for
all 1 E I} .
By condition (i) of Definition (1.1), the set T is closed. Let
W
= {fE
LI/(t) = 0, tE T}.
Clearly, lew, let now 1 E W, and assume, by contradiction that 1 Ii 1. Since I is a '?6(X; IK)-module, and X is compact, it follows from [8, Th.
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J.B. Prolla / Existence of Best Approximants
1.30], that for some point x EX; f(x) does not belong to the closure of I(x) in Ex' Now the closure of I(x) is a closed two-sided ideal in Ex, and therefore reduces to {O}. Hence f(x) -::I- O. Since fEW; x g T. However, we have just seen that I(x) = {O}. Hence x E T. This contradiction shows that in fact W C 1. 0 Corollary (3.2). Let X, L and each Ex be as in Theorem (3.1), and assume
that (3.3) is true. Then every proper closed two-sided ideal in L is proximinal. Proof. We have seen that, since X is compact, (3.3)=? (3.2)=? (3.1). Corollary (3.3). Let X be a compact Hausdorff space. Every proper closed
ideal in C6?(X; K) is proximinal.
References [1] R.c. Buck, Approximation properties of vector-valued functions, Pacific J. Math. 53 (1974) 85-94. [2] F. Cunningham jr. and N.M. Roy, Extreme functionals on an upper semicontinuous function space, Proc. Amer. Math. Soc. 42 (1974) 461-465. [3] G. Gierz, Bundles of Topological Vector Spaces and Their Duality, Lecture Notes in Math. 955 (Springer, Berlin, 1982). [4] R. Godement, Theorie generate des sommes continues d'espaces de Banach, c.R. Acad. Sci. Paris 228 (1949) 1321-1323. [5] 1. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951) 219-255. [6] S. Machado and J.B. Prolla, An introduction to Nachbin spaces, Rend. Circ. Mat. Palermo 21 (1972) 119-139. £7] E. Michael, Continuous selections I, Ann. of Math. 63 (1956) 361-382. [8] J.B. Prolla, Approximation of Vector Valued Functions (North-Holland, Amsterdam, 1977). [9] J.B. Prolla and S. Machado, Weierstrass-Stone theorems for set-valued mappings, J. Approx. Theory 36 (1982) 1-15. [10] G. Shilov, On continuous sums of finite-dimensional rings, Sbornik N.S. 27 (1950) 471-484. [11] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401-485.