Nearest points and support points in Banach spaces

JOURNAL

OF MATHEMATICAL

ANALYSIS AND APPLICATIONS 48, 794-800 (1974)

Nearest Points and Support

Points in Banach Spaces

CLIFFORD KOTTMAN Oregon State Universaty,

Cornallis,

Oregon 97331

AND

BOR-LLJH LIN University

of Zowa, Iowa Czty, Iowa 52240 Submttted by Ky Fan

1. INTRODUCTION The following proposition theorem of R. C. James [3].

is well

known

and is a consequence

of a

PROPOSITION. A Banach space, X, is rejlexive if and only if for each closed convex subset (or for each closed subspace), A, of X each point in X admits a nearest point in A. For a proof see [7] or [lo, p. 2531. In this paper we establish some related characterizations of reflexivity for Banach spaces in terms of the support properties of closed convex subsets. The unifying theme for our discussion will be the following decomposition, which was motivated by a paper of M. 2. Nashed [9]. We use the symbol PA(u) to denote the (possibly empty) set of points in 9 nearest to 7.4. DEFINITION. Let Y be a family of closed convex subsets of a Banach space X. For a real number, (Y, we shall say X possesses property D(or, 9) if for each ,4 E 9 and each ICE X there is a pomt u E X such that x E u + aPA( (We shall use the convention that u + o = o .) Nashed, in the paper cited above, showed that every Hilbert space possesses D( 1, YO) where Y0 is the famdy of all closed nonempty convex subsets of the space. Our investigation is divided into three sections which consider, respectively, the cases when (Y > - 1, 01 = - 1, and ar < - 1. In each case we obtain a characterization of reflexivity (Theorem 1, Theorem 2, and

794 CopyrIght AI1 rights

0 1974 by Academic Press, Inc. of reproductmn m any form reserved.

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Corollary 1.) In Theorem 3 a characterization for finite dimensional Banach spaces is obtained. We remark that if 01and /3 are real numbers (a # 0) and one defines a Banach space X to have property D(or, p, 9) if for each A E Y and each N E X there is a II E X such that x E OLU+ /U’,(U) then property D(ol, p, 9’) is equivalent to property D@/a, 9). We shall use the following notations. X ~111always denote a real Banach space, B(x, I) represents {y E X: /j x - y I/ < I}, and U = B(0, 1). If A is a subset of X and .r E X we define d(x, A) = inf{(( r - y 11:y E A}. We say a functional f~ X* (the space of all continuous linear functionals on X) supports d if there is a point x0 E =2 such that f(x,,) = sup{f( y): y E =1}. Such a point x0 is called a support point of A. Notice that if u is not an element of a closed convex set A then each point of Ps4(u)is a support point of both -q and B = B(u, d(x, A)). (The Hahn-Banach theorem guarantees the existence of a functional f~ X* which separates -4 and B. If f supports ,-I at the pomts of PA(u) then -f supports B at the same points.) Let .‘cbe a support point of -4. A, will denote {f~ X*: lifll = I and f supports -4 at x). The conjugate of a point .1cis defined by .rc =: (f E X*: ll.fll = 1 and f (.r) = 11x II}. A nonempty subset -4 of X IS called a sun if for each ZIE PA(u) and each t 2 0, ZIE P,JzI -t t(u - ~1)).We ~111use the fact that each convex subset of X is a sun (see [2] or [6]). A subset of X is said to be Einearly bounded if it meets each line in a bounded set. We shall be interested in the following subsets of the family ,YO of all nonempty closed convex subsets of X: Sp = all linearly bounded sets m yI, , Ya = all bounded sets m .YO, Ya = all linear subspaces of Yi _

2. THE CASE WHEN 01> - 1

The following theorem characterizes those Banach spaces which possess D(ol, YZ) for i = 0, 1, 2, 3 and CL> -1. Statements (c), (d), and (e) are included in this theorem to emphasize the difference between this case and the cases when oi = - 1 and OL< -1. 1. Let 01be a real number such that 01 > -1.

THEOREM

are equivalent:

(a)

X is reflexive,

(b)

X possessesD(oL,YO),

Then the following

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KOTTMAN

(c)

X possesses D(ar, 5$),

(d)

X possessesD(cy, Y2),

(e)

Xpossesses D(or, 9?J.

AND LIN

Proof. (a) => (b). Let A E 9s and let “Vbe a point in X. Since X is reflexive P,(x/(l + a)) # o . Let z, E P,(x/(l + m)) and u = x - (YV. Then since u = w + t(x/(l + a) - w) for t = 1 + OL> 0 and since A is a sun we have v E PA(u) and thus x E ZJ+ (ypA(u).

(b) * (c) * (d) and (b) ti (e). These implications Sp,C 9r C 9a and 9s C 9a . (d) * (a) and (e) * (a). above, it suffices to show such that (1 + 01)xE u + to verify that (1 + a)x -

are trivial since

Let A E 9s u 9a and let x E X. By the Proposition that PA(x) # 0. By the hypothesis, choose u E X PA(u). Using the fact that A is a sun it is easy u E oQA(x). 1

3. THE CASE WHEN a = -1

No Banach X has property D(- 1, 9) unless Y C 9r . This is because if A is not linearly bounded, A contains a ray, say {x + ty: t > 0}, where y # 0. Now y $ u - PA(u) f or any point u E X, otherwise u E PA(u) + y, which by the fact that A is closed, convex, and contains the ray would imply u E A, contradicting y # 0. The following lemma shows that the only sets which allow the decomposition for 01= --I are those which are supported by a collection of functionals rich enough to support U at each point of norm 1. LEMMA

1. X possesses D(-

each x E X (x # 0) there is an

1,9)

f

if and only ;f for E xc which supports A.

each A E 9’

and

Proof. Suppose X has property D(- 1, 9’). Let A E 9 and x E X (x # 0). Now x = u - ZJ for some o E PA(u). Thus if f is a functional of norm 1 which separates A and B(u, 11u - w 11)then f supports A at z, and f E XC. Conversely, given A E 9 and x E X (x # 0) choose f E xc such that f supports A at a point y. If we define u = x + y then x = u - y and it remains only to show y E PA(u). This follows from the fact that if z is any point of A then 11u - z 11> f (u - z) = f(x) + f (y) - f(z) > f(x) =

II.?cII= II’I--Yll.

I

In view of the previous lemma, the following theorem states that it is precisely the reflexive Banach spaces in which each closed convex bounded

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set is supported by a collection of functionals large enough to support the unit ball at each of its boundary points. In this light it is reminiscent of J. D. Pryce’s version of James’ theorem [l 11. THEOREM

2.

X is rejexiwe if and onZy if X possesses D( - 1, yZ).

Proof. If X is reflexive, then U and each A E yZ are weakly compact and therefore are supported by each f E X*. Conversely, suppose X is not reflexive. Let Y be a separable nonreflexive closed linear subspace of X. By a theorem of Mazur [S, p. 781 there is a point x in Y with 11x 11= 1 such that there is a unique functional f E I-* supporting the unit ball of Y at .1cwith 11 f II = 1. Since Y is not reflexive there exists [5, p. 161 a symmetric bounded closed convex body rZ in Y such that the distance between A and f -I( 1) is zero and iz n f -‘( 1) = o . Since A is symmetric, f cannot support A. Furthermore, if g E xc, that is if g is any Hahn-Banach extension of f to X, then g cannot support A for otherwise g restricted to Y (which is f) would also support -4. Since iz E 9’ , Lemma 1 shows X does not possess property D(- I, Sp,). u

It should be noted that Theorem 2 does not imply that a subset il of X is weakly compact if X possesses D(- 1, {A}); for example, let rl he the unit ball of a nonreflexive Banach space. THEOREM

3.

X is finite dimensional if and only if 9 possesses D( - I, Y;).

Proof. The necessity follows from the fact that in finite drmensional spaces one has both reflexivity and the property that 9i = 9a . To show sufficiency, let X be infinite dimensional and choose a closed separable subspace Y of X such that Y is infinite dimensional. As m the proof of Theorem 2, let x be a point of 1’ with II x /I = 1 such that there is a unique f E Y* with I/f 11= 1 supporting the unit ball of Y at x. B> [7, Theorem 51 there is a symmetric linearly bounded closed convex set A such that the distance from A to f -‘( 1) is zero and d n f -‘( 1) = ~5. The remainder of the proof is similar to that of Theorem 2. 1

4. THE

CASE WHEN 01 <

-1

If x is not an element of the closed convex set A, then y is a nearest point in A to x if (x - Y)~ n A, # a,. This motivates the following DEFINITION. Let A be a nonempty closed convex set in X and let x E X. We say a point y E A is a far point in A from x if (y - x)~ n A, # 0.

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Equivalently, we could say y is a far point in A from x provided there is an f~ X* such that A and B(x, 11.v - y 11)are both supported by f at y. Let -4 be a closed convex set in X and let x E X. Notice that if y is a far point in A from x and z is any point in {tx + (1 - t)y: t < 0}, then y is a point of -4 nearest to z. If x $ A, y E P,(x), and z is a point in {tx + (1 - t)y: t < 0} then y is a far point in A from a. Also, if there is a farthest point, y, in A from x, then y is a far point in A from x. The following theorem shows that the existence of the decomposition for 01< - 1 depends on the existence of far points. THEOREM 4. Let 9’ be a family of closed convex subsets of X and let OL be a real number with 01 < - 1. Then X possesses D(ol, 9) if and only if for each ,4 E 9’ and each x $ A there is a far point in A from x. Proof. Suppose X possessesD(cz, Y), A E Y, and x $ A. By the decomposition there is a point u in X such that (1 + a)x E u + oQA(u). If v is any element of PA(u) we have x = tu + (1 - t)v for t = l/(1 + a) so by the remarks preceding this theorem, v is a far point in A from x. To show the converse, let A E Y and let x E X. If x/(1 + a) is in A we have x E u + oQA(u) where u = x/(1 + a). Otherwise, let y be a far point in A from x/(1 + a). Thus x = (x - ay) + ay and it only remains to show that y E PA(x - ay). But this follows from the remarks preceding this theorem since x - ay = t(x/l + a) + (1 - t)y for t = 1 + (11. 1 COROLLARY 1. If 01 is a real number with OL< -1 if and only if X possesses D(ol, Sp,).

then X is reflexive

Proof. For closed linear subspaces of X, nearest points and far points agree, so the result follows from the proposition above. 1 COROLLARY 2. If a < - 1, then X possesses D(a, Ye) where Ye is the family of all nonempty compact convex subsets of X. Proof. Since each A E Yd is compact, it admits a farthest point from each point x E X, and each farthest point is a far point in A from x. i

Corollary 2 implies that each finite dimensional Banach space possesses D(or, Yz) for each 01< - 1. We do not know whether an infinite dimensional Banach space exists which possesses D(ar, Y;) for a < - 1. The following

two theorems assert that a large class of Banach spaces fail to possess that property. In addition, M. Edelstein and A. C. Thompson [l] have recently shown that the space co fails to have D(cu, Y;) for 01< --I in a strong way by constructing a closed convex bounded subset, A, of c,, which has no support functionals which also support the unit ball of c,, ; thus for every

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point x E c,, there are no far points in A from X. In the following, 1, denotes the usual space of p-summable sequences. THEOREM

5.

For each p with 1 < p < co, I, does not possess D(or, LQ

for any 01 < -1. Proof. We shall exhibit a closed convex bounded subset A of I, such that there is no far point in A from the origin. Define A = ((x,) E 1,: X, 2 2-nl* for n = 1,2 )... and cf, (1 + l/n) x,,* < 4) so that A is a closed bounded convex subset of 1, . If x = (x1 , x, ,...) E 1, with x, > 0 for all n and 11x 11=: 1, then the unique element L E xc is 3 = (xlpP1,xl-‘,...). Letting y = (yf-‘, y2p-l,...) for each point y = (yl , yz ,...) in 1, with yn > 0 for all n, it is easy to seey is a far point in tl from the origin if and only if y E -4 and y(y) = sup{ j(x): x E A}. Let .L”= (x1 , x2 ,...) be a point in A. If XI*=, (1 + l/n) x,* < 4, choose X > 1 such that zrSp=,(1 + l/n)(X.Q = 4 and let y = hi. Now y E A and .G(y) = G(x) > G(x), th us x is not a far point in A from the origin. Suppose that I:=1 (1 + 1,/n) x, P = 4. Choose a positive integer k such that

(1) XI, > 2-h/*, (2)

Xk z

xL+l>

and define y = (yl , y2 ,...) as follows:

yn

=

&

- <)1/P

1(XRl + wp

if if if

n#k, n#k+l, n = R, n=k+l,

where E > 0 is chosen small enough that (xk” - ~)l/p > 2-“/e and 6 is defined by (1 + l/(k + 1))6 = (1 + l/K)<. Then

--1 f (1+~)x,l’-(1+~)r+(1+&)S=4soyEd. n=1 Moreover, f(y) - i(x) = (x,” - q/1’ “E-1 + (xf+1 + S)llP cc,“;: - XL* - x,p+, ) which is strictly positive if and only if (q-‘/q;:)((x,p

- 6)1/P- Xk) > xk+l - (x;+l + syp.

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AND LIN

Using condition (2), this happens if (xk* - Eyp + (xi+1 + 8)lip > xk + ‘h+l ’ Definingf(c) = (xIc” - ,)l/P + (x!+~ + 8)llP it is easy to check thatf(0) xk + xLfl and that f’(0) > 0 and thus for small c, f(c) > f(0). 1

=

THEOREM 6. For each nonrefrejciere Banach space X there is a Banach space Y isomorphic to X such that Y fails to possess property D(oL, z??~)for

a < -1. Proof. Let f E X* with (1f (( = 1 and for each x E X define [ x [ = max(lj x Ij,2f (x)), where 11. Ij is the original norm on X. Let Y be the vector space X equipped with the norm 1 . /. The set C = {y E Y: 4 < 2f (y) = 1y 1 < l} has nonempty interior, so by a construction of Klee [5] one may construct a bounded closed convex set A which is contained in the interior of C, such that the distance from A to f-l(&) is zero, and such that Anf-I(+) = o. Th ere is no far point in A from the origin. The proof is completed by applying Theorem 4. 1 COROLLARY 3. If X is a Banach space such that ewery space isomorphic to X possesses D((Y, Y;) for some CY< - 1, then X is reflexive.

REFERENCES 1. M. EDELSTEIN AND A. C. THOMPSON, Some results on nearest points and support points m co, Paczjic J. Math. 40 (1972), 553-560. 2. N. V. EFIMOV AND S. B. STECHKIN, Approxlmatwe compactness and Chebychev sets, Dokl. Akad. Nauk SSSR 140 (1961), 522-524. 3. R. C. JAMES, Characterlzatlons of reflexiwty, Studia Math. 23 (1964), 205-216. 4. V. L. KLEE, JR., The support property of a convex set in a linear normed space, Duke Math. J. 15 (1948), 767-772. 5. V. L. KLEE, JR., Some characterrzatlons of reflexiwty, Rev. Ci. (Lima) 52 (1950). 15-23. 6. V. L. KLEE, JR., Remarks on nearest pomts m normed hnear spaces, Proc. of the Colloquium on Convexity, Copenhagen 1965, pp. 168-176. 7. BOR-LUH LIN, Distance sets m normed vector spaces, Nieuw Arch. Wisk. 14 (1966), 23-30. 8. S. MAZUR, Uber konvexe Mengen m linearen nornuerten Ratimen, Studia Math. 4 (1933), 70-84. 9. M. 2. NASHED, A decomposition relatwe to convex sets, Proc. Amer. Math. Sot. 19 (1968), 782-786. 10. R. R. PHELPS, Umqueness of Hahn-Banach extensions and unique best approxunation. Trans. Amer. Math. Sot. 95 (1960), 238-255. 11. J. D. PRYCE, Weak compactness in locally convex spaces, Proc. Amer. Math. Sot. 17 (1966). 148-155.