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A duality theory for Banach spaces with the Convex Point-of-Continuity Property Hare, David Edwin George 1987

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A Duality Theory for Banach Spaces with the Convex Point-of-Continuity Property by David Edwin George Hare B.Math., The University of Waterloo, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA July 1987 © David Edwin George Hare, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M A T H E M A T I C S The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J U L Y 1 0 , 1 9 8 7  DE-6(3/81) Abstract A no rm ||-|| on a Banach space X is Frechet differentiable at x £ X if there is a functional / 6 X* such that M m l l l « + » I I - I N - / M l _ 0 . «-° Ml This concept reflects the smoothness characteristics of X. A dual Banach space X* has the Radon-Nikodym Property [RNP) if whenever C C X* is weak*-compact and convex, and e > 0, there is an x E X and an a > 0 such that d i ame te r{ / 6 C : f(x) > sup g(x) — a} < e . g€C This property reflects the convexity characteristics of X*. Culmina t ing several years of work by many researchers, the following theorem established a strong connection between the smoothness of X and the convexity of X*: Every equivalent norm on X is Frechet differentiable on a dense set if and only if X* has the RNP. A more general measure of convexity has been recently receiving a great deal of attention: A dual Banach space X* has the weak* Convex Point-of-Continuity i i Property (C*PCP) if whenever 0 ^  C C X * is weak*-compact and convex, and e > 0, there is a weak*-open set V such that V n C ^ 0 and diam V n C < e. In this thesis, we develop the corresponding smoothness properties of X which are dual to C*PCP. For this, a new type of differentiability, called cofinite Fre-chet differentiability, is introduced, and we establish the following theorem: Every equivalent norm on X is coGnitely Frechet differentiable everywhere if and only if X* has the C*PCP. Representing joint work with R. Deville, G. Godefroy and V . Zizler, an alter-nate approach is developed in the case when X is separable. We show that if X is separable, then every equivalent norm on X which has a strictly convex dual is Frechet differentiable on a dense set if and only if X* has the C*PCP, if and only if every equivalent norm on X which is Gateaux differentiable (everywhere) is Frechet differentiable on a dense set. This result is used to show that if X * does not have the C*PCP, then there is a subspace Y of X such that neither Y* nor (X/Y)* have the C*PCP, yet both Y and X/Y have finite dimensional Schauder decompositions. The corresponding result for spaces X* failing the RNP remains open. iii Table of Contents Abstrac t i i Acknowledgement v I. Introduction 1 1. Basic Definitions and Nota t ion 2 2. His tory 8 3. Pre l iminary Results on CPCP and C*PCP 11 II. The Dua l i ty Theorems 20 1. A New Type of Differentiability 21 2. The M a i n Theorems 26 3. The Strong Case 32 III. The Separable Case 40 1. Phelps Spaces and CPCP 41 2. Some Geometrical Lemmas 43 3. Proofs of Theorems III.2 and III.3 52 IV. Some Structure Theorems 67 1. A Structure Theorem for Non-CPCP Spaces 69 2. A Decomposit ion Theorem for Non-Phelps Spaces 74 References 84 iv I would like to thank my supervisor, N . Ghoussoub, as well as J . Fournier, V . Zizler and R . Devi l le , for helpful discussions during the preparation of this thesis. I would also like to thank the Na tu ra l Sciences and Engineering Research Counc i l and the H . R . M a c M i l l a n Fami ly for their generous financial assistance. Th i s work would never have been completed without my wife's continual support, and might never have been started if not for D . J . Albers . To them, I dedicate this thesis. v Chapter I Introduction This is a thesis about the Geometry of Banach Spaces, that is, the geometric structure of sets from the various topologies on Banach spaces: norm, weak and weak*. We are concerned in this dissertation w i t h the linear structure of convex sets. We begin by defining the terms and concepts which we w i l l require. The reader w i l l find the books of R u d i n [27], Diestel [9] and Diestel and U h l [10] to be good sources of background material for basic Funct ional Analysis and more specialized mater ial on the geometry of Banach spaces. In Section 2, we present some of the history of the subject of this thesis, namely the Convex Point-of-Continuity Property of Banach spaces, and in Section 3 we develop some of the basic tools which we w i l l need in the subsequent chapters. 1 1.1. Basic Definitions and Notat ion A l l Banach spaces considered in this thesis are over the Rea l field, R , and are infinite dimensional unless otherwise specified. Let X be a Banach space, X* its dual . The action of a functional / G X* on an element i £ l w i l l be denoted by (/ , x). The convention (a;, / ) = (/ , x) defines the canonical isometric embedding of X into its double dual , X**, and we w i l l always consider X as a subspace of X**. The weak topology on X is the topology defined by X*: It is the weakest top-ology on X such that a l l the functionals in X* are continuous. Similarly, the weak* topology on X * is the topology defined by X. A base for the weak topology is the family of finite intersections of half-spaces, i.e., sets of the form V = {y e X : \(fk, x - y)\ < 6, k = 1, . . . , m } , where x € X, 8 > 0 and { / j f e } ™ C X*. These sets are called the elementary weak-open sets in X. The elementary weak*-open sets are defined s imilar ly in X*. For purposes of clarity, the dual no rm on X* to a norm ||-|| on X w i l l often be denoted by ||-||*. The notat ion F < X indicates that F is a closed, linear subspace of X. In this case, X/F is again a Banach space, whose elements w i l l be denoted by x, where x is 2 a coset representative of x i n X . The norm in X/F is defined by | |x| | = ^nf ||x + h\\. If F < X, the annihilator of F is F1 = {/ £ X* : ( / , F ) = o}. If F < X*, the weak*-annihilator of F is F ± = {x £ X : (x, F ) = o}. Note F ± = F1- n X if F < X * . We recall the well known isometric relations X * / F ± = F * , ( X / F ) * = F±, and also the fact that {F^)1 is the weak*-closure of F when F < X * . The open and closed unit balls and the unit sphere of X w i t h respect to a given no rm w i l l be denoted by Ux, Bx and Sx> respectively. If C C X , then C is convex if x , y £ C and 0 < t < 1 imply £ x + ( l — i ) y £ C . For example, elementary weak- and weak*-open sets are convex. A functional / £ X * supports C at x 0 £ C if sup( / , C ) = (/ , x 0 ) ; / exposes C at x 0 if / supports C at, and only at, x 0 ; / strongly exposes C at xQ if / exposes C at x 0 and also / defines the relative norm topology on C at x 0 , i.e., if {xn} C C and (/ , xn) —>• ( / , x 0 ) , then x w —> x 0 . Respectively, / is called a supporting, exposing or strongly exposing functional for C , and x 0 is called a support, exposed or strongly exposed point of C. A s suggested by the notation used in the previous paragraph, convergence in norm is denoted by xn —• x 0 ; convergence in the weak topology by x n x 0 ; and convergence in the weak* topology by x n x 0 . A s/j'ce of C , C 7^  0, is a relatively weak-open subset of C determined by a 3 single functional: S1{C, f,<x) = {xeC: </, x) > sup( / , C)-a}, where / G X * and a > 0. C is dentable if it has slices of arbi tar i ly small (norm) diameter, where the diameter of a set A is defined by We w i l l use the convention diam0 = —oo, so the statement di&mA > 0 implies, in part icular , that A is non-empty. Note that a slice is always non-empty. If the slices of C containing some (fixed) point x0 G C have arbi t rar i ly small diameter, then x0 is a denting point of C. Tha t is, x0 is a denting point of C if it has a relative no rm neighbourhood base in C consisting of slices. Note that a strongly exposed point is a denting point, w i t h al l of the slices determined by a single (fixed) functional, namely the corresponding strongly exposing functional. If X is a dual space, say X = Y*, then the prefix weak* applied to any of the definitions given above indicates that the corresponding functional is taken from the predual space, Y, and not from Y** . For example, a weak*-slice of a set 0 ^  C C X * is a set of the form S1(C, x, a) = {/ G C : (x, / ) > sup(x, C) - a] , where x G X and a > 0. 4 If C C X is convex, then x £ C is an extreme point of C if the relation x = il+f-, where y, z £ C , implies y = 2 = x. The set of extreme points of C is denoted by E x t ( C ) . If C is an arbitrary subset of X, we denote the closure of C in the norm, weak and weak* topologies by C, C™ and C*, respectively. In the last case, the meaning is clear if X is a dual space. Otherwise, we mean by this the weak*-closure of C i n X**. The convex hull of C is co(C) = f\{Cl D C : Cx convex}. The notat ion co(C) or co* (C) indicates the closure of the convex hul l in the corresponding topology. A s a convex set has the same closure in the weak and norm topologies, we do not need to consider c o ^ C ) . Similar ly , we denote the linear span, closed linear span and weak*-closed linear span of C by s p ( C ) , sp(C) and sp*(C) , respectively. Turning from geometry to calculus, let <p: X —• R be a continuous, convex function, i.e., (p is continuous, and <p(tx + ( 1 - t)y) < ttp{x) + ( 1 - t)<p(y) for a l l x , y £ X and 0 < t < 1 . Then for a l l x, h £ X, the one-sided l imi t (*) ^ = f l i m exists, and defines a continuous function <p'(x)(-):X —• R . If, in addit ion, <p'(x)(-) is 5 a linear function, so <p'(x) E X*, then <p is said to be Gateaux differentiable at x. If is Gateaux differentiable at x, and, moreover, the l imi t in (*) is uniform i n h E , then is Frechet differentiable at x. Since p is assumed to be convex, we may reformulate this as follows [5]: ip is Frechet differentiable at x i f and only if Observe that i f <p is either Gateaux or Frechet differentiable at x, then since <p is convex and continuous, we have that.for a l l h E X, (<p'(x), h-x) < <p(h) - <p(x) . Geometrically, this means that the hyperplane {(h, (<p'{x), h - x) + <p(x)): h e X} c X x R supports the epigraph of ip, that is, the set {(h, r) E X x R : h E X, <p(h) < r] , at the point (x, <p(x)). Note that a norm can never be differentiable, in any sense, at the origin. Thus, i n those results i n this thesis where the expression differentiable everywhere is used to describe a norm, this means everywhere except at 0, which is equivalent to being differentiable everywhere in Sx. 6 We w i l l have much recourse to the following theorem [28]: Theorem 1.1 (Smuljan) Let ||-|| be a norm on X, ||-||* its dual on X*, and let xe X, f E X*. Then: a) ||'|| is Frechet differentiable at x, with derivative f, if and only if x weak*-strongly exposes Bx* at / . b) \\-\\* is Frechet differentiable at f, with derivative x, if and only if f strongly exposes Bx at £• It is important to realize that while the Frechet derivative of an arbitrary norm is an element of the dual space, the Frechet derivative of a dual no rm is an element of the predual space. F ina l ly , we recall that a closed, bounded, convex and balanced neighbourhood C of 0 i n X defines the unit ba l l of an equivalent no rm on X. If X is a dual space, then, by Alaoglu 's Theorem, C defines an equivalent dual n o r m on X if and only if C is weak*-compact. 7 1.2. H i s t o r y A m o n g al l of the properties of abstract Banach spaces which have been stud-ied in the past half-century, there is one which stands out as a testimony to the richness and mystery of the subject: the Radon-Nikodym Property, RNP for short. Or ig inal ly introduced i n the 1930's to provide integral representations of operators on Banach spaces, it describes those spaces on which the classical Radon-Nikodym Theorem can be applied to measures valued i n the space. Dur ing the late 1960's and the 1970's, there was a growing awareness that the RNP is as much a geometric property of Banach spaces as it is an analytic property. One of the most s t r iking examples of this is the fact that a Banach space X has RNP if and only i f every closed, bounded set is dentable, if and only if every closed, bounded, convex set is dentable [10]. We w i l l make extensive use of this characterization of RNP spaces. In 1968, A s p l u n d [l] began studying the class of Banach spaces X on which every continuous, convex function is Frechet differentiable on a dense Qs subset of X. Such Banach spaces are now known as A s p l u n d spaces. A s p l u n d and others realized that the condit ion " § g n is unnecessary, and that it is sufficient to posit that every equivalent norm on X is Frechet differentiable on a dense set. 8 The connection between Asplund spaces and RNP dual spaces suggested by Smuljan's Theorem (Theorem 1.1) motivated considerable research throughout the 1970's, and, proceeding through the work of Asplund [1], Namioka and Phelps [23], Huff and Morris [17], John and Zizler [18], and finally Stegall [29], the following classification theorem was obtained: Theorem 1.2 A Banach space X is an Asplund space if and only if X* has RNP. Dually, X has RNP if and only if X* is a weak*-Asplund space (every equivalent dual norm on X* is Frechet differentiable on a dense set). In the 1980's, many researchers have begun studying larger classes of spaces than RNP spaces, often in an attempt to better understand the RNP itself. One such class consists of the spaces X which have the Point-of-Continuity Property (PCP): Every closed, bounded, non-empty subset C of X has relatively weak-open subsets of arbitrarily small diameter (the terminology will be made clear in the statement of Lemma 1.3). Since a slice of C is an example of a relatively weak-open subset of C, it is clear that a space with RNP also has PCP. The converse is not true [12]. The PCP has been quite successfully studied by, among others, Edgar and Wheeler [12] and Ghoussoub and Maurey [13], [14]. However, in enlarging the study of RNP spaces to include PCP spaces, the often desirable characteristic of convexity was sacrificed. In particular, differentiability properties implicitly depend upon convexity, and there are no results relating PCP and differentiation. Also, 9 Namioka and Phelps showed in [23] that the dual version of PCP, in which one considers relatively weak*-open subsets of weak*-compact sets in X*, is the same as the corresponding dual version of RNP, which in tu rn is the same as RNP itself, and hence provides no new information. In an attempt to remedy this si tuation, the class of Banach spaces under consid-eration was enlarged further: A Banach space X has the Convex Point-of-Continuity Property (CPCP) if every closed, bounded, non-empty, convex subset C of X has relatively weak-open subsets of arbi trar i ly small diameter. It is clear that PCP implies CPCP, and, again, the converse is not true [15]. More interesting, perhaps, is that the dual version of CPCP, denoted C*PCP, is distinct from RNP [16]. The thrust of this thesis is to obtain classification theorems in the style of the A s p l u n d - i J i V P Dua l i ty Theorem (Theorem 1.2) for Banach spaces w i t h CPCP or w i t h C*PCP. 10 1.8. Pre l iminary Results on CPCP and C*PCP CPCP was introduced by Bourgain in 1980 [3], and is a good i l lustrat ion of our remarks in the preceding section: His purpose in defining CPCP was to assist i n the study of RNP. Bourga in proved several key results about CPCP, and, bui ld ing on this work, several researchers are now studying CPCP for its own sake. It has come to be viewed as an important tool in the study of convexity properties of Banach spaces. The definition of the dual version of CPCP is credited to Godefroy in [16]: A dual Banach space X* has the weak* Convex Point-of-Continuity Property (C*PCP) if every weak*-compact, convex subset C of X* has relatively weak*-open subsets of arbi trar i ly small diameter. It is perhaps wor th noting that the connection between CPCP and C*PCP is much closer than the definitions, as stated, suggest: A convex set is closed if and only if it is weakly closed, and a subset of X* is weak*-compact if and only if it is weak*-closed and bounded. Hence CPCP is a property of weakly closed, bounded, convex sets, and C*PCP is a property of weak*-closed, bounded, convex sets. In [15] and [16], Ghoussoub, Maurey and Schachermayer exhibited: (a) A B a -nach space which has CPCP but not PCP; (b) A dual space which has C*PCP but not RNP; and (c) A Banach space which does not contain an isomorphic copy of lx 11 and whose dual does not have C*PCP. Regarding example (c), Pelczynski [24] has shown that if X contains an isomorphic copy of tx, then X* contains an isomorphic copy of Lx. If X* has C*PCP, then it is immediate that X* has CPCP, and also that any closed subspace of X* must have CPCP. Since Lx does not have CPCP, it follows that X* having C*PCP implies that tx does not isomorphically embed i n X. It is unknown at this t ime if, for dual spaces, the properties C*PCP and CPCP are distinct . L e m m a 1 . 3 (Bourgain [3]) Let X be a Banach space with norm \\-\\. Then: a) The following are equivalent: i) For every closed, bounded, convex 0 ^ C c X and for every e > 0, there is a weak-open U C X such that 0 < d iam U n C < e. ii) For every closed, bounded, convex 0 ^  C C X, the identity map id: (C, wk) —> (C, ||-||) has a point of continuity. b) The following are equivalent: i) For every weak*-compact, convex 0 ^ C C X* and for every e > 0, there is a weak*-open U C X* such that 0 < d iam U fl C < e. ii) For every weak*-compact, convex 0 ^  C C X*, the identity map id: (C, w*) —• (C, ||-||*) has a point of continuity. 12 A point x E C at which id: (C, wk) —• ( C , ||-||) is continuous is called a point of weak-to-norm continuity of C. We define a point of weak*-to-norm continuity correspondingly. Observe that since the weak and weak* topologies are not locally bounded, a point of weak- or weak*-to-norm continuity of C can never be in the norm interior of C. This is clearly also true for denting points, exposed points, etc. A n immediate consequence of L e m m a 1.3 is that if every closed, bounded, con-vex, non-empty subset of X has a point of weak-to-norm continuity, then every such set has a weakly dense, weak Qs subset of points of weak-to-norm continuity [3]. The dual statement for points of weak*-to-norm continuity is also true. L e m m a 1.4 (Bourgain [3]) Le t X be a Banach space. Then: a) If X does not have CPCP, then there is an open, bounded, convex set C C X and an e > 0 such that if x G C, F < X has finite codimension, and V is a weak-open neighbourhood of x in X, then d i a m C D (x+ F) fl V > e. b) If X* does not have C*PCP, then there is an open, bounded, convex set C C X* and an e > 0 such that C is weak*-compact and if f G C, F < X* has finite codimension and is weak*-closed, and V is a weak*-open neighbourhood of f in X*, then d i a m C n (/ + F) n V > e. Bourga in i n fact stated and proved only part (a) of L e m m a 1.4. However, the proof of part (b) is essentially the same as for part (a), and it has become standard 13 usage to refer to L e m m a 1.4, in its entirety, as Bourgain 's L e m m a (sometimes as Bourgain 's Bubble Lemma) . We shall also use this nomenclature. We include a proof of L e m m a 1.4, as our proof is substantially different from Bourgain 's proof, and we wish to record some corollaries of the method itself. P r o o f o f L e m m a 1.4 Observe that by contraposition and induct ion on the codimension of F, to prove (a) it suffices to prove the following: Let C C X be bounded and convex, e > 0, / £ X*, and a E R such that i n f ( / , C) < a < sup( / , C). Let V be a weak-open set such that 0 < d iam V n C n / _ 1 (a) < e. Then there is a weak-open U such that 0 < d iam U D C < e. Withou t loss of generality, we may assume that V is an elementary weak-open set. B y translating C, if necessary, we may assume a — 0. For (3 > 0, let Up=vn r 1 (-/?, p). Let xx, x2 G V n C such that ( / , xx) — —(/, x2) > 0. For i = 1, 2, let Kt = {xi + t{y-xi):t>0,y£VnCn f ' 1 (0)} . if t- is the positive cone generated by xt- and V C\ C n / _ 1 (0). Note that the choice of xx and x2 implies V D C C Kx U K2, and so Up n C c i K t U K J n 0 ) . 14 We c l a im that we can choose P sufficiently small so that diam( JfC 1 U K2) n f-1 (-/?, P) < e . F r o m this, we obviously have d iam Upf)C < e, which w i l l finish the proof of part (a). To establish the c la im, let ui = xi + hivi ~ *,•)> «2 = xj + '2(2 /2 - *y) e u # 2 ) n / - 1 (-/?, p), where j = 1 or 2 and yk G V D C n f~l (0), Ar = 1, 2. Then | ( / , ufc)| < p,k = 1,2. Note that since ( / , yk) = 0, k = 1, 2, *! = 1 - (/, « !> / ( / , xt-> and i 2 = 1 - </, u2>/(/, x3-) . Thus I K - " 2 I I < \\Vi ~ 2/2II + - ( * < + h(Vi - xi))\\ + II2/2 - K + i 2 (y 2 - * y ) ) l l (*) < d i a m F n C n / ' 1 (0) + |1 - * i 1112/x ~ *J + I 1 ~ *al lls/a -< d i a m V n C n r 1 (0) + 2pM/(f, xx) , by the choice of xx and x 2 > where M = sup{ | |y — x f c | | : y G V PI C f l / _ 1 (0), A; = 1, 2} < 00 . 15 Since d i a m F n C n / 1 (0) < e and (/ , x x ) is fixed, it is clear that we can choose /? sufficiently small so that the last expression in (*) is less than e, as required. The proof of part (b) is s imilar . • Remark 1.5 It is important to realize that in the statement of L e m m a 1.4, e depends only on the set C, and not on F or F . Corollary 1.6 a) Let C C X be closed, bounded and convex, let f G X*, and let a G R such that i n f ( / , C) < a < sup( / , C). Then x G C n / _ 1 (a) is a point of weak-to-norm continuity of C D f~l (a) if and only if x is a point of weak-to-norm continuity of C. b) Let C C X* be weak*-compact and convex, let x G X, and let a G R sucA that inf(x, C) < a < sup(x, C). Then f G C Pi x - 1 (a) is a point of weak*-to-norm continuity of C H x _ 1 (a) if and only if f is a point of weak*-to-norm continuity of C. Corollary 1.7 a) Let X be a Banach space which does not have CPCP. Then there is an equivalent norm, \\-\\, on X and an e > 0 such that if | |x | | < 1, F < X is finite codimensional, and V is a weak-open neighbourhood of x, then 1 6 d i a m B j r n (x + -F) n V > e, where "Bx" and "diameter" both refer to the norm ||-||. b) Let X be a Banach space such that X* does not have C*PCP. Then there is an equivalent norm, \\-\\, on X and an e > 0 such that if \\f\\* < 1, F < X* is finite codimensional and weak*-closed, and V is a weak*-open neighbourhood of f, then d iam Bx* H (/ + F) n V > e, where "Bx* " and "diameter" both refer to the norm ||-||*. P r o o f The proofs are immediate from L e m m a 1.4, given the observation that if C is the set produced by L e m m a 1.4, then C + (—C) is a bounded, convex, balanced neighbourhood of 0 which also satisfies the conclusion of the lemma (with the same e). • L e m m a 1.8 Let X be a Banach space, let 0 / C C X be closed, bounded and convex, and let x £ C be a point of weak-to-norm continuity of C. Then x is a point of weak*-to-norm continuity of C* C X**. P r o o f This follows from two observations: (a) C is weak*-dense in C*; and (b) a dual no rm is weak*-lower semi-continuous. For suppose V is an elementary weak-open neighbourhood of x in X, say V = {y G X : \{fk, x - y)\ < 6, k = 1, ..., m] , 17 where 8 > 0 and {f^}™ C X * . Consider the set V* = {y** eX** :\(fk,x-y**)\<6,k = l,...,m}. V * is an elementary weak*-open neighbourhood of x in X**. If y**, z** G V* D C , then there are nets {ya} and {za} in V* n C = V Pi C such that y a y** and za ^ z**. Thus , by the weak*-lower semi-continuity of the double-dual no rm ||-||**, | | y " - z " i r < l i m i n f | | y a - z j r a = l i m i n f | | y Q - za\\ < d i a m i n e . Thus , d i a m V * D C < d i a m V n C , which establishes the result. • L e m m a 1.9 Let X be a Banach space and let 0 ^  C C X be closed, bounded and convex. Let y, z G C and 0 < t < 1 and let x = ty + (1 — t)z. If x is a point of weak-to-norm continuity of C, then so are y and z. The corresponding result for points of weak*-to-norm continuity also holds. P r o o f Let e > 0 and let V be an elementary weak-open neighbourhood of x such that d i a m V n C < te. Note that y = jx - ^j-z, so W = jV - ^-z is a weak-open neighbourhood of y. B y the convexity of V and C , d i a m W f l C < } d i a m F n C < e . • 18 Fina l ly , we w i l l require two last results, due to Bishop and Phelps [2], which we w i l l use several times to prove theorems involving density. The first result relates the proximi ty of two functionals to, roughly, the near-parallelness of their corresponding hyperplanes. The second is the well known Bishop-Phelps Theorem. L e m m a 1.10 Let X be a Banach space, f,g(z Sx* and e > 0. If for any x G X the conditions (/, x) = 0 and (g, x) = 1 imply \\x\\ > e - 1 , then \\f + g\\* < 2e. o r | | / - 0 | | * <2e. T h e o r e m 1.11 Let X be a Banach space, and let 0 ^ K C X be closed, bounded and convex. Then the set S = {/ G Sx* • 3x G Sx such that (f, x) = sup ( / , K)} is norm dense in Sx*. 19 Chapter II The Duality Theorems Since a Banach space w i th RNP automatically has CPCP, but the converse is not true, there are evidently two possible directions one could take in developing an A s p l u n d - R N P type duali ty theory for Banach spaces w i t h CPCP: One could require only that certain norms, and not a l l , be densely Frechet differentiable; or, one could consider a different (weaker) definition of differentiability. In this chapter, we w i l l pursue the latter alternative, and we w i l l find that there is a natural definition of differentiability which exactly characterizes Banach spaces which have CPCP. We w i l l consider the former alternative i n Chapter III. After introducing the new type of differentiability i n Section 1, and proving our duali ty theorems i n Section 2, we turn, i n Section 3, to a stronger definition of CPCP, in which the distinguished elements of convex sets are points of strong weak-to-norm continuity, which are convex analogues of strongly exposed points. 20 I I . l . A New T y p e of Differentiability Definition I I . l Let X be a Banach space. A norm, ||-| |, on X is said to be cofinitely Frechet differentiable at x G X if for every e > 0 there exists a finite dimensional subspace F < X such that P + HxiF + II* - Hx/F ~ 2\\x\\x/F I imsup '• = 1 '— < e . fc-»o \\n\\x/F (Observe that necessarily x ^ F.) Examples II.2 a) The usual (supremum) no rm on c 0 is cofinitely Frechet differentiable ev-erywhere: To see this, let x = (xk) G c 0 , w i t h H x ^ = 1. T h e n there is an n0 G N such that n > nQ implies \xn\ < 1 /2 . A l so , there is an nx < n0 such that | | x | | = | x n i | = 1. Let F = s P { e * : 1 < k ^ no> * ^ n i } » where {ek} is the standard basis of c 0 . Then | | £ | | C o / f = H x ^ = 1, and if h G c 0 w i t h 0 < H&II < 1 / 2 , then | | x ± h\\ = \xni ± h n i \ = l ± h n i s g n ( x n i ) , 21 so ||x + A7|| + \\x-h\\ - 2 | | x | = 0 I N I Thus ||•||C o/j? is, in fact, Frechet at x . b) The usual norm on lx is cofinitely Frechet differentiable nowhere: Let x = (xk) £ t-i, w i t h \\x\\x = 1. Let F < lx be finite dimensional and such that HxH^y^i / 0. Let e > 0. Since d i m F < oo, BF is compact, and so for a l l m £ N there is an index nm £ N such that \xnm \ < and if y £ SBF, then \ynm \ < Let 0 < 6 < 1/2 and, for each m £ N , let hm = b~enm, where {ek} is the s tandard basis of £ x . T h e n llx ± hm II = inf ||x ± hm + y\\ = inf ||x ± hm + v l l i II mil y e F II m tfll y & B F II m since | | x ± / ^ l ^ < 3/2 , and i f y £ 3Bp, then \\x±hm+y\\1= \ x k + V k \ + ± S + V n m \ > + y\U — + 6 — 11 tfH1 2m 2 m = llx + y l ^ + 6~ — m > \\x\\ + S- — . m Since m can be chosen independently of 6, and since F is finite dimensional, it follows that ||x + X|| + l l x - A l l -2||x|l 2e sup - L 0 — > sup 2 - — - = 2 . ||fc||=* I N I ™ e N 5 m 22 Thus H ' l l j cannot be cofinitely Frechet differentiable at x. A s in the case of a Frechet differentiable norm, there is a strong connection between the cofinite Frechet differentiability of a norm on X and the size of weak*-slices i n X * (compare Smuljan's Theorem (Theorem 1.1)). We start w i t h a slight strengthening of a result of John and Zizler [18]: P r o p o s i t i o n I I . 3 Let X be a Banach space with norm \\-\\, let e > 0, and let x € X . Then the following are equivalent: a ) l i m a u p l l ^ l M l E f H I W I < e . h—>0 b) There exists a > 0 such that d iam S1(BX* > x-> a) < e-P r o o f (a) =>• (b) Suppose (b) fails. Let a, 6, A > 0 be such that 0 < a < 6\e. B y hypothesis, we can choose f,g(z S1(BX*, a) such that | | / — g\\ > e ( l — 6). Let u £ Sx such that ( / — g, u) > e ( l — 6). Then \\x + Xu\\ + \\x — Xu\\ > (/, x + Xu) + (g, x — Xu) + x) + X(f-g, u) > 2\\x\\ - 2a + Xe(l - 6) > 2\\x\\ - 26Xe + Xe - 6Xe = 2\\x\\ + Xe(l - 36) . Thus sup t±miE_3-m>e[1^s) ||fc||=A 23 Since 6 and A can be chosen to be arbitrari ly smal l , \\x + h\\ + \\x - h\\ - 2\\x\\ hmsup ^ U L — — > e , so (a) fails. (b) =>• (a) Suppose (a) fails. Then there is an e > 0 and a sequence {hn} C X such that \\hn || -* 0, but | |x + hn\\ + \\x - hn\\ > e\\hn\\ + 2\\x\\ for a l l n. Choose { / B } , {gn} c Sx* such that (fn, x+hn) = \\x + hn\\ and (gn, x-hn) = \x-hn\\. Then </«, * + ^n> + (»n. * - K) > 4KW + 2||x|| , hence </» - 9n, K) > e\\hj + 2\\x\\ - (fn + gn, x) > e\\hn\\ . Thus | | / B - f l f B | | > e for al l n. Since ( / n , x) —»• | |x | | and (gn, x) —»• it is clear that no slice of Bx* which is determined by x can have diameter less than e. • Corollary II.4 Let X be a Banach space with norm \\-\\, and let x 6 X. The following are equivalent: a) \\-\\ is cofinitely Frechet differentiable at x. b) For every e > 0 there is a finite dimensional F < X and an a > 0 such that d i a m S ^ B ^ x , x, ct) < e. 24 P r o o f (a) => (b) Let e > 0 and choose F < X w i t h d i m F < oo and such that hmsup - ij-=-— ] L J L < e . h^o \\h\\ Since (X/F)* — F 1 , (b) follows by applying Proposi t ion II.3 to the space X/F. The proof of (b) => (a) is similar. • 25 II.2. The M a i n Theorems In this section we w i l l prove the advertised classification theorems, namely: Theorem II.5 Let X be a Banach space. The following are equivalent: a) Every equivalent norm on X is coRnitely Frechet differentiable at some point. b) Every equivalent norm on X is coRnitely Frechet differentiable everywhere. c) X*has C*PCP. Theorem II.6 Let X be a Banach space. The following are equivalent: a) Every equivalent dual norm on X* is coRnitely Frechet differentiable at some point. b) Every equivalent dual norm on X* is coRnitely Frechet differentiable ev-erywhere. c) X has CPCP. P r o o f of Theorem II.5 We w i l l establish the chain of implications (a) =>• (c) =>• (b) =>• (a). The last of these is clear. (a) =J> (c) Suppose X* does not have C*PCP. B y Corol lary 1.7(b), there is an e > 0 and an equivalent norm, | | - | | , on X such that if | | / | | * < 1, F < X is finite 26 dimensional, and V is a weak*-open neighbourhood of / in X*, then d iam 8^ * f l if + F1) nV > e. We w i l l show that this norm is nowhere cofinitely Frechet differentiable. Let xx G X, F < X w i th d i m F < oo and x x £ F. Wi thou t loss of generality, \\xi Wx/F ~ 1- Let 0 < a < 1, and choose / G UF± such that (f,x~i) = 1 — a / 2 . Let {ek}™ be a basis for F, and let xk = xx + ek, k = 2, . . . , m . Let V = {g G X* : \(g - f, xk)\ < a/2, k = 1, . . . , m] . Then , by the choice of the norm, we have d iam 8X* n (/ + F-1) n V > e . Now / G F1-, so / + F1- = F - 1 . I f j e F 1 , then (ff, x f c ) = (g, xx) = (g, xx), thus 8x.n(f + F±)nV = {geBF± :\{g-f,xl)\< a/2} . B u t |(<7 — / , X j ) ! < a/2 if and only if 1 - a = </, x x ) - a / 2 < (0, x 2 ) < (/ , x x ) + a / 2 = 1 . Since | | x x || = 1, the first inequality says that &X* n (/ + F ± ) n V c Sl(Sjr± , X i , a) , and so d i a m S i ( S j p ± , x l 5 a) > e. Since 0 < a < 1 was arbitrary, the result now 27 follows from Corol lary II.4. (c) => (b) Let ||-|| be an equivalent no rm on X , and assume that X * has C*PCP. Let xx e Sx, e > 0, and 0 < a < 1. Then 8X* n xf1 (a) is weak*-compact and convex, so by Corol lary 1.6, there is an / £ Bx* PI xf1 (a) which is a point of weak*-to-norm continuity of Bx*. Let V be an elementary weak*-open neighbourhood of the point / such that d i am Bx* n V < e, say V = {gEX* :\(g-f, xk)\<6, k = 1, . . . , m } , where {xk}™ C X , xx is as above and 8 > 0. Let -F = sp{</ , xx)xfc - {/, xk)xx : fc = 1, . . . , m } . Note that f e Fx and UxJ > (/, xx) = (/, xx) = a. Let M = max | ( / , xk)\. Then M > a. Suppose g 6 BJTJL such that (</, ^i) = (g, Xi) > a - a8M~x , where 8 is as in the definition of V. Then , by the definition of F, \{f~9, xk)\ = \(f, xk) - (g, Xl)(f, xk)/(f, xx)\ = |(/> x k ) \ I 1 - <0> xi)/a\ < M(6M~l) = 6, 28 k = 1, . . . , ra. Since | | x x || > a, it follows that si(BF±, xx, aSM'1 ) c B P n y , and so d iam Sl(BFx , xx, a.8M~X) < e . The result now follows from Corol lary II.4. Theorem II. 5 is proven. • The proof of Theorem II.6 is essentially the same as that for Theorem II.5. One begins by establishing a dual version of Proposi t ion II.3, namely: P r o p o s i t i o n I I .7 Let X be a Banach space with norm \\-\\, let e > 0, and let f G X * . Then the following are equivalent: < e. b) There exists a > 0 such that d iam S1(BX, f, « ) < e. P r o o f (a) (b) A s for Proposi t ion II.3, (a) (b). (b) =>• (a) Th i s proof is the same as for Proposi t ion II.3, (b) (a), except i n one part icular: We cannot necessarily choose sequences {xn} and {yn} in Sx such that (f + hn, xn) = | | / + hn\\ and ( / - hn, yn) = \\f - hn\\. However, we can 29 choose {xn} and {yn} in Sx such that + 11/+ M - y l l M and (f-hn,yn)>\\f-hj-ej-\\hn\\, where 8 > 0 is arbitrary. One then obtains {hn, xn - yn) > e ( l - <5)||hn||, whence \\xn - yn\\ > e(l-6). The rest of the proof is the same. • C o r o l l a r y I I . 8 Let X be a Banach space with norm ||-||, and let f £ X*. The following are equivalent: a) \\-\\* is coRnitely Frechet differentiable at f. b) For every € > 0 there is an F < X*, with d i m F < oo, and an a > 0 such that d i am Sl(BFx, /, a) < e. Note that the "near" differentials, that is the functionals x £ (X*/F)* = F-1 such that {x, f) « | | / | | , are actually elements of Fj_, that is, the predual of X* jF. This reflects the si tuat ion of a Frechet differentiable dual norm, as established by Smuljan's Theorem (Theorem 1.1). P r o o f o f T h e o r e m I I .6 The proof now proceeds as for Theorem II.5, using Corol la ry II.8 in place of Corol lary II.4, where appropriate. • 30 R e m a r k I I . 9 Observe that in the proof of Theorem II.5, (c) =>• (b), the a chosen can be arbi trar i ly close to 1. Th i s means that if ||-|| is cofinitely Frechet differentiable at x £ Sx, then for every e > 0 and 6 > 0 there is a finite dimensional F < X such that \\x + h\\ + \\x-h\\ -2\\x\\ l imsup ^ U ^—^ < e and \\x\\ > l - 6 = \\x\\ - 6. This latter condit ion implies that the translate z + F of the subspace F is nearly tangent to Bx at x. Indeed, if there is an / £ Bx* such that {/, x) = 1 and f is a point of weak*-to-norm continuity of Bx*, then we can choose F in Theorem II.5 so that x + F is tangent to Bx at x, and so | | z | | = | |x | | . Th i s requires / to be in bo th the set of points of weak*-to-norm continuity of Bx* and the set of f u n c t i o n a l which at tain their n o r m i n X. The former set is a weak*-dense Qs subset of Sx*, while the latter set is norm dense, by the Bishop-Phelps Theorem (Theorem 1.11). It is unknown if these two sets have non-t r ivia l intersection. C o r o l l a r y 11.10 Every equivalent norm on the James Tree space (JT) is co-finitely Frechet differentiable everywhere. P r o o f It is shown in [16] that the dual space, JT*, has C*PCP. A p p l y Theo-rem II.5. H 31 II.3. The Strong Case A s mentioned in Section 2 of Chapter I, a Banach space X has RNP i f and only if every closed, bounded, convex set in X is dentable, which is equivalent to saying that every such set has a denting point. In 1974, Phelps [26] proved an apparently much stronger result, namely that X has RNP if and only if every closed, bounded, convex subset of X has a strongly exposed point. Bourga in [4] proved a local version of this result: A closed, bounded, convex subset of X has a strongly exposed point if and only i f each of its subsets is dentable. In pursuit of an analogous result for CPCP, we offer the following: Definition 11.11 Let 0 ^ C c X be weakly closed and bounded. A point x G C is a point of strong weak-to-norm continuity of C i f it has a relative norm neighbourhood base in C consisting of relatively weak-open sets, a l l of which are determined by a fixed (finite) set of functionals. Tha t is, there are functionals { / j t} m C X* such that l i m d i a m C n {y E X : \(fk, x — y)\ < a, k = 1, . . . , m } = 0 . If 0 ^ C C X* is weak*-compact, then a point of strong weak*-to-norm continuity of C is defined analogously. 32 Observe that a point of strong weak- (respectively, weak*-) to-norm continuity is a point of weak- (respectively, weak*-) to-norm continuity. D e f i n i t i o n 11.12 A Banach space X has Strong CPCP if every closed, bounded, convex, non-empty subset of X has a point of strong weak-to-norm continuity. Dual ly , X* has Strong C*PCP if every weak*-compact, convex, non-empty subset of X* has a point of strong weak- to-norm continuity. Our a i m is to prove duali ty characterizations of these properties analogous to Theorems II.5 and II.6. For this, we need a stronger version of cofinite Frechet differentiability. D e f i n i t i o n 11.13 A no rm ||-|| on a Banach space X is strongly cofinitely Frechet differentiable at x 6 X if there is a finite dimensional subspace F < X such that | |* | |_X/F 1 8 Frechet differentiable at x. E x a m p l e 11.14 Example II.2 shows that the usual no rm on c 0 is strongly co-finitely Frechet differentiable everywhere. Indeed, since lx = C Q has RNP, it has strong C*PCP, and so it w i l l follow from the next theorem that every equivalent norm on c 0 is strongly cofinitely Frechet differentiable everywhere. 33 Theorem II.15 Let X be a Banach space. The following are equivalent: a) Every equivalent norm on X is strongly coRnitely Frechet differentiable at some point. b) Every equivalent norm on X is strongly coRnitely Frechet differentiable everywhere. c) X* has Strong C*PCP. Theorem 11.16 Let X be a Banach space. The following are equivalent: a) Every equivalent dual norm on X* is strongly coRnitely Frechet differen-tiable at some point. b) Every equivalent dual norm on X* is strongly coRnitely Frechet differen-tiable everywhere. c) X has Strong CPCP. L e m m a 11.17 If a Banach space X has a closed, bounded, convex subset C which has no points of strong weak-to-norm continuity, then it has an equiv-alent norm whose unit ball has no points of strong weak-to-norm continuity. The dual result also holds. P r o o f Let B = C + (-C) + Bx- B is the unit ba l l of an equivalent no rm on X. Suppose x £ B is a point of strong weak-to-norm continuity, so that there are 34 functionals {/j.}™ C X* such that i f Va = {yeX: \(fk,x-y)\ < a, k = 1, . . . , m) , then l i m d i am BnVn = 0 . For each a > 0, choose u a , va G C, wa G such that ua — va + wa G B n Va. Now C * is weak*-compact in X**, so by passing to a subnet i f necessary, we may assume that there is a u** G C * such that ua —> u**. For each a > 0, let V:={y** eX** :\{fk,x-y**)\<a,k = l , . . . , m } . Then , as i n the proof of L e m m a 1.8, diam(V a * + va - wa) D C* < d i a m ( V a + va - wa) n C = diamVan(C-va + wa) < d i a m V a n S , so u** is a point of strong weak*-to-norm continuity of C * . B y the weak*-density of C i n C* and the fact that C is norm closed, u** G C , which contradicts the assumption that C has no points of strong weak-to-norm continuity. • L e m m a I I . 18 Let C C X be closed, bounded and convex, let f e X* and let a G R such that i n f ( / , C) < a < sup ( / , C). Then x G C n / _ 1 (a) is a point of strong weak-to-norm continuity of C fl f~l (a) if and only if x is a point of 35 strong weak-to-norm continuity of C. The dual result also holds. P r o o f It suffices to observe that i n the proof of Bourgain 's L e m m a (Lemma 1.4) only one functional, namely / , is added in the construction. Thus if x has arbi t rar i ly smal l relatively weak-open neighbourhoods i n C fl f~l (a) requiring m functionals, then, in C, x has arbi trar i ly small relatively weak-open neighbourhoods requiring no more than m + 1 functionals. • P r o o f o f T h e o r e m 11.15 (b) => (a) Clear. (a) =>• (c) B y L e m m a 11.17, we must show that the dual unit ba l l of every equivalent no rm on X has a point of strong weak*-to-norm continuity. Thus , let ||-|| be an equivalent norm on X and let x x e X b e a point at which ||-|| is strongly cofinitely Frechet differentiable. Let F < X be finite dimensional and such that ||-||X/F i s Frechet differentiable at \\x~i\\. B y Smuljan's Theorem (Theorem 1.1), xx weak*-strongly exposes BF± at some / £ SF± . We c la im that / is a point of strong weak*-to-norm continuity of Bx* • Let {ej.}™ be a basis for F, let xk = xx +ek, 2 < k < m, and define ip: X* —+ R by ^(d) = m a x \{f — Qt xk)\- Since the sets {I/J < ct}a>0 are elementary weak-open neighbourhoods of / in X*, to show that / is a point of strong weak*-to-norm continuity of Bx* it suffices to show that i f {fn} C Bx* and rj>{fn) —> = °> 36 then fn - /. Since d i m F < oo, we can find a bounded projection P:X* F1-. Suppose { / J C Bx* such that ib{fn) 0. Then (fn, xk) - (/, = (/, xx) for a l l 1 < k < m , since / 6 F - 1 . For A; > 2, this means ( /n. *Jfc) = </n> xl) + </n. eJfc> = </ . .* i> + < ( ^ - - P ) / . , e * > . where J : X —• X is the identity operator. S i n c e (/»» **) - ( / „ ) * i ) - » °> w e must have ( ( / - P)fn, ek) -»• 0, 2 < k < m. Since d i m F = d i m range(7 — P) < oo and {e^.} is a basis for F, (I — P)fn —• 0 in norm. Thus ( / „ , - (Pfn, xt) - 0, so {Pfn, xj - (/ , Since Pfn e F1-and / G F - 1 , this is equivalent to {Pfn,xl) —>• ( / , x x ) . Since x t weak*-strongly exposes BF±, Pfn —> / . Thus / n = P / n + (I — P)fn —> f in norm, as required. (c) =>• (b) This proof is essentially the same as for Theorem II.5, where we use L e m m a 11.18 in place of Corol lary 1.6. Observe that the same subspace F can then be used for every c > 0. The result then follows from Smuljan's Theorem (Theorem 1.1). • Proof of Theorem 11.16 Th i s is essentially the same as the proof of Theo-rem 11.15. B 37 P u t t i n g together the result of Phelps ' mentioned at the beginning of this section and the A s p l u n d - R N P Dual i ty Theorem (Theorem 1.2), we have the following result: Every closed, bounded, convex subset of X* has a weak*-strongly exposed point if and only if every equivalent norm on X is Frechet differentiable on a dense set. Compar ing this w i t h Theorem 11.15, we see that in the Strong C*PCP case, the words "on a dense set" are replaced by "everywhere", yie lding an apparently stronger result. The source of the difference is L e m m a 11.18. There is no corre-sponding result for strongly exposed points, because a strongly exposed point must be strongly exposed by a single functional. However, we easily have the following: C o r o l l a r y 11.19 Let X be a Banach space such that X* has RNP. Then every equivalent norm on X is 1-coSnitely Frechet differentiable everywhere, meaning that for every x £ Sx there is a 1-dimensional subspace F of X such that 1 S Frechet differentiable at x. The corresponding dual result also holds. R e m a r k This notion of Strong CPCP is indeed very strong. It is not, for ex-ample, impl ied by PCP. It is possible that Strong CPCP implies RNP. In [14], Ghoussoub and Maurey discuss a notion of Strong PCP, whose corresponding ver-sion for CPCP is s t r ict ly weaker than the Strong CPCP presented here. The 38 Ghoussoub-Maurey approach may eventually prove to be more fruitful. However, we have not been able to characterize this weaker notion in terms of differentiability at this t ime. 39 Chapter III The Separable Case In [25], Phelps exhibited a Gateaux differentiable, nowhere Frechet differen-tiable no rm on a Banach space. The main result of this chapter is that if X is a separable Banach space, then the existence of such a no rm is equivalent to X* not having C*PCP. Thus we arrive at an Asp lund- i ? iVP type duali ty theorem for Banach spaces w i t h CPCP or C*PCP which does not require any new concept of differentiability. However, it does require the addit ional hypothesis that X be separable. We w i l l discuss this hypothesis further. In Section 1, we introduce a new class of Banach spaces, analogous to A s p l u n d spaces, which we cal l Phelps spaces, and we state our main results. Section 2 is devoted to the development of some geometry needed for the proofs of our main theorems, wh ich are presented i n Section 3. The results of this chapter were obtained i n collaboration w i t h R . Devil le , G . Godefroy and V . Zizler , and w i l l appear i n [7]. 40 I I I . l . P h e l p s S p a c e s a n d CPCP D e f i n i t i o n I I I . l A Banach space X is called a Phelps space if every continuous, convex, Gateaux differentiable function on X is Frechet differentiable on a dense subset of X. X* is a weak*-Phelps space if every continuous, convex, Gateaux differen-tiable dual function on X* is Frechet differentiable on a dense subset of X*. A dual function on X* is one which is weak*-lower semi-continuous. A dual norm on X* is an example of such a function. Recal l that a norm on X is strictly convex i f every element of norm 1 is an extreme point of the corresponding unit ba l l . We have the following classification theorems: T h e o r e m III .2 Let X be a separable Banach space. The following are equiv-alent: a) X* has C*PCP. b) Every equivalent norm on X with strictly convex dual is Frechet differen-tiable on a dense set. 41 c) Every equivalent Gateaux differentiable norm on X is Frechet differen-tiable on a dense set. d) X is a Phelps space. Theorem III.3 Let X be a separable Banach space. The following are equiv-alent: a) X has CPCP. b) The dual of every equivalent strictly convex norm on X is Frechet differ-entiable on a dense set. In the case that X* is also separable, (a) and (b) are equivalent to: c) Every equivalent Gateaux differentiable dual norm on X* is Frechet dif-ferentiable on a dense set. d) X* is a weak*-Phelps space. We point out that as separable dual spaces have RNP [10], the Banach spaces of Theorem III.3, (c) and (d) are, of course, A s p l u n d spaces, and hence are Phelps spaces. Theorem III.3 allows us to decide i f their duals are weak*-Phelps spaces. We do not know if the requirement that X* be separable in Theorem III.3, (c) and (d), is necessary. Note that there exist non-separable Banach spaces which have no equivalent Gateaux differentiable norm (and (c) w i l l be used to prove (d)). We w i l l prove these theorems in Section 3. 42 III.2. Some Geometrical Lemmas In order to prove Theorems III.2 and III.3, we need to develop some basic results about sets which have points of continuity. The first of these, together w i t h Smuljan's Theorem (Theorem 1.1), provides the motivat ion for the equivalences (a) -O- (b) of Theorems III.2 and III.3. Par t (a) of this lemma is due to L i n , L i n and Troyanski [21]. We include a complete proof because the methods are representative of the methods used to prove the ma in theorems. Lemma I I I . 4 a) Let 0 ^ C C X be closed, bounded and convex, and let x G E x t ( C ) . I f x is also a point of weak-to-norm continuity of C, then x is a denting point ofC. b) Let 0 7^  C C X* be weak*-compact and convex, and let f G E x t ( C ) . If f is also a point of weak*-to-norm continuity of C, then f is a weak*-denting point of C. Proof We prove part (b) first. b) Le t V be an elementary weak*-open neighbourhood of / . Then C \ V is n a finite union of weak*-compact, convex sets, say C \ V — [j Ck. (The sets Ck are l 43 actually closures of weak*-slices of C.) Let A = { ( A J G R r e : \ k > 0, 1 < A; < n , and f > A = *} • n Define <p:A. x Y\Ck X* by <p({Xk), (ck)) = Yl^kck- *P 1S dea r ly continuous from the no rm x (product weak*) topology to the weak*-topology, and the domain of ip is compact in this topology, so the range of <p is a weak*-compact, convex subset of C which does not contain / , since / is extreme in C. Thus, by the Hahn-Banach Theorem (applied to the weak*-topology), there is a weak*-slice S of C such that / G S and S C C \ range tp CV. Tha t is, the weak-slices of C containing / form a neighbourhood base for the relative weak*-topology on C at / . Since / is a point of weak*-to-norm continuity of C, this is also a base for the relative norm topology on C at / . Tha t is, / is a weak*-denting point . a) B y L e m m a 1.8, a; is a point of weak*-to-norm continuity of C* C X**. We c la im that x is also an extreme point of C*. For suppose x = y — , where y** , z** EC*. B y L e m m a 1.9, y** and z** are also points of weak*-to-norm continuity of C*. Since C is weak*-dense in C* and C is no rm closed, y**, z** 6 C . Since x G E x t ( C ) , y** = z** = x. B y (b), x is a weak*-denting point of C*. If 5 is a weak*-slice of C* containing x, then S fl C is a (weak) slice of C containing x. Since the norm topology on X is the no rm topology on X** restricted to X, these slices have arbi t rar i ly small diameter, 44 and so x is a denting point of C, as required. • To illustrate the remarks at the beginning of this section, suppose that ||-|| is an equivalent no rm on X which has a str ict ly convex dual . Then for the dual unit ba l l , Bx* , we have Sx* = E x t ( B x * ) . If X* has C*PCP, L e m m a 111.4(b) says that 8X* is weak*-dentable. A p p l y i n g the methods of Phelps [26], we can estab-lish that BX* in fact has a weak*-strongly exposed point . B y Smuljan's Theorem (Theorem 1.1), ||-|| is Frechet differentiable at some point . The motivat ion for Theorem III.3, (a) (b), is s imilar , using L e m m a 111.4(a). Notice that the preceding discussion is isometric, that is, it deals w i th a specific norm. In order to establish the complete equivalence of (a) and (b) of Theorems III.2 and III.3, we must study a class of equivalent norms on X, which we may view in X* as a class of weak*-compact, convex sets. The next lemma w i l l form the cr i t ical step in establishing our results about density. Lemma III. 5 a) Let X be a Banach space with CPCP, e > 0, and f E X*. Let K <£ f ' 1 (0) be closed, bounded and convex, and let C = c o ( ( / _ 1 (0) D e'1 Bx) U K) . Then there is an x G K which is a point of weak-to-norm continuity of C. 45 b) Let X be a Banach space such that X* has C*PCP, e > 0, and x G X. Let K <£ x~x (0) be weak*-compact and convex, and let C = c o ( ( x - 1 (0) n e _ 1 Bx* )UK). Then there is an f G K which is a point of weak*-to-norm continuity of C. R e m a r k I I I . 6 There are two important observations to make w i t h respect to L e m m a III.5: F i rs t of a l l , note that the point of continuity produced by the lemma (in ei-ther (a) or (b)) is an element of K, which is generally a proper subset of C, yet it is a point of continuity of the larger set C. Secondly, observe that i f in case (a) we assume K C S^-, and we take a func-t ional g of norm 1 such that sup(<7, C) = a < 1, then / and g must satisfy the hypotheses of the Bishop-Phelps L e m m a (Lemma 1.10). For in this case, the entire hyperplane g~x (1) must lie to one side of C, and so the relations (g, y) = 1 and (/, y) = 0 must imply | |y|| > c - 1 , as required. Thus one of | | / ± g\\* < 2e. A similar obervation can be made for part (b). 46 P r o o f o f L e m m a I I I . 5 a) Wi thou t loss of generality, we may suppose that sup ( / , K) > 0. Choose 0 < a < sup{ / , K). If the set C l = c o ( ( / " 1 (0) n e _ 1 Bx) U K) were closed, then we could obtain, by Corol lary 1.6, a point of weak-to-norm con-t inui ty of C j in the set Cx fl / - 1 (a), and then using convexity and L e m m a 1.9, we could slide this point up to K and be done. In general, however, Cx is not closed. Thus we w i l l transfer the problem to X**, where we can take advantage of compactness in the weak*-topology. Let D = c o ( ( / _ 1 (0) D e _ 1 Bx**) U ~K*) , where we consider / G X*** (and recall K* denotes the weak*-closure of K in X**). Note that since K* and / _ 1 (0) n e _ 1 Bx** are bo th convex, D is exactly the range of the map ( / _ 1 (0) n e _ 1 Bx~ ) x K* x [0,1] —-> X* (x, y, t) i—> tx + (1 - t)y . This map is clearly continuous from the product topology weak* x weak* x norm to the weak*-topology on X**. Since the domain is a compact space in this topol-ogy, it follows that D is weak*-compact (note that I? is a convex hu l l , not a closed convex hul l ) . A l so , K is weak*-dense in K* and / _ 1 (0) n e _ 1 Bx is weak*-dense in 47 f-1 (O)ne-1 Bx** , so D = C*. Now C n / _ 1 (a) is a closed, bounded, convex subset of X, so it has a point of weak-to-norm continuity, say z. B y Corol lary 1.6, z is a point of weak-to-norm continuity of C. B y L e m m a 1.8, z is a point of weak*-to-norm continuity of D. Since ( / , z) = a > 0, there are, by the definition of D, elements xeK*, y G f~l (0) D e" 1 Bx** , and 0<t<l such that 2 = tx + (1 — t)y. B y L e m m a 1.9, x is also a point of weak*-to-norm continuity of D. Now K is weak*-dense i n K* and is norm closed, so the fact that x is a point of weak*-to-norm continuity of K* implies x G K. x is clearly a point of weak-to-norm continuity of C , as required. b) A s was shown i n part (a) for the set D, the set C i n this case is weak*-compact and convex. B y Corol lary 1.6, we can choose a g G C n x - 1 (a) which is a point of weak*-to-norm continuity of C, and slide it up to / G K using convexity and the definition of C. Then , by L e m m a 1.9, / is a point of weak*-to-norm con-t inui ty of C, as required. • Remark III.7 Observe that if, in L e m m a 111.5(a), we take K to be the unit bal l of an equivalent s t r ict ly convex norm on X, then the point x G K produced by the lemma is, i n fact, an extreme point of C also. 48 To see this, suppose x = ^Y~, where y , z G D. Then there exist yx, zx G K*, y2,z2€ f~l (0) n e" 1 Bx** and 0 < a, (3 < 1 so that y = ctyj + ( l — oi)y2 and 2 = + (1 — ff)z2. Note that D has non-— * empty interior, since it contains K , which is the unit ba l l of an equivalent norm on X**. Thus , by the Hahn-Banach Theorem, there is a g 6 X*** such that (g, x) = sup(y, D). Suppose, by way of contradiction, that 0 < a, f3 < 1. Then since D is convex and g is linear, (9, x) = (g, y) = (g, z) = (g, yx) = (g, y 2 ) = {g, zx) = (g, z2) . B y L e m m a 1.9, the points y, z, yx and zx are a l l points of weak*-to-norm con-t inui ty of D. B y the weak*-density of C i n D, these points must a l l be in C. Thus, yx, zx G K. Since x E K also, since K is str ict ly convex, and since (g, x) = (g, yx) = (g, zx), we must have yx = zx = x. Thus Now (/, x) 7^  0, and {/, y 2 ) = (f,z2) = 0. Thus, applying / to the above equation, we get = 1, which is impossible if 0 < a < 1 and 0 < /? < 1. Therefore, x = y = z, so x is extreme in I?, hence also in C. A n interesting application of L e m m a III.5 is the following remarkable version of the Bishop-Phelps Theorem (Theorem 1.11) for Banach spaces w i t h CPCP. Recal l 49 that Bourga in has shown that if K is a closed, bounded, non-dentable subset of a Banach space, then the set of support functionals for K is of first category i n X* [4]. P r o p o s i t i o n I I I .8 Let X be a Banach space (not necessarily separable) which has CPCP, and let 0 ^  K C X be closed, bounded and convex. Then the set of functionals in X* which support K at a point of weak-to-norm continuity of K is norm dense in X*. P r o o f We may assume, without loss of generality, that K C \&x- Let e > 0 and let / G Sx* • We w i l l show that there is a g G Sx* such that | | / — g\\* < 2e and g supports i f at a point of weak-to-norm continuity of K. If / is constant on K, we can clearly choose g = f. Otherwise, there exist u,v,w = »±£- G K such that (/ , u) < (/ , w) < (/, v). Let Kx — K — xu. Then Kx C Bx and it suffices to establish the proposit ion for the set Kx. B y L e m m a III.5, there is an x G Kx which is a point of weak-to-norm continuity of C = c o ( ( r 1 ( 0 ) n e - 1 S x ) U i f 1 ) . B y the choice of Kx, we may assume (/ , x) > 0. Since Kx <£. / _ 1 (0), sp C = X, and so C must have non-empty interior. Thus by the Hahn-Banach Theorem, there exists age Sx* such that (g, x) = sup{g, C) > 0. Since Kx C Bx, {d, x) < 1. 50 B y Remark III.6, / and g satisfy the hypotheses of the Bishop-Phelps L e m m a (Lemma 1.10). Since / and g are both positive at z , we must have | | / — g\\* < 2e if e is sufficiently smal l . B 51 III.3. Proofs of Theorems III.2 and III.3 P r o o f of Theorem HI.2 (d) =>- (c) Th i s is immediate, since a norm is a continuous, convex function. (c) =>• (b) Th i s follows from the well known result that i f ||-|| is a norm on X such that ||-||* is s tr ict ly convex, then ||-|| is Gateaux differentiable everywhere i n Sx [20]. (a) =>• (c) Let ||-|| be an equivalent Gateaux differentiable norm on X. Then for each x G X there is a unique fx G Sx* such that (fx, x) — 1, and the mapping I H / J is norm-to-weak* continuous [5]. For each positive integer n , let Un = {x G Sx : 3 weak*-open W, fx G W, and d iam VP n Bx* < n~l } . We w i l l show that Un is norm open and dense in Sx for a l l n. G iven this, the set U = P| Un is dense in Sx, by the Baire Category Theorem. Now suppose that x0 G U. T h e n / has weak*-open neighbourhoods Wn such that d i a m V P B n Bx* < n~l, for every n . Tha t is, fXQ is a point of weak- to-norm continuity of Bx* • Let {^fc} C Sx such that xk x0. Since the mapping x H-> fx is norm-to-weak* continuous, / ^ . Since fXQ is a point of weak*-to-norm continuity of Bx*, fXk —• fXo. Tha t is, the mapping x (-»• / z is norm-to-norm continuous at x 0 . Th i s , in turn, 52 establishes that ||-|| is Frechet differentiable at x 0 [5]. Un being open obtains from the norm-to-weak* continuity of the support map-ping I H / X . For density, let 0 < e < 1/2, x 0 E Sx, and, as in L e m m a 111.5(b), let C = C O ( ( X Q 1 (0) n e _ 1 Bx*)UBP). Then , as i n L e m m a III.5, C is weak*-compact and convex, and there is an / 0 E Bx* which is a point of weak*-to-norm continuity of C, w i t h ( / 0 , x 0 ) > 1/2. Let V be an elementary weak*-open neighbourhood of / 0 such that d iam V"PiC < ( 2 n ) _ 1 . We may clearly assume that (g, x0) > 1/2 if g EV n C , Now C is the unit ba l l of an equivalent dual norm, say |||-|||*, on X*. Le t B be the corresponding unit ba l l in X, so B = { x E X : \\\x\\\ < l } (S is the weak*-polar of C). B y the Bishop-Phelps Theorem (Theorem 1.11), there is a g E V n C and an xx E B w i t h 1 = llklll* = sup($, B) = (g, X i ) = HtaHl = s u p ( x i , C) . Tha t is, xx supports C at g (considering xx E X * * ) . B y the definition of C, there are fl e S X * > h e xo1 (°) n e _ 1 &X* a n d 0 < < < 1 53 such that g = tfx + ( l — t)f2. It follows that 1 = (g, xx) = t(fu xx) + (1 - t)(f2, x x) < * i i / i i r i k i i + ( i - o i i i / 2 i i r i i k i i n < * l k i l l + ( i - 0 -Now t / 0 and B C B x, since C D Bx*. Thus HxJI = 1, and, by convexity, x x supports Bx* at / j . Tha t is, x x) = 1. B y the choice of V, t > 1/2. Thus, by convexity, W = \V - ^ - / 2 is a weak*-open neighbourhood of fx w i t h d i a m W n C < 2 • d i a m F n C < n _ 1 . Therefore, x x EUn. It remains only to show that xx is sufficiently close to x 0 . Th i s follows from Remark III.6: B y the definition of C and the choice of xx, if / G X * , (/, x x) = 1, and (/, x 0) = 0, then | | / | | * > e - 1 . B y the Bishop-Phelps L e m m a (Lemma 1.10), H^i + xo\\ ^ 2e or ||xx — x 0 | | < 2e. Since xx + x 0) > 3/2 > 2e, we must have l l x i ~~ x o l l — 2c, as required. Thus, Un is dense, and we are done. (b) =>• (a) Th i s proof was motivated by examples of Phelps [25] and Edel -stein [11]. Suppose X * does not have C*PCP. B y Corol lary 1.7, there is an equivalent 54 norm, ||-||, on X and an e > 0 such that the corresponding dual unit ba l l , Bx*, has no relatively weak*-open subsets of ||-||*-diameter less than e. Throughout this proof, "diameter" w i l l always mean w i t h respect to this dual norm, ||-||*. Since X is separable, X* has a countable norming set, { x y } i ° , in Sx. Tha t is, for / E X*, | | / | |* = sup ( / , Xj). We define a new dual norm, ||-||*_, on X * by 3 oo i i/nf = n / i r 9 + £ 2 - ' < / , * , . > * That ||• | | i is a dual norm follows from the fact that ||-||* is a dual norm and Xy is weak*-continuous for a l l j . A l so , since we clearly have > ||-||*, is an equivalent no rm on X * . We w i l l show that | |- | |J is a s tr ict ly convex norm whose unit bal l is not weak*-dentable, and thus, by Smuljan's Theorem (Theorem I . l ) , ||-1jx cannot be Frechet differentiable anywhere. To see that is s t r ict ly convex, suppose f,g£X* such that 55 Then 0 = 2 | | / | | f + 2\\g\\f -\\f + g\\f = 2\\ff +2\\gf -\\f + gf 00 + £ 2->' (2(/, z , . ) 2 + 2(g, Xjf -(f + g, x,-) 2) . 3=1 Since ||-||* is a norm, 2 i i / i r 2 + 2 i H r 2 > n / + g i r 2 , and since Xy is linear for al l j, 2(f, xj)2+2(g, Xj)2 > ( / + <7 ,x y ) 2 . Since the sum is 0, we must have that for a l l j, 2(f, Xj)2 + 2(g, x y } 2 -{f + g, x y > 2 = (f - g, x y) 2 = 0. B y the choice of {xy}, f = g, and so is s tr ict ly convex. It remains to show that the unit ba l l , B{, of is not weak*-dentable. For this, let m = i n f { | | / | | * : | | / | | J = l } and let 6 > 0. Let x G X, f0 G X* such that ( / 0 , X) = | | / 0 | | J = HXIIJ = 1, where |j - 1 | x denotes the corresponding predual norm in X. We w i l l show that there is an fx G X* such that \\fx — /0||*_ > me/3, (fx — / 0 , x) < 6 and | | / j | | J < 1 + 3£. Since 6 is arbitrary, it follows that there is a sequence {fn}™ in X * such that for a l l n = 1, 2, | | / J | i = | | / 0 | | i = 1, I I / » ~~ / o i l ! > M E / 6 , and l i m (/n, x) = (/0, x) = 1. Therefore, every slice of the 56 unit ba l l , Bx*, of | |- | |J determined by x has ||-||J-diameter at least me/6. Since x was arbitrary, Bx is not weak-dentable, completing the proof. Thus , choose an integer n 0 such that 2 2 _ n ° < 6. Consider the weak*-open set U = {/ G X* : \{f - /o, x)| < 6 and\(f - f0, Xj)\ < 6, 1 < j < n0} . Let 8 = \\f0\\*Bx* ={feX*: \\f\\* < | | / 0 | | * } . Since f0eUnB, l W { U n B ) = (K¥u>B**^> and so d iamC/ D 8 > ||/ol|*£ > me, since ] | / 0 | | i = 1- Thus there is an fx G 8 such that I I / ! - / o | | * > \\fx - / o | | * > m e / 3 , \(fx - f0, x)\ < 6, and \(fx - / 0 , Xj)\ < 6 for 1 < j < n 0 . Now oo ii/oiif = ii/oir 2+E2" y(/o,^)2 = i , y=i and fx GS (so | | / 0 | | * > | | / i | |*), so oo ii/iiif = n/iir 2+E 2 _ y(A^y) 2 00 oo y=i y=i no = l + £ 2 - ' ( ( / 1 , x y > a - ( / 0 , x y > 8 ) oo + £ 2 ^ ( ( / 1 , x J . ) 2 - ( / 0 , x y ) 2 ) y=no+l 57 < 1 + E 2 ~ j ( K / i . * y > l + K/o. * y ) l ) ( K / i , * y > l - K/o. * y > l ) y=i oo + E 2 - J ' ( | ( / 1 , x y ) | + | ( / 0 , x y ) | ) 2 ;=«o+i no < 1 + J V ' ( | ( / l f x y ) | + | ( / 0 , x ^ D f l ^ , x y ) - ( / 0 , x y ) | ) y=i 00 + E 2 _ y ( K / i ^ y ) l + K / o ^ y ) l ) 2 j'=no+l wo <i+ 6 x:2 - j ' i i x y i i ( i i / 1 i r + n/0ir) y=i 00 + E 2 ^ i i x y i i 2 ( i i / 1 i r + n/ 0ir) 2 y=no+l < 1 + 26 + 2 _ n° • 22 < 1 + 36 , w i t h the penultimate inequality following from the facts that | | x y | | = 1 for a l l j, and ii/ i ir < i i / o i i * < i i / o i i ; = 1. The proof of (b) =>• (a) is complete. (c) =>• (d) The method we w i l l use here is adapted from Namioka and Phelps [23, Theorem 6]. Having already established (a) =>• (c) =>• (b) =£> (a), we w i l l make use of the fact that (a) and (c) are equivalent. Let <p: X —+ R be convex, continuous and Gateaux differentiable. It is well known that such a function ip is locally Lipschi tz . B y translat ion and shifts, we may assume <p(0) = —1, and it suffices to show that given 6 > 0, there is an 58 i £ l such that | |x | | < 6 and <p is Frechet differentiable at x. Let 6 > 0. Wi thout loss of generality, we may assume that 6 is sufficiently small so that <p(x) < —1/2 if | |x | | < 6, and that <p is Lipschi tz w i th constant K > 1 on ( x G X : \\x\\ < S}. Let U = { (x , r) 6 X x R : | |x | | < 6, <p(x) <r< -<p{-x)} . Since <p is convex and continuous, U is a closed, bounded, convex and balanced neighbourhood of 0 in X x R , hence defines a norm, |||-|||, on X x R which is equiv-alent to the no rm | | (x , r ) | | = max{ | | x | | , | r | } . A t r iv ia l consequence of Bourgain 's L e m m a (Lemma 1.4) and the equivalence of (c) and (a) is that X* x R = (X x R ) * has C*PCP. Now HI • HI is not necessarily Gateaux differentiable everywhere, but the assump-tions on <p imply that it is Gateaux differentiable on the set { (x , <p(x)) : | |x| | < <?}. Combin ing this fact w i t h the observation that the proof of (a) =^ (c) can be carried out locally, that is, on any relatively open subset of Sx, we obtain that |||-||| has a point of Frechet differentiability of the form ( x 0 , r 0 ) , r 0 = "P^o) , ||z01| < m i n { £ / 2 , 1 / ( 2 M ) } , where M = sup{ | | / | | * : | | | ( / , *)|||* < l } . Note r 0 < 0. Let ( / 0 , s0) be the Frechet derivative of |||-||| at ( x 0 , r 0 ) , so that lll(/o» 5 o) l l l * = 1 = <(/o. ^ o ) . ^ . r o)) = </o. Xo) + S0rQ . 59 and for a l l e > 0 there is an r\ > 0 such that |||(x, r) — ( x 0 , r0)\\\ < rj implies I IIK*, r)||| - | | | (x 0 , r0)||| - <(/0> s0),(x-x0, r - r 0)> | (*) < e | | | ( x - x 0 , r - r 0 ) | | | < e-)max{\\x- x0\\, \r - r 0 | } for some (fixed) 7 > 0 (which gives the equivalence between the norms |||-||| and ||-| |). Note that sQ < 0, since otherwise the choice of xQ implies 1 = (/o>*o>+ 5o ro < (fo, *o) < M\\x0\\ < 1 /2 , which is impossible. We w i l l show that gQ = — S Q 1 f0 is the Frechet derivative of <p at XQ, i.e., that (g0, x - xQ) < <p{x) - <p{x0) for a l l x, and for a l l e > 0 the inequality \<p(x) -<p{x0) - {g0, x-x0)\ < e | | x - x 0 | | holds for x sufficiently close to xQ. Let e > 0 and choose 0 < n < 8/2 such that (*) holds for n and ex = —s0e/iK (recall that <p is Lipschi tz w i th constant K on {||x|| < 8}). Suppose that 0 < n' < 60 min{?7, and \\x — xQ\\ < n'. Then | |x| | < 6 and |||(x, <p(x)) - (x0, <p(x0))\\\ < 7 m a x { | | x - x 0 | | , \<p{x) - <p(x0)\} < 7 m a x { | | x - x 0 | | , / c | | x - x 0 | | } = 7 / c | | x - x 0 | | so |||(x, £>(x))||| = 1, and (*) implies | (( /o» s0),{x-x0, V>ix) < — ^ 7 m a x { | | x - x 0 | | , \<p(x) - <p(x0)\} < — ^ ^ m a x { | | x — x 0 | | , K\\X — x 0 | | } = -*oe\\x-xo\\ » since K > 1. D i v i d i n g by — s0(> 0), we get I ( # 0 , x - x 0 ) -<p{x) +<p{x0)\ < e | | x - x 0 | | , as required. This completes the proof of Theorem III.2. • P r o o f o f T h e o r e m I I I . 3 (d) =>• (c) This follows from the fact that a dual norm is a continuous, convex and weak*-lower semi-continuous function. (b) (c) Th i s follows from the well known result that i f ||-|| is a norm on X whose dual is Gateaux differentiable, then ||-|| is s tr ict ly convex [20]. 61 (a) =>• (b) Let ||-|| be an equivalent strictly convex norm on X. For each positive integer n, let Un = {/ G Sx* : 3 a > 0 such that diamS1(BX, f, a) < } , where Bx is the unit ball of ||-||. We claim that Un is norm open and dense in Sx* 5 and hence, by the Baire Category Theorem, U = f] Un is dense in Sx*. Given the claim, it is clear that each / G U is a strongly exposing functional for X. B y Smuljan's Theorem (Theorem L l ) , each / G U is a point of Frechet differentiability of ||-||*. To see that Un is open, let / G Un with corresponding a > 0, so that diam S1(BX, f, a) < n~l. Let (3, e > 0 such that (3 < a — e. Suppose g G Sx* such that ||/ - 0||* < e. Let x G S1[BX, 9, 0)- Then (f,x) = (g,x)-{g^f,x)>l-/3-e>l-ct. Thus x G S1(BX, f, a). Therefore, S1(BX, g, 0) C SI(8X, /, a), and it follows that diamS1(BX, g, /?) < rTl. Tha t is, g£Un. To see that C/„ is dense, let / G Sx*, let e > 0, and let C = co((/- 1 {0)ne-1Bx)uBx). B y L e m m a 111.5(a), there is an x G Bx which is a point of weak-to-norm continuity 62 of C. Since ||-|| is s tr ict ly convex, x G E x t ( B x ) , so, by Remark III.7, x E E x t ( C ) . B y L e m m a 111.4(a), a; is a denting point of C . Let g G Sx* and a > 0 so that d iam S1(C, g, a) < n _ 1 and x G S1(BX, 9, oc) C S1(C, g, a). Since C is symmetric (i.e., C = —C), we have that g G Un and —g(EUn. B y Remark III.6, / and g satisfy the hypotheses of the Bishop-Phelps L e m m a (Lemma 1.10). Thus one of + < 2e or | | / — g\\* < 2e must hold. Since bo th g and — g are in Un, and e was arbitrary, Un is dense in Sx*, as required. (c) (a) Th i s proof is a slight modification of the proof of Theorem III.2, (b) =• (a). Suppose X does not have CPCP and X * is separable. B y Corol lary 1.7, there is an equivalent norm ||-|| on X and an e > 0 such that Bx has no relatively weak-open subsets of ||-||-diameter less than e. Since X * is separable, X has a countable norming set {/•} C S^* which is also norm dense. We define a new norm, |[-1ja, on X by oo NMl*lf + £ 2 - > </,.,*>' We c la im that the corresponding dual norm, is Gateaux differentiable every-where in Sx*. To see this, it is sufficient [5] to prove that 11 -1| is weakly uniformly rotund, that is, if {xn} and {yn} are bounded sequences in X such that (*) K m 2 | | x J | 2 + 2 | | y J | 2 - | | x w + y j | 2 = 0 , n—>oo 63 then xn — yn —• 0. Note, however, that if (*) holds, then the same convexity properties as used in the proof of Theorem III.2, (b) =>- (a), to show strict convexity can be used to show that K m 2 ( / y , xn)2 + 2{fj, yn)2 - </., xn + yn)2 = l i m ( / . , xn - yn)2 = 0 for a l l j = 1,2, — Since {xn} and {yn} are bounded sequences and since s p { / y } = X*, xn — yn ^ 0. Thus , is Gateaux differentiable. The proof that the uni t ba l l of || -|J x is not dentable, and hence that ||-||*_ is not Frechet differentiable anywhere, proceeds exactly as in the proof of Theorem III.2, (b)-=* (a). (c) => (d) Using the equivalence of (c) and (a) in the case that X * is separ-able, this proof is just a dualized version of the proof of Theorem III.2, (c) =>• (d). Theorem III.3 is proven. • Combin ing results of Phelps [26] and Davis and Phelps [6], we have the following very strong results for RNP spaces: Proposit ion III.9 Let X be a Banach space. The following are equivalent: a) X* has RNP. b) The unit ball of every equivalent dual norm on X* is weak*-dentable. 64 c) The unit ball of every equivalent dual norm on X* is the weak*-closed convex hull of its weak*-strongly exposed points. Proposit ion III.10 Let X be a Banach space. The following are equivalent: a) X has RNP. b) The unit ball of every equivalent norm on X is dentable. c) The unit ball of every equivalent norm on X is the closed convex hull of its strongly exposed points. A s a consequence of our Theorems III.2 and III.3, we have the following results, which highlight the strong geometric connections between the properties CPCP and RNP, at least i n the separable case. Proposit ion III. 11 Let X be a separable Banach space. The following are equivalent: a) X* has C*PCP. b) The dual unit ball to every equivalent Gateaux differentiable norm on X is weak*-dentable. c) The dual unit ball to every equivalent Gateaux differentiable norm on X is the weak*-closed convex hull of its weak*-strongly exposed points. 65 Proposit ion III.12 Let X be a separable Banach space. The following are equivalent: a) X has CPCP. b) The unit ball of every equivalent strictly convex norm on X is dentable. c) The unit ball of every equivalent strictly convex norm on X is the closed convex hull of its strongly exposed points. 66 C h a p t e r I V Some Structure Theorems In this chapter, we study the structure of spaces which do not have CPCP and of spaces which are not Phelps spaces. The pr incipal concept used in this chapter is that of a finite dimensional Schauder decomposition of a Banach space. Th i s is a generalization of the notion of topological basis. D e f i n i t i o n I V . 1 A finite dimensional Schauder decomposition (FDD) of a B a -nach space X is a family {Fn} of finite dimensional subspaces of X such that for every x £ X there is a unique sequence {xn} C X such that xn £ Fn for a l l n, and x = x n (convergence in norm). In Section 1, we show that a Banach space which fails CPCP has a subspace which fails CPCP but has an FDD. Th i s extends results of Bourgain , who proved the corresponding theorems for non-RNP and n o n - P C P spaces. Section 2 represents joint work w i t h R . Devil le , G . Godefroy and V . Zizler, and w i l l appear in [8]. In that section, we use the results of Chapter III to show that 67 if X is a separable non-Phelps space, then X has a subspace Y such that neither Y nor X/Y is a Phelps space, yet bo th Y and X/Y have FDWs. The analogous result for A s p l u n d spaces is s t i l l open. 68 I V . l . A S t r u c t u r e T h e o r e m f o r N o n - C P C P S p a c e s In [3] and [4], Bourgain showed that if a Banach space X does not have RNP (respectively, PCP), then X has a subspace Y such that Y does not have RNP (respectively, PCP), yet Y does have an FDD. In this section, we w i l l extend this result to non-CPCP spaces. Our proof is a modification of Bourgain 's proof in [3]. In part icular , we w i l l require the following lemma, which he proved in [4]: L e m m a I V . 2 Let X be a Banach space, n G N , e > 0 , A <Z X and let x G A™. Then there exists a finite subset J of A such that if {fk}i is a set of n functionals in Bx*, then J n {y G X : \(fk, x - y)\ < e, k = 1, . . . , n} ^ 0 . The result follows quickly from the compactness of the space [Bx* ,u>*)n. T h e o r e m I V . 3 Let X be a Banach space which does not have CPCP. Then there is a subspace Y < X such that Y does not have CPCP, but Y does have an FDD. P r o o f For n G N , let be the set of rationals in [0, 1] w i t h denominators at 69 most ra. For a subset C of X, let co„(C7) = { ± \ i X i : X i € C, X{ G $„, = l} . L l l J Let ||-|| be the equivalent norm produced by Corol lary 1.7, w i t h correspond-ing e > 0. Le t { 7 W } C (0, oo) be such that + 7n) < ° ° - We define a sequence {An} of finite subsets of Ux and a sequence {En} of finite codimensional subspaces of X as follows: Let x € - ^ l = a n < l - ^ l — kerx*, where x* G S x * is chosen such that (x*, x) = \\x\\. Now suppose that An and En have been defined for some ra. Let x G An. B y the choice of the norm, d i a m U x H (x + En) n V > e for every weak-open neighbourhood V of x . Thus if A=(Uxn(x + En))\U(x, 6 / 2 ) , where I7(x, e/2) is the open ba l l of radius e/2 centred at x, then x G A*" . B y L e m m a I V . 2 , we can choose a finite subset J% of A such that J * n {y G X : \(fk, x - y)\ < n~l,k = 1, . . . , ra} ^ 0 for every set of n functionals {/fc}™ C Bx*. Note that dist(x, J*) > e/2. 70 Let A n + l — con[An U ( (J Jn))- Then A n + 1 is a finite subset of Ux, so there is a finite subset An C Sx* such that if x £ sp then there is an x* £ An such that ||a;|| < ( l + ln)(x*, x). Let E n + 1 = f] ke rx* . Th i s completes the construction of the sequences {An} and {En}. Now let Xn = spAn, for n £ N , and let Y = IJ-^JI- We c l a im that this subspace of X satisfies the conclusions of the theorem. Firs t we w i l l show that Y has an FDD. Observe that the definition of AnJrX ensures that Gcon{An+En)=con{An) + E n , and so Xn+1 = sp A n + 1 C sp(An +En)=Xn+En. Thus x n + i = ( x n + E n ) n x n + 1 = x n + ( E n n x n + 1 ) . B y the definition of En, i f x £ Xn and y £ En D X R + 1 there is an x* £ sp An = F ^ such that N I < ( l + 7 n ) < x * , *> = {l + ln)(x*,x + y) < ( l + 7 j | | x + y| | , and hence there is a projection 7rn: Xn+1 —• Xn w i th \\irn\\ < 1 + 7 n - Thus ' there is a projection oo 1 71 w i t h 00 00 i ^ i i<n(i + 7 j<n(i + 7 f c ) < o o , n 1 and so P„ extends to a l l of Y. Note P„Pm = P „ , P „ = Pm if m < n. Tb Tb l i b III Tb fib Let Fn = (Pn - Pn_x )Y. It is elementary that {Fn} is an FDD for Y. Now we tu rn to the question of showing that Y does not have CPCP. Let 8 {jAn. F i rs t observe that 8 is convex. For if x , y G 8, 0 < t < 1 and 6 > 0, then choose n G N sufficiently large so that there is an integer k, 0 < k < n, such that | £ — t\ < 6/3, and there are xx,yx G An such that | |x — xx \\ < 6/3 and \\y-yi\\<6/3. Then £ * i + (i - $)yi e A „ + 1 c 8 , by the definition of A n + l , and | | ( < x + ( i - < ) y ) - ( J x 1 + ( i - ^ ) y i ) | | < - txx || + ||te! - J*! || + || (1 ~ - (1 - || + l l ( i - * ) y i - ( i - * ) y | | < t8/3 + | | x x ||<5/3 + | | y i \\6/3 + (1 - i)<5/3 <6, since x 1 ? y i G 8 C 8x- Since 6 > 0 was arbitrary, te + (1 — £)y G 8, and so 8 is convex. 72 Now suppose x £ B and V = {y e Y : I (fk, x - y) | < 8, k = 1, . . . , m} , where 6 > 0 and {/j.}™ C By*. B y the Hahn-Banach Theorem, we may assume {fk}™ C Bx* • Choose n > m sufficiently large so that n~l < 6 and V Pi An ^ 0. Then by the definition of J * , J%nV ^ 0. Since x e B, J% C B and dist(x, J*) > e/2, we must have d i a m F n B > dist(x, V n J*) > e/2 , and so F does not have CPCP. • 73 I V .2. A D e c o m p o s i t i o n T h e o r e m fo r N o n - P h e l p s S p a c e s In [19], Johnson and Rosenthal showed that if X is any separable Banach space, then X has a subspace Y such that both Y and X/Y have FDD's. For separable non-Phelps spaces, we have the following result: T h e o r e m IV.4 Let X be a. separable Banach space which is not a Phelps space. Then X has a subspace Y such that neither Y nor X/Y is a Phelps space, yet both Y and X/Y have FDD's. We point out that at this t ime it is not known whether a similar theorem holds for non-Asplund spaces. The proof we present here ul t imately depends on Corol la ry 1.6. A s that corollary stands, one cannot s imply replace the words "point of weak-to-norm continuity" w i t h "denting point" , because the result is a clearly false statement. P r o o f We w i l l use the terminology "C is not weak*-<5-dentable" as an abbrevi-at ion for "C has no weak*-slices of diameter less than 8". Let X be a separable, non-Phelps space. B y Theorem 111.2, there is an equiv-alent norm, ||-||, on X and a 8 > 0 such that the dual unit ba l l , Bx*, is str ict ly convex and not weak*-35-dentable. 74 Also , since X is separable, we can choose a biorthogonal system {(x,-, x * ) } i e N in X x X* [22]. Tha t is, {(x t-, x , * ) } , e N s a t i s f i e s : a) (x*, x y ) = 6{j; b) sp{x t } = X; and c) sp*{x*} = X*. We w i l l par t i t ion the positive integers, N , into two sets, a and A , obtaining the FDD for Y from the vectors { x t } , e o . , so that Y = s p { x i } l & r , and obtaining the corresponding constructs for X/Y from { x * } t £ A . This par t i t ioning is produced by the following lemma: L e m m a TV.5 With the notation and assumptions described above, there exist increasing sequences {o~n} and { A n } of finite subsets of N such that: a) on n An — 0 and {1, 2, . . . , n} C on U An for each n G N . bj For each n , the set c n satisfies: i) For every non-zero x* G sp{x* : i G A , ^ } there exists x G spfx^ : t" G on U A n _ j } such that | |x | | = 1 and |(x*, x ) | > (1 - n _ 1 )| |x*||. ii) For every x G sp{x t- : t G a n _ 1 } with | |x | | = 1, there exist y*, z* G (sp{x t- : t G c r n })* of norm 1 such that (y*, x) > 1 — n _ 1 , (2*, x) > 1-n" 1, and ||y* - || > 5. 75 c) For each n, the set An satisfies: i) For every non-zero x G sp{x t- : i G on} there exists x* G sp{x* : i £ on U A „ } such that \\x*\\ = 1 and \(x*, x)\ > (1 - ra_1)||x||. ii) For every x G sp{xj : i G on U A w _ 1 } of norm 1 satisfying {y* e sp{x* : » G A B _ ! } : | |y*|| < 1 and (y*, x) > 1 - ^n)'1 } ^ 0 , there exists y* G sp{x* : i G A n } of norm 1 such that dist(y*, sp{x* : i G A ^ ! }) > 6 and (y*, x) > 1 - n _ 1 . P r o o f o f T h e o r e m I V . 4 (continued) Given the lemma, we can complete the proof of Theorem I V . 4 , as follows: 00 oo Let o — \}on and A = | J A n , where {on} and { A n } are as in L e m m a I V . 5 , 1 l and let Y = sp{x1 : i G a } . We w i l l show that Y has the desired properties. Fi rs t of a l l , Johnson and Rosenthal showed in [19] that the conditions (b.i) and (c.i) guarantee that both Y and X/Y have FDD's. To show that neither Y nor X/Y is a Phelps space, it suffices, by Theorem III.2, to show that both Y* and (X/Y)* are str ict ly convex (with respect to their corre-sponding induced norms), but that neither By* nor B^x/Y)* 1S weak*-dentable. Since a subspace of a str ict ly convex space is t r iv ia l ly s tr ict ly convex, it is clear that (X/Y)* = Y1- < X* is s tr ict ly convex. Suppose now f,g£ X*/Y1 S Y*, w i th | | / | | = | |^| | = 1 and \\f+ g\\ = 2 . Let Q: X* —> X*/Y1- be the canonical quotient map. Since Y1- is weak*-closed, Q is weak*-to-weak* continuous. Also , Q{BX*) is dense in BY*, so there exist sequences {/*}» {^fc} c S X * s u c h t h a t Q(fk) -* 1 a n d Q{9k) ~* 0 - I f n o w / a n d 5 are weak*-accumulation points of {fk} and {ff^}, respectively, then the weak*-to-weak* continuity of Q implies Q{f) = f and Q(g) = g. Clear ly | | / | | * = \\g\\* = 1 , since = 1 and f,gE Bx* • Since ||-||* is s tr ict ly convex on X*, and since 2 = H 7 I I + llffll = H/H* + \\B\\* > 11/ + g\\* > \\7 + g\\ = 2 , we must have / = g, and so / = g. Thus X* /Y1- = Y"* is s tr ict ly convex. We now use condit ion (b.ii) of L e m m a IV.5 to show that BY* is not weak*-6-dentable. Let a > 0 and let x £ SY • Since sp{x t- : i E a} is norm dense in Y, we may assume that x E sp{xi : i E } for some n 0 E N , and that U Q 1 < a. Condi t ion (b.ii) then implies the existence of elements y*, z* £ (sp{x t- : i £ crno})* of no rm 1 such that (y*, x) > 1 — a, (z*, x) > 1 — a, and ||y* — z*\\ > 6. B y the Hahn-Banach Theorem, we may assume y*, z* £ Y*. Then the above estimates show that y*, z* £ S1(BY*, x, a) and d i am S1(BY*, x , a) > 6. Since x and a were arbitrary, BY* is not weak*-£-dentable . 77 Last ly , we show that condit ion (c.ii) implies that B^X/Y)* — BY± is not weak*-6-dentable. Let x £ Sx and note that Y1 = sp*{x t* : i £ A } . Let a > 0. B y the density of sp{x t- : i G a U A } in X, we may assume that x G sp{x{ : i G o~no U A r e Q } for some n 0 G N such that UQ 1 < a . B y choosing ra0 sufficiently large, we may also assume that S = {z* G sp{x* : t G A ^ . ! } : \\z*\\ < 1, (z*, x) > 1 - a} ± 0 . T h e n by (c.ii) , d i a m S 1 ( 8 Y ± , x, <*) > d i a m S > 6. Since x and a were arbitrary, By± is not weak*-6-dentable. The theorem is proven. • It remains to prove L e m m a I V . 5 . For this, we need the following elementary lemmas. L e m m a I V . 6 Let X be a Banach space such that X* is strictly convex and 8X* is not weak*-6-dentable, for some 6 > 0. Then for every finite dimen-sional Y < X, 8yx is not weak*-S-dentable. P r o o f B y Theorem III.2, X* does not have C*PCP. B y L e m m a 111.4(b), the given str ict ly convex dual norm on X* is easily seen to satisfy the conclusion of Corol lary 1.7(b). Since Y1- is weak*-closed and finite codimensional in X*, the lemma follows from Corol lary 1.7(b). • 78 The hypothesis that X* be str ict ly convex is necessary. Indeed, if X is a separable Banach space such that X* has C*PCP but not RNP, we can choose, by Proposi t ion III.9, an equivalent norm on X whose dual unit ba l l is not weak*-dentable. B y Theorem II.5 and Corol lary II.4, for every 6 > 0 there is a finite dimensional subspace Y of X such that By± has weak*-slices of diameter less than 8. L e m m a I V . 7 Let X be as in Lemma IV.6. Then for every finite dimen-sional Y < X*, BX*/Y J S n o * weak*-6/2-dentable. P r o o f A s in the proof of L e m m a I V . 6 , the strict convexity of X* gives that it is enough to show that if Bx*/y w e r e weak*-<5/2-dentable, then Bx* would have a relatively weak*-open subset of diameter less than 8. Thus let U C Bx*jy be a relatively weak*-open subset w i t h d i a m t / = 7 < 8/2. Let Q: X* —> X* jY be the canonical quotient map, and let Q — Q\gx„ . Then Q: Bx* —• Bx* JY is weak*-to-weak* continuous and surjective, so Q (U) is a rela-tively weak*-open subset of Bx*. Let {xn} be norm dense in Y and let y £ Q~l (U). Then Q~l (u) = U [(*» + B(y> 1))n (u) • Each set (xn +B(y, 7)) C\Q~l (U) is weak*-closed in Q~x (U), which is a Baire space i n the relative weak*-topology on Bx*, so by the Baire Category Theorem there is an ra0 £ N such that ( i ^ + B(y, 7)) Pi Q~l (U) has non-empty interior V in this 79 topology. Since Q (U) is relatively weak*-open in Bx* > V is relatively weak*-open in Bx* • Clearly, diamV" < 2y < 6. • P r o o f o f L e m m a I V . 5 We define an and An by induction. Let ox = {1} and = 0, and suppose that for some n > 1, and A n _ 1 have been constructed satisfying (a), (b), and (c) of the statement of L e m m a I V . 5 . Fi rs t we w i l l explain the construction of an: Since A n _ 1 is finite, the unit sphere of sp{x t- : i £ A w _ 1 } is compact, so we can choose a finite subset a'n of N \ A n _ 1 satisfying { l , 2, . . . , n) C a'n U A J l _ 1 and also condit ion (b.i) (with a'n in place of on). Next observe that since sp{x* : * £ A „ _ x } C (sp{xi : i ^ A n _ 1 }) J * and since these two spaces have the same dimension (namely l A , ^ |), they must in fact be equal. Let H = sp{x* : i £ An_x }, so that H± = sp{x t- : i ^ A f l _ 1 }. Now let u £ sp{Xj : i £ o ^ J i }, w i t h | |u | | = 1. Since u £ H± and (-HjJ* — X* jH, we have, by L e m m a I V . 7 , that diam{x* £ BrH±y : (x*, u) > 1 — ra-1 } > £ . (Recal l that Bx* is not weak*-35-dentable.) Thus there exist y*u, z*u £ B^H±y satis-fying ( y ; , u> > 1 - n " 1 , « , u ) > l - n - 1 and | | y £ - < | | > 5 . Choose tu £ sp{x t- : i £ A n _ ! } of norm 1 such that |(y* - 2*, t t t ) | > (5, and denote 80 b y °~n,u t h e support of tu (i.e., if tu = ^2jeK a j X j , w i t h a y ^ 0 for al l j , then a n u = K). Note a n u is a finite subset of N \ A n _ x . The compactness of the unit sphere, S x , of s^>{xi : i £ } w i l l allow us to work w i t h a finite set of such elements u. Namely, for each u £ S x , let Uu = {xeS1:{y*l,x)>l-n-1, « , x) > 1 - n~l } . Since for a l l u chosen as above, the sets Uu are relatively open in S x , the compactness k of S x gives that there exists a finite set {«y}* C S± such that | J UUj = S x . If we denote the corresponding objects chosen as above by {y^}* , {ZJ}I, {tj}i and { a n y }*, we have that for each x £ S x , there is a 1 < j < k such that <yj, x) > 1 - n _ 1 , (zjf, x) > 1 - n _ 1 and (yj* - z j , t y ) > (5 . k Now set a n = < U (IJ c r^- ) . If 5 » : (SP{*.- = * £ A n _ x })* -> (sp{z f. : t £ * n } ) * is the natural restriction map, we have by the above estimates that \\SM) - Sn[z})\\ > 6 , so ||y* — Zj || > 6, since = 1, and condit ion (b.ii) is fulfilled. We now proceed to the construction of An. 81 B y the compactness of the unit ba l l of sp{x t- : i G crn}, we can choose A'n such that conditions (a) and (c.i) are satisfied (with A ' R in place of A n ) . We c l a im that for each u G sp{x,- : i G an U A r i _ 1 } of no rm 1 which satisfies (*) {y* G sp{x t- : G A n _ x } : ||y*|| < 1, (y*, u) > 1 - ( 2 n ) - x } ^ 0 , we can choose, by L e m m a IV .6 , a y* G sp{x* : z ^ o^} of norm 1 and satisfying <y£. «> > i - a n d d i s t ( y u ' s PK * = * e A w - i } ) > 6. Indeed, otherwise U={y* G sp{x? :» i an} : | |y*|| = 1, (y*, u) > 1 - n~l } would be non-empty and included in (rjnsp{xt : i G An-i }) + 8BX* , by the choice of u . Now W = U f~l sp{x* : i G A f l _ 1 } is a separable subset of Sx*, so choose a countable dense subset {x*} C W. Then 3 and each set (xy + 6BX*) H 17 is relatively weak*-closed in J7. Since X is separable, Sx* is a Bai re space in the weak*-topology, hence so is U, so by the Baire Category Theorem, there is a j0 G N such that (x*^ + 6BX*) D U has non-empty relative weakMnterior V. Obviously, d i a m V < 28. Since U is clearly weak*-open in Sx*, 82 it follows that Sx*, and hence Bx*, has a relatively weak*-open subset of diameter < 26. B y L e m m a III.4, Bx* is weak*-26-dentable, which is a contradict ion. Thus let K = { u G sp{xj : i G on U A n _ x } : | |u | | = 1, u satisfies (*)} . For each u G K, let A n t t be the support of the corresponding ?/* and let Vu = {xeK:(y*u,x)>l-n-1}. B y the compactness of K, we can choose finitely many such elements u, say { « y ) i , such that | J V„. = K. 1 J k Now set A r e = A ' n U ( | J A W U f c ) . Then condit ion (c.ii) is satisfied, and the proof of L e m m a IV .5 is complete. • 83 References E . A s p l u n d , Frechet differentiability of convex functions, A c t a M a t h . 121 (1968), 735-750. E . Bishop and R . R . Phelps, The support functionals of a convex set, P roc . Sym-pos. Pure M a t h . , V o l . 7, Amer . M a t h . S o c , Providence, R . I., 1963, 27-35. [3] J . 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U h l , Jr . , Vector Measures, M a t h Surveys, N o . 15 , Amer ican M a t h Society, Providence, R . I., 1977. 84 [11] M . Edelstein, Concerning dentability, Pacific J . M a t h . 46 (1973), 111-114. [12] G . A . Edgar and R . H . Wheeler, Topological properties of Banach spaces, Pacific J . M a t h . 115 (1984), 317-350. [13] N . Ghoussoub and B . Maurey, §s-embeddings in Hilbert space, J . Funct . A n a l . 61 (1985), 72-97. [14] , ~Hs-embeddings in Hilbert space and optimization on Qs-sets, M e m . A m e r . M a t h . Soc. 349 (1986). [15] N . Ghoussoub, B . Maurey and W . Schachermayer, A counterexample to a prob-lem on points of continuity in Banach spaces, to appear. [16] , Geometrical implications of certain infinite dimensional decomposi-tions, to appear. [17] R . E . Huff and P . D . Mor r i s , Geometric characterizations of the Radon-Nikodym property in Banach spaces, Studia M a t h . 56 (1976), 157-164. [18] K . John and V . Zizler, A note on strong differentiability spaces, Comment . M a t h . U n i v . Carolinae 17 (1976), 127-134. [19] W . B . Johnson and H . P . Rosenthal, On w*-basic sequences and their applica-tions to the study of Banach spaces, S tudia M a t h . 43 (1972), 77-92. [20] V . Klee , Some new results on smoothness and rotundity in normed linear spaces, M a t h . A n n . 139 (1959), 51-63. [21] B . - L . L i n , P . - K . L i n and S. L . Troyanski , A characterization of denting points of a closed, bounded, convex set, Longhorn Notes, U . T . Funct ional Analysis Seminar, 1985-1986, The Universi ty of Texas at A u s t i n . [22] J . Lindenstrauss and L . Tzaf r i r i , Classical Banach Spaces, Springer-Verlag Lec-ture Notes i n Mathemat ics , 338, Be r l i n - Heidelberg - New Y o r k , 1973. [23] I. Namioka and R . R . Phelps, Banach spaces which are Asplund spaces, Duke M a t h . J . 42 (1975), 735-750. 85 [24] A . Pelczyriski , On Banach spaces containing Lx(p), Studia M a t h . 30 (1968), 231-246. [25] R . R . Phelps, A representation theorem for bounded convex sets, P roc . Amer . M a t h . Soc. 11 (1960), 976-983. [26] , Dentability and extreme points in Banach spaces, J . Funct . A n a l . 17 (1974), 78-90. [27] W . R u d i n , Functional Analysis, M c G r a w - H i l l Inc., 1973. [28] V . Smuljan, Sur la derivabilite de la norme dans Vespace de Banach, D o k l . A k a d . Nauk S S S R 27 (1940), 643-648. [29] C . Stegall, The Radon-Nikodym property in conjugate Banach spaces, II, Trans. A m e r . M a t h . Soc. 206 (1975), 213-223. 86 

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