Convex Functions, Monotone Operators and Differentiability (Lecture Notes in Mathematics, 1364)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zi.irich F. Takens, Groningen

1364

Robert R. Phelps

Convex Functions, Monotone Operators and Differentiability 2nd Edition

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Author Robert R. Phelps Department of Mathematics GN-50 University of Washington Seattle, WA 98195-0001, USA

Cover Graphic by Diane McIntyre

Mathematics Subject Classification (1991): 46B20, 46B22, 47H05, 49A29, 49A51, 52A07

ISBN 3-540-56715-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56715-1 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data Phelps, Robert R. (Robert Ralph), 1926 Convex functions, monotone operators, and differentiabilityIRobert R. Phelps. 2nd. ed. p. cm. - (Lecture notes in mathematics; 1364) Includes bibliographical references and index. ISBN 0-387-56715-1 1. convex functions. 2. Monotone operators. 3. Differentiable Functions. I. Title. II. Series: Lecture notes in mathematics (Springer Verlag); 1364. QA3.L28 no. 1364 (QA331. 5) 510 s-dc20 (515'.8) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. V iolations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989, 1993 Printed in Germany

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PREFACE

In the three and a half years since the first edition to these notes was written there has been progress on a number of relevant topics. D. Preiss answered in the affirmative the decades old question of whether a Banach space with an equivalent Gateaux differentiable norm is a weak Asplund space, while R. Haydon constructed some very ingenious examples which show, among other things, that the converse to Preiss' theorem is false. S. Simons produced a startlingly simple proof of Rockafellar's maximal monotonicity theorem for sub differentials of convex functions. G . Godefroy, R. Deville and V. Zizler proved an exciting new version of the Borwein-Preiss smooth variational prin­ ciple. Other new contributions to the area have come from J. Borwein, S. Fitzpatrick, P. Kenderov, I. Namioka, N. Ribarska, A. and M. E. Verona and the author. Some of the new material and substantial portions of the first edition were used in a one-quarter graduate course at the University of Washington in 1 991 (leading to a number of corrections and improvements) and some of the new theorems were presented in the Rainwater Seminar. An obvious improvement is due to the fact that I learned to use 'JEX. The task of converting the original MacWrite text to 'JEXwas performed by Ms. Mary Sheetz, to whom I am extremely grateful. Robert R. Phelps February 6, 1992 Seattle, Washington

PREFACE TO THE FIRST EDITION

These notes had their genesis in a widely distributed but unpublished set of notes Differentiability of convex functions on Banach spaces which I wrote in 19 7 7-78 for a graduate course at University College London (UCL). Those notes were largely incorporated into J. Giles' 19 82 Pitman Lecture Notes Convex analysis with application to differentiation of convex functions. In the course of doing so, he reorganized the material somewhat and took advantage of any simpler proofs available at that time. I have not hesitated to return the compliment by using a few of those improvements. At my invitation, R. Bourgin has also incorporated material from the UCL notes in his extremely comprehensive 19 83 Springer Lecture Notes Geometric aspects of convex sets with the Radon -Nikodym property. The present notes do not overlap too greatly with theirs, partly because of a substantially changed emphasis and partly because I am able to use results or proofs that have come to light since 19 83. Except for some subsequent revisions and modest additions, this material was covered in a graduate course at the University of Washington in Win­ ter Quarter of 19 88. The students in my class all had a good background in functional analysis, but there is not a great deal needed to read these notes, since they are largely self-contained; in particular, no background in convex functions is required. The main tool is the separation theorem (a.k.a. the Hahn-Banach theorem); like the standard advice given in mountaineering classes (concerning the all-important bowline for tying oneself into the end of the climbing rope), you should be able to employ it using only one hand while standing blindfolded in a cold shower. These notes have been influenced very considerably by frequent conversa­ tions with Isaac Namioka (who has an almost notorious instinct for simplifying proofs) as well as occasional conversations with Terry Rockafellar; I am grate­ ful to them both. I am also grateful to Jon Borwein, Marian Fabian and Simon Fitzpatrick, each of whom sent me useful suggestions based on a preliminary verslOn. Robert R. Phelps October 5, 19 88 Seattle, Washington

IN TRODUCTION

The study of the differentiability properties of convex functions on infinite dimensional spaces has continued on and off for over fifty years. There are a couple of obvious reasons for this. Aside from the intrinsic interest of inves­ tigating the many consequences implicit in something as simple as convexity, there is the satisfaction (for this author, at least) in discovering that a number of apparently disparate mathematical topics (extreme points - rather, strongly exposed points - of noncompact convex sets, monotone operators, perturbed optimization of real-valued functions, differentiability of vector-valued mea­ sures) are in fact closely intertwined, with differentiability of convex functions forming a common thread. Starting in Section 1 with the definition of convex functions and a funda­ mental differentiability property in the one-dimensional case [right-hand and left-hand derivatives always exist], we get quickly to the first infinite dimen­ sional result, Mazur's intriguing 1933 theorem: A continuous convex function on a separable Banach space has a dense Go set of points where it is Gateaux differentiable. In order to go beyond Mazur's theorem, some time is spent in studying the subdifferential of a convex function Ii this is a set-valued map from the space to its dual whose image at each point x consists of all plausible candidates for the derivative of I at x. [The function I is Gateaux differ­ entiable precisely when the subdifferential is single-valued, and it is Frechet differentiable precisely when its sub differential is single-valued and norm-to­ norm continuous.] Since a sub differential is a special case of a monotone operator, Section 2 starts with a detailed look at monotone operators. These objects are of inde­ pendent origin, having been extensively studied in the sixties and early seven­ ties by numerous mathematicians (with major contributions from H. Brezis, F. Browder and G. J. Minty) in connection with nonlinear partial differential equations and other aspects of nonlinear analysis. (See, for instance, [Bre], [De), [pa-Sb) or [ZeD. Also in the sixties, an in-depth study of monotone op­ erators in fairly general spaces was carried out by R. T. Rockafellar, who established a number of fundamental properties, such as their local bound­ edness. He also gave an elegant characterization of those monotone operators which are the sub differentials of convex functions. [The connection between monotone operators and derivatives of convex functions is readily apparent on the real line, since single-valued monotone operators coincide in that case with monotone nondecreasing functions, as do the right-hand derivatives of convex functions of one variable.) Mazur's theorem was revisited 30 years later by J. Lindenstrauss, who showed in 1 963 that if a separable Banach space is assumed to be reflexive, then Mazur's conclusion about Gateaux differentiability could be strengthened to Frechet differentiability. In 1 96 8, E. Asplund extended Mazur's theorem in two ways: He found more general spaces in which the same conclusion holds, and he studied a smaller class of spaces (now called Asplund spaces) in which Lindenstrauss' Frechet differentiability conclusion is valid. Asplund used an

VIII

ingenious combination of analytic and geometric techniques to prove some of the basic theorems in the subject. Roughly ten years later, P. Kenderov (as well as R. Robert and S . Fitzpatrick) proved some general continuity theorems for monotone operators which, when applied to subdifferentials , yield Asplund's results as special cases. In Section 2 we follow this approach, ircorporating recent work by D. Preiss and 1. Zajicek to obtain the major differentiability theorems. The results of Section 2 all involve continuous convex functions defined on open convex sets. For many applications, it is more suitable to consider lower semicontinuous convex functions, even those which are extended real valued (possibly equal to + 00). (For instance, in many optimization problems one finds just such a function in the form of the supremum of an infinite fam­ ily of affine continuous functions.) Lower semicontinuous convex functions also yield a natural way to translate results about closed convex sets into results about convex functions and vice versa. (For instance, the set of points on or above the graph of such a convex function - its epigraph forms a closed con­ vex set). In Section 3 one will find some classical results (various versions and extensions of the Bishop-Phelps theorems) which, among other things, guaran­ tee that subdifferentials still exist for lower semicontinuous convex functions. A nonconvex version of this type of theorem is I. Ekeland's variational prin ­ ciple, which asserts that a lower semicontinuous function which nearly attains its minimum at a point x admits arbitrarily small perturbations (by trans­ lates of the norm) which do attain a minimum, at points near x . This result, while simple to state and prove, has been shown by Ekeland [Ek] to have an extraordinarily wide variety of applications, in areas such as optimization, mathematical programming, control theory, nonlinear semigroups and global analysis. In Section 4, we examine variational principles which use differentiable perturbations. The first such result was due to J. Borwein and D. Preiss; subsequently, this was recast in a different and somewhat simpler iorm by G. Godefroy, R. Deville and V . Zizler; we follow their approach. Some deep theorems about differentiability of convex functions fall out as fairly easy corollaries, and it is reasonable to expect future useful applications. This is followed by the generalization (to maximal monotone operators) of Preiss' theorem that Gateaux differentiability of the norm forces every continuous convex function to be generically Gateaux differentiable. Section 5 describes the duality between Asplund spaces and spaces with the Radon-Nikodym property (RNP). These are Banach spaces for which a Radon-Nikodym-type differentiation theorem is valid for vector measures with values in the space. Spaces with the RNP have an interesting history, starting in the late sixties with the introduction by M. Rieffel of a geometric property ( dentability) which turned out to characterize the RNP and which has led to a number of other characterizations in terms of the extreme points (or strongly exposed points) of bounded closed convex subsets of the space. A truly beautiful result in this area is the fact that a Banach space is an Asplund space if and only if its dual has the RNP. (Superb expositions of the RNP may be -

IX

found in the books by J. Diestel and J. J. Uhl [Di-U] and R. Bourgin [Bou].) In Section 5, the RNP is defined in terms of dent ability, and a number of basic results are obtained using more recent (and simpler) proofs than are used in the above monographs. One will also find there J. Bourgain's proof of C. Stegall's perturbed optimization theorem for semicontinuous functions on spaces with the RNPj this yields as a corollary the theorem that in such spaces every bounded closed convex set is the closed convex hull of its strongly exposed points. The notion of perturbed optimization has been moving closer to center stage, since it not only provides a more general format for stating previously known theorems, but also permits the formulation of more general results. The idea is simple: One starts with a real-valued function f which is, say, lower semicontinuous and bounded below on a nice set, and then shows that there exist arbitrarily small perturbations 9 such that f + 9 attains a minimum on the set. The perturbations 9 might be restrictions of continuous linear functionals of small norm, or perhaps Lipschitz functions of small Lipschitz norm. Moreover, for really nice sets, the perturbed function attains a strong minimum: Every mimimizing sequence converges. The brief Section 6 is devoted to the class of Banach spaces in which every continuous convex function is Gateaux differentiable in a dense set of points (dropping the previous condition that the set need be a G6). Some evidence is presented that this is perhaps the "right" class to study. Even more general than monotone operators is a class of set valued maps (from a metric space, say, to a dual Banach space) which are upper semicon­ tinuous and take on weak* compact convex values, the so-called useo maps. In Section 7, some interesting connections between monotone operators and usco maps are described, culminating in a topological proof of one of P. Kenderov's continuity theorems.

CONTENTS

1 . Convex functions on real Banach spaces

1

Subdifferentials of continuous convex functions, Gateaux and Frechet differen­ tiability, Mazur's theorem.

2. Monotone operators, subdifferentials and Asplund spaces

17

Upper semicontinuity of set-valued monotone operators, characterization of Gateaux and Frechet differentiability in terms of continuous selections, Preiss­ Zajicek generic continuity theorem for monotone operators into separable dual spaces, Asplund spaces and subspaces with separable duals, weak*-slices, subdif­ ferentials of convex functions as maximal monotone operators, local boundedness of monotone operators, Kenderov's generic continuity theorems for maximal monotone operators, weakly compactly generated dual spaces and Asplund spaces. 3.

Lower semicontinuous convex functions

38

Extended real-valued convex functions and their subdifferentials, support points of convex sets. minimal points with respect to special cones, Ekeland's variational principle, Brondsted-Rockafellar theorem, Bishop-Phelps theorems, maximal mono­ tonicity of the subdifferential, maximal cyclically monotone operators are subdiffer­ entials, subdifferentials of saddle functions.

4. Smooth variational principles, Asplund spaces, weak Asplund spaces

58

,a-differentiability, smooth bump functions and the Godefroy-Deville-Zizler variational principles, density of ,a-differentiability points, Borwein-Preiss smooth variational principle, Banach-Mazur game, generically single-valued monotone op­ erators on Gateaux smooth spaces. 5.

Asplund spaces, the RNP and perturbed optimization

79

Slices and weak* slices and dentability, RNP, infinite trees, E is an Asplund space means E' has the RNP, duality between weak* strongly exposed points and Frechet differentiability, perturbed optimization on RNP sets, bounded closed convex RNP sets are generated by their strongly exposed points, Ghoussoub-Maurey theorems.

6. Gateaux differentiability spaces

95

Gateaux differentiability spaces, equivalence with M-differentiability spaces, du­ ality characterization in terms of weak* exposed points, stability results.

102 7. A generalization of monotone operators: Usco maps Upper semicontinuous compact valued ( usco) maps, maximal monotone oper­

ators are minimal usco maps, topological proof of Kenderov's generic continuity theorem, the Clarke subdifferential.

References

110

Index

115

Index of Symbols

1 17

Section 1

1. Convex functions on real Banach spaces.

The letter E will always denote a real Banach space, D will be a nonempty open convex subset of E and f will be a convex function on D. That is, f: D --t R satisfies

f[tx + (1

-

t)y] ::; tf( x) + ( 1

-

t)f(y)

whenever x, y E D and 0 < t < 1 . IT equality always holds, f is said to be affine. A function f : D --t R is said to be concave if f is convex. We will be studying the differentiability properties of such functions, assuming, in the beginning, that they are continuous. (The important case of lower semicontinuous convex functions is considered in Sec. 3.) -

1.1. Examples. ( a) The norm function f(x)

IIxll is an obvious example. More generally, if C is a nonempty convex subset of E, then the distance function dc(x)

=

=

inf{lIx - yll:yE C},

is continuous and convex on D= E. ( Note that dc(x)

xE E, =

Ilxll if C = {O}.)

( b ) The supremum of any family of convex functions is convex on the set where it is finite. In particular, if A is a nonempty bounded subset of E, then the farthest distance function x -+ sup{lIx - yll:yEA} is continuous and convex on D= E. ( c) The norm function is also generalized by sublinear functionals, that is, func­ tions p: E -+ R which satisfy p(x + y) ::; p(x) + p(y) and p(tx) = tp(x) whenever t 2: O. Obviously, the supremum of a finite family of linear functionals is sublinear. A sublinear functional p is continuous if and only if there exists M > 0 such that p(x) ::; Mllxll for all x.

(d ) The Minkowski gauge functional is another generalization of the norm func­ tion: Suppose that C is a convex subset of E, with 0 E int C. Define pc(x) = inf{>. > o:xE >'C},

xE E.

The functional pc is sublinear and nonnegative. Moreover, pc(x) {>.x: >. 2: O} c C, and bdry C {x:pc(x) = I}; in fact =

=

0 if and only if

2

Section 1 int C

=

{x:pc(x) < 1} C C C {x:pc(x) ::; 1} = C.

There exists M > 0 such that pc(x) ::; Mllxll for all x ( take M = 1/r, where the ball of radius r centered at 0 is contained in C), hence pc is necessarily continu­ ous. Conversely, any positive-homogeneous, subadditive, nonnegative and continuous functional p on E is of the form pc, simply take C = {x:p(x) ::; 1}. If p fails to be a seminorm, that is, if p(x) f:. p( -x), then C is not symmetric with respect to O. Conversely, if C is either open or closed and is not symmetric, then p is not a seminorm.

The following elementary lemma is fundamental to the study of differen­ tiability of convex functions.

Lemma 1.2. If X o E D, then for each x E E the "right hand " directional derivative f(xo + tx) - f(xo) d+ f(xo )(x) = lim t-+O+ t exists and defines a sublinear functional on E. Proof. Note that since D is open, f( Xo + tx) is defined for sufficiently small t > O. Figure 1 . 1 below shows why d+ f(xo ) exists; the difference quotient is nonincreasing as t -+ 0+, and bounded below, by the corresponding difference quotient from the left.

)Co +tx

Fig. 1.1 To prove this, we can assume that Xo = 0 and f(xo ) = O. IT 0 < t < s, then by convexity t ( s - t) t f(tx) ::; - f( sx) + -- f(O) = -f( sx), s s s which proves monotonicity. Applying this to -x in place of x, we see that - [f(xo - tx) - f(xo)]/t is nondecreasing as t

-+

0+. Moreover, by convexity again, for t > 0

2f(xo) ::; f(x o - 2tx) + f(xo + 2tx), so that

- [f(x o - 2tx) - f(xo )] < [f(xo + 2tx) - f(xo )] 2t 2t which shows that the right side is bounded below and the left is bounded above. Thus, both limits exist; the left one is - d+ f(xo)( - x) and we obviously have

1. Convex functions on real Banach spaces.

3

-at f(xo )( - x ) ::; d+ f(xo )(x). It is also obvious that d+f( x) is positively homogeneous. To see that it is subadditive, use convexity again to show that for t > 0, [f(x + t( u + v )) - f(x)] < f(x + 2tu ) - f(x) f(x + 2tv) - f(x) + t 2t 2t and take limits

as

t -+ 0+.

Definition 1 .3. The convex function f is said to be Gateaux differentiable at Xo E D provided the limit df(xo)(x) = limo t-+

f(xo + tx) - f(xo) t

exists for each x E E. The function df( xo ) is called the Gateaux derivative (or Gateaux differential) of f at Xo . It is immediate from this definition (requiring the existence of a two-sided limit ) that f is Gateaux differentiable at Xo if and only if -d+f(xo)( - x) = d+ f(xo )(x) for each x E E. Since a sublinear functional p is linear if and only if p( - x) = -pC x) for all x , this shows that f is Gateaux differentiable at Xo if and only if x -+ d+f(xo)(x) is linear in x ; in particular, if this is true, then df(xo ) is a linear functional on

E.

1.4. Examples. ( a) If f is a linear functional on E ( not necessarily continuous ) , then df(xo)(x) = f(x) for all Xo and x. For an example of a discontinuous linear functional on a normed linear space, let f(x) = x/CO), for x in the space of all polynomials on [-1, 1] with supremum norm. ( It is easy to construct a sequence of polynomials Xn converging uniformly to 0 such that x�(O) = 1 for all n. ) Thus, x � df(xo)(x) need not be continuous. ( b) Th e nor m IIxll = E lxnl in £1 is Gateaux differentiable precisely at those points x = (xn) for which Xn ;/; 0 for all n. In this case, the Gateaux differential is the bounded sequence ( sgn xn) E £00. The norm in £1(r) (r uncountable) is not Gateaux differentiable at any point. Proof. If x E £1 and Xn = 0 for some n, let cn = (0,0, ... ,0, 1,0,... ) be the sequence whose only nonzero term is a 1 in the n-th place. It follows that IIx+ tcnlh -lIxlh = Itl, so dividing both sides by t shows that the ( two-sided) limit as t � 0 does not exist. [This observation shows how to prove the second assertion, since any element of £l(r) vanishes at all but a countable number of members of r.] Suppose, on the other hand, that for every n, Xn ;/; 0, that (; > 0 and that Y E £1. We can choose N > 0 such that En>NIYnl < (;/2 . For sufficiently small C > 0 we have sgn(xn + tYn) = sgn Xn if 1 �

Consequently,

n

�

N,

ItI < C.

4

Section 1

ll x + tYl h - II X l h - LJ.,Yn sgn Xn I < t I E�= l C l { I Xn + tYn l - I X n l - tYn sgn xn } 1 + 2En>NI Yn l < f provided I t I < c . I

_

IT f i s a continuous convex function which i s Gateaux differentiable at a point, then its differential is in fact a continuous linear functional. This is a consequence of the following basic result. 1.5. Notation. IT x E E and r > 0, the closed ball centered at x is denoted by B(x; r) = { y E E: II x - y ll � r } . Proposition 1.6. If the convex function f is continuous at Xo E D, then it is locally Lipschitzian at Xo , that is, there exist M > 0 and 8 > 0 such that B(xo; 8) c D and I f(x) - f(y)1 � M ll x - y ll whenever x, y E B(xo; 8). Proof. Since f is continuous at Xo, it is locally bounded there; that is, there exist Ml > 0 and 8 > 0 such that I f I � Ml on B(xo ; 28) C D. IT x, y are distinct points of B(xo; 8), let a = Il x - YII and let z

=

y + (8/a)(y - x);

see Fig. 1 .2 below.

Fig. 1.2 Note that z E B(xo; 28). Since y = [a/(a + 6)]z + [6/(a + 6)] x is a convex combination ( lying in B(xo ; 28)), we have

f(y) � [a/(a + 8)]f(z) + [6/(a + 8)]f(x)

so

f(y) - f(x) � [a/(a + 6)] U (z) - f(x)} � (a/6) · 2Ml = (2Mt / 8) ll x - yli . Interchanging x and y gives the desired result, with M = 2M1 / 8. Remark. Note that we only used local boundedness of f; hence the latter property is equivalent to continuity for a convex function on an open convex

1. Convex functions on real Banach spaces.

5

set. In fact, it suffices that f be merely locally bounded above: If f � M on the ball B(xo; r), then for all x E B(xo; r) we have 2xo - x E B(xo; r) and hence < f(2xo - x) + f(x) < M + f(x) f(x o ) ' 2 2 so -f(x) � M + 2 I f(xo) l; that is, I f I � M + 2If(xo)1 on B(xo; r).

Corollary 1.7. If the convex function f is continuous at Xo E D, then d+ f(xo) is a continuous sublinear functional on E, and hence df(xo) (when it exists) is a continuous linear functional. Proof. Given Xo E D there exists a neighborhood B of Xo and M > 0 such that, if x E E, then f (xo + tx) - f (xo) � Mtll x ll provided t > 0 is sufficiently small that Xo +tx E B. Thus, for all points x E E, we have d+f(xo)(x) � M llxll, which implies that d+f(xo) is continuous.

Proposition 1.8. The continuous convex function f is Gateaux differentiable at Xo E D if and only if there exists a unique functional x* in E* satisfying (x*, x - xo) � f(x) - f (xo),

x E D,

or equivalently Y

E E.

(1.1)

( 1 .2)

Proof. We first show that ( 1 . 1) and (1.2) are equivalent. If x* satisfies (1.1), then for any y E E we have Xo + t y E D for sufficiently small t > 0 hence t (x*, y ) = (x*, (xo + ty) - xo) � f (xo + t y) - f(xo) which implies that x* satisfies (1.2). Conversely, if x* satisfies (1 .2) and x E D, let y = x - Xo; then Xo + t(x - xo) E D if 0 < t � 1 so (x*, x - xo) � d+f(xo)(x - xo) �

c

1 (f(xo + t(x - xo)) - f (xo ) ) .

Setting t = 1 yields (1.1 ) . If df(xo) exists, then df(xo )(x - xo) � f(x) - f(xo) as above, so df(xo) satisfies ( 1 . 1 ) . Moreover, if x* satisfies ( 1 . 1 ), then it satisfies (1.2); linearity of d+f(xo) = df(xo) implies that x* = df (xo ). Conversely, suppose that x* is the unique element of E* satisfying ( 1 .1), hence the unique element satisfying (1.2). We now apply the general fact that if a continuous sublinear functional p majorizes exactly one linear functional, then p is itself linear. Indeed, if p is not linear, then it dominates many linear functionals (see the sketch below); the proof is an easy consequence of the Hahn-Banach theorem: If -pc -x) < p(x), find p-dominated extensions of the linear functionals

fl (rx)

=

rp(x)

and

f2(rx) = -rp( -x).

Section 1

6

Fig. 1.3 The functionals x* which satisfy ( 1 . 1 ) play an important role in the study of convex functions, so they are singled out for special attention.

Definition 1.9. IT f is a convex function defined on the convex set C and x E C, we define the subdilJerential of f at x to be the set 8f( x) of all x* E E* satisfying (x * , y - x ) � f(y) - f(x) for all y E C. Note that this is the same as saying that the affine function x * + a, where f(x) - (x* , x ), is dominated by f and is equal to it at y = x, as indicated in the sketch. a =

Graph o f

x

Fig. 1.4 The Hahn-Banach argument we used above shows quickly that if f is continuous at Xo, then 8f(xo) is nonempty: d+ f(xo) is continuous and sub­ linear, so (as above) there exists x* such that (x* , y ) � d+ f(xo )( Y ) for all y E E. Using the fact that the right-hand difference quotients for d+ f(xo) are decreasing, replacing y by y - Xo and letting t = 1 , we get

(x * , y - xo ) � � f(xo)(Y - xo) � f ( y - xo) + xo ) - f(xo) for all y E C. As we will see later, it is still possible to have 8f(xo) nonempty for certain convex f which are not continuous at Xo.

1. Convex functions on real Banach spaces.

7

1.10. Exercise. Prove that for any convex function f the set 8f(xo) (possibly empty!) is convex and weak* closed. (Note that a continuous convex f is Gateaux differentiable at Xo if and only if 8f(xo ) is a singleton.)

Proposition 1.11. If the convex function f is continuous at Xo E D, then af(xo) is a nonempty, convex and weak* compact subset of E*. Moreover, the map x --+ af(x) is locally bounded at Xo, that is, there exist M > 0 and neighborhood U of Xo in D such that II x* 11 :::; M whenever x E U and x* E af(x). Proof. The fact that af(xo) is nonempty, weak* closed and convex follows from the preceding remarks and Exercise 1 .10. The fact that it is weak* compact will follow from Alaoglu's theorem, once we have shown the local boundedness property. Since, by Proposition 1.6, f is locally Lipschitzian at Xo , there exist M > 0 and a neighborhood U of Xo such that I f(y) - f(x) 1 :::; M ll y - x II whenever x, y E U. IT x E U and x * E af(x), then for all y E U we have ( x* , y - x ) :::; fey) - f(x) :::; M ll y - x II , which implies that II x* 1 I :::; M.

Definition 1.12. Suppose that E and F are normed linear spaces, that U is a nonempty open subset of E and that

(1.3)

Another way of stating this is to say that

0, there exists 8 > 0 such that

1 I
<

8.

(1 .4)

We call
8

Section 1

1.13 Facts. (a) If f is a continuous function which is Frechet differentiable at Xo , then it is Gateaux differentiable there and tj>'(xo) = d
1.14 Examples. (a) The norm in £1 is not Frechet differentiable at any point.

Proof By Example 1.4(b), we need only consider a point x = (x n) for which X n i- 0 for all n. Given such an x, for each m 2: 1 let y

m= (0, 0, ... , 0, -2xm, -2xm+1 , -2xm +2 '''.)'

Then lIym l h --+ 0 as m --+ 00. Of course, the sequence (sgn Xn) is our only candidate for the Frechet differential. But

(b) The square of the norm in Hilbert space H is everywhere Frechet differ­ entiable. By the chain rule, the norm is therefore differentiable at every point other than the origin.

Proof. If x , y E H, then Il x + Y l1 2 - II x l1 2 - 2(x, y) = IIYl1 2 j it follows readily that y --+ 2(x, y) is the Frechet derivative of II . 11 2 at x . .

(c) There exists an equivalent norm on £1 which is Gateaux differentiable at every point (except the origin), but is nowhere Frechet differentiable. This striking example will be easy to prove after we have developed a few tools in later sections, so it will be postponed until Sec. 5 (following Theorem 5.12). (d) In Hilbert space H let C be a nonempty closed convex set and denote by P the Lipschitz continuous nearest point mapping (or metric projection) of H onto Cj that is, for all x E H, P( x) is the unique point satisfying

II x - P(x) 1I = inf{ l Ix - yll:y E C}. Define f on H by then f is continuous, convex and everywhere Frechet differentiable, with !,(x) = P(x) for all x.

Proof We first prove the fundamental fact that the mapping P satisfies (in fact, it is characterized by) the following variational inequality: For all x E H, (x - P(x), z - P(x)) � 0 for all z E C.

(1.5)

1 . Convex functions on real Banach spaces.

9

Indeed, if Z E C and 0 < t < 1 , then Zj == tz + ( 1 - t)P(x) E C and hence II x - P(x) 1I ::; II x - Zjll = lI (x - P(x)) - t(z - P(x)) � . Squaring both sides of this inequality, expanding and then cancelling II x - P(x) 1I on both sides yields

0 ::; -2t(x - P(x), Z - P(x)) + t 2 11z

_

p (x) 1I 2 .

If we then divide by t and take the limit as t ..... 0 we obtain ( 1 .5). Moreover, if y E H and we write down (1.5) again, using y in place of x, then take Z = P( y ) in the first equation, Z P(x) in the second one and add the two, we obtain =

(x - y ,P(x) - P( y) } � II P(x) - P( y) 1I 2 for all X, y E H,

( 1.6)

Since (x - y, P(x) - P(y ) ) ::; II x - yll · II P(x) - P(y ) lI , we see that P is a contraction: II P(x) - P( y ) 1 I ::; II x - y ll for all X, y E H. Returning now to the function f, note that since

2f(x)

=

II x ll 2 - inf{ lI x - Y1 l 2 : y E C}

=

sup{2(x, y) - IIYIl 2 : y E C},

it is the supremum of affine functions, hence is convex ( and it is clearly continuous ) . To see the differentiability property, fix x E H j then for any y E H we have

lI (x + y ) - P(x + y) 1I ::; lI (x + y) - P(x) lI , so lI (x + y ) - P(x + y)1 1 2 ::; II x + Yll 2 - 2(x + y , Px) + IIP(x) 1I 2 = II x + Yll 2 + II x - P(x) 1I 2 - lI x ll 2 - 2(y, P(x)) , hence f(x + y ) - f(x) - (P(x), y) � o . O n the other hand, since II x - P(x) 1I < II x - P(x + y ) 1I we get f(x + y) - f(x) - (P(x), y) ::; (y ,P(x + y ) - P(x)) ::; 2 ::; lIyll · IIP(x + y) - P(x) 1I ::; lIylI , which implies the differentiability assertion.

1.15. Exercises. ( a) P rove that for continuous convex functions in finite dimensional spaces, Gateaux differentiability implies FrEkhet differentiability. ( Hint: Use the Fact 1.13 ( b ) , the local Lipschitz property and compactness of the unit ball in finite dimen­ sional spaces. ) ( b) (A calculus student's delight. ) In R n , Gateaux ( hence Frechet ) differentia­ bility of a continuous convex function f at a point Xo is equivalent to the existence of the partial derivatives (8f/8x i)(XO), i = 1 , 2, 3 , ... ,no ( Hint: With basis {ei}, use the linearity of d+ f(xo) on each line Rei to show that 8f(xo) contains only the single linear functional x ..... E Xilf(xo ) .) . Convex functions on the real line have many points of differentiability, as shown by the following result. Despite its elementary nature, it lies at the heart of the proof of Mazur's theorem for separable Banach spaces (Theorem 1.20, below).

Theorem 1 .16. If f is convex on an open interval D c R, then f'(x) exists for all but (at most) countably many points of D .

10

Section 1

Proof. We first show that d+ f(x)(1) [for simplicity, we will write d+ f(x) ] is a nondecreasing function of x. Suppose, then, that X l < X 2 ; we want d+ f(x I ) ::; d+ f(X 2 )' Without loss of generality, we may assume that X l = 0 and f(x I ) = O. [If necessary, we can translate both D and f.] Since d+ f (0) ::; f ( X2 ) / X2 , it suffices to show that the latter expression is dominated by [J (X 2 + t) - f(X 2 ) ]/t whenever t > O. But if we take .\ = X 2 /(X 2 + t), then we have X2 = .\(X 2 + t) + ( 1 - .\)0, so by convexity,

which is equivalent to the desired inequality. A geometrical proof ofthe mono­ tonicity of d+ f can be obtained from the following sketch: p

XI

X2

X2+t

Fig. 1.5 Denoting the various chords to the graph of f by PQ, QR, etc., it is clear that slope PQ ::; slope PR ::; slope QR ::; slope QS ::; slope RS. Expressing the first and last of these in terms of f, we see that (for any s > 0 such that X l + s < X 2 , and any t > 0)

[f(X I + s) - f(x I ) ]/ s ::; [f(X 2 + t) - f(X 2 ) J1t , which shows that d+ f(X I ) � d+ f(X2 ) . We next show that any point where f fails to be differentiable is a point where the monotone function x -+ d+ f(x) has a jump. There are, of course, at most countably many such points. Now, if f'(xo ) fails to exist, then

so it suffices to show that the latter inequality implies that d+ f( x) has a jump at X = Xo, that is, that

Since the right side of this expression dominates d+ f(xo), it suffices to show that the left side is dominated by - d+ f(xo)( - 1), that is, if x < Xo, then d+ f(x)(1) < - d+ f(xo)( - 1). In view of the monotonicity of the limits which define these two quantities, we need only show that, letting to = (!)( Xo - x) we get [J(x + to ) - f(x ) ]/to ::; [f(xo - to ) - f(xo)]Jto. But this i s easily seen to be equivalent to the convexity inequality

1. Convex functions on real Banach spaces.

11

and the proof is complete. The above conclusion no longer holds in R2 ; for instance, the function (XI, X2) -+ I X1 1 fails to be differentiable at each point of the x2-axis. It i8 dif­ ferentiable almost everywhere (Lebesgue) , and this is generally true in R n, as shown in the next exercise. Notice that we obtained a differentiability prop­ erty without assuming f to be continuous. This suggests that continuity of convex functions is automatic; indeed, as we show below, this is true in finite dimensional spaces. The existence of discontinuous linear functionals on any infinite dimensional normed linear space shows that convexity does not imply continuity in the general case.

1.17. Exercise.

Prove that a continuous convex function f on an open convex subset D of Rn is differentiable almost everywhere. ( Hint: Show that for fixed basis element ek, the function x -+ d + f(x )(ek) is a pointwise limit of continuous functions, hence is Borel measuraable and there the set B of points xED where -I!;(x) does not exist is a Borel set. Use Fubini's theorem and Theorem 1.16 to show that B has measure zero. Then use Exercise 1. 15(b ) . )

A more general (and more difficult) result than this is given by a theorem of Rademacher; for perhaps the simplest proof, see [S-P].

Theorem 1 .18 (Rademacher). Let U be a nonempty open 8ub8et of Rn and 8upp08e that f : U -+ Rm i8 locally Lip8chitzian. Then 1 i8 (Frechet) differentiable alm08t everywhere (Lebe89ue). Proposition 1.19. Supp08e that f i8 convex on a nonempty open convex 8ub8et D of R n. Then f i8 continuoU8 at each point of D . Proof. It suffices t o assume that 0 E D and to show that f i s continuous at o. By the remark following Prop. 1.5 it suffices to show that f is bounded above in some neighborhood of o. We can assume that the topology of Rn is given by the (1 norm: I Ixl l = E19�n lxk l . Since D is open, there exists r > 0 such that x E D whenever IIxl l � r. Letting e k denote the k-th coordinate vector, k = 1, 2, . . . , n, we can express any such x as x = EXk ek = E(x k /r)(re k ), where Elxk /r l = I Ix/rl l � 1 . Thus, we can write x = AO · 0 + EAk(rek) + EP.k ( - re k ) , where A k 2:: 0, k = 0, 1 , 2, . . . , n and P. k 2:: 0 , k = 1 , 2, . . . , n and E Ak + EP.k = 1. By the convexity of f, f( x) � Ao f(O) + E>'k!(rek) + Ep.k!( - rek), which implies that f( x) � M == max{ l f(O) I , If(rek ) l , I f( - rek ) l } in the ball B(O; r). We now return to our primary interest, the fact that continuous convex functions on certain Banach spaces are necessarily generically ( = "in a dense

12

Section 1

Go subset" ) differentiable. The following theorem of Mazur motivated much subsequent work in this direction. As we will see, it reduces the problem to the one-dimensional theorem.

Theorem 1.20 ( Mazur, 1933). If E is a separable Banach space and f is a continuous convex function defined on a convex open subset D of E, then the set of points x where df(x) exists is a dense Go set in D. Proof. We first show that the set of x E D where df( x) does not exist i s a (relative) Fu subset of D. Let {xn} be a sequence which is dense in the unit sphere of E and for each n � 1, m � 1 let An,m denote all those x in D for which there exist x*, y* E of(x) satisfying (x * - y * , x n) ;::: 11m. Since df(x) doesn't exist if and only if of(x) contains more than one point, it is clear that df(x) fails to exist if and only if x E UAn, m. To see that each An, m is relatively closed, suppose that {zk } C An,m with Zk - z, where z E D. For each k we can choose x i. Since E is separable, the bounded subsets of E* are metrizable in the weak* topology, so by local boundedness and weak* compactness there is no loss of generality in assuming that there exist x* and y* such that x i - x* and Yk - y* (weak*). It follows that for any y E D, we must have

(x* , y - z)

=

lim (x i, y - Zk) ::; lim[f(y) - f( Zk) )

=

f(y) - f(z),

so that x* (and similarly, y*) is in of(z). Since

we see that Z E An,m. Finally, to see that D \ An, m is dense in D for each n , m, suppose that Xo E D. By Theorem 1.16, the function

ft (r) = f(xo + rxn) defined on 1 = { r E R: Xo + rXn E D} is differentiable with the exception of at most countably many points. In par­ ticular, we can approximate Xo by points of the form x, = Xo + rXn where f{ (r) exists. If x*, y* E of(x'), then their restrictions to the line Xo + RX n yield sub differentials to II at x'. Differentiability of II at x' implies that these two restrictions must coincide on the entire line, hence (x* , x n) = (y* , x n). It follows that x' E D\An,m, for m = 1 , 2, 3, . . . . We conclude that n(D\An, m ) is a dense Go subset of D , since open subsets of a Banach space have the Baire property. It has been shown [ C-K) that in certain nonseparable C(X) spaces (for instance, if X is the Cech compactification of the rationals) the set G of points where the supremum norm is Gateaux differentiable need not be a Go; indeed, G can be dense yet not even contain a dense Gfj. While these examples are

1. Convex functions on real Banach spaces.

13

complicated, the following simple example shows that some hypotheses are required for the validity of a generic differentiability theorem.

1.21. Example. For x

= ( X n ) in £"'" define a semi norm p by p( X ) = lim sup lxn l ;

then p is continuous, but nowhere Gateaux differentiable.

Proof Clearly, p( x ) ::; I l x l i""" so p is a continuous seminorm. If p(x) Xn ...... 0, so taking y = (1, 1, 1, . . . ) we have

0, then

1 c (P (X + ty ) - p(x)] = I t lit, which shows that dp(x) does not exist. If p(x) > 0, we can assume that p(x) = 1. Choose a subsequence {X n (i)} of {Xn } such that I X n (i) 1 ...... 1 . By taking a further subsequence we can assume that the X n (i) have the same sign and, since p(x) = p( -x), it suffices to consider the case X n(i) > 0 for all i. Define Yn = 0 if either n i= n( i) or n = n( i) with i odd, while Yn = 1 if n = n(i) with i even. Then

c1 (P ( X + t y) - p( x)]

=

1 if t > 0, = 0 if - 1 < t < O.

Despite the foregoing example, there are nonseparable spaces in which the conclusion to Mazur's theorem remains valid, for instance, the class of weakly compactly generated (WCG) Banach spaces (to which we will return later). The attempt to characterize those spaces in which convex continuous functions are always generically differentiable has motivated the following terminology.

Definition 1.22. A Banach space E is said to be an Asplund space ( weak As ­ plund space) provided every continuous convex function defined on a nonempty open convex subset D of E is Frechet differentiable (Gateaux differentiable) at each point of some dense G6 subset of D. Note that the term "weak" does not refer to the weak topology; it arose because Gateaux differentiability has sometimes been called "weak differentiability" (since it is weaker than Fnkhet - sometimes called "strong"- differentiability).

Some 30 years after Mazur's paper, J. Lindenstrauss [Li] obtained the next result of this nature, when he showed that if a separable Banach space was also reflexive, then the Gateaux differentiability in Mazur's theorem could be changed from Gateaux to Frechet. Five years later, E. Asplund renewed the study of such questions [Asp], making some impressive contributions. Mazur's theorem can be restated in the form "Separable Banach spaces are weak As­ plund spaces" (while Example 1.21 shows that £00 is not a weak Asplund space and Example 1. 14(a) shows that £1 is not an Asplund space). We will even­ tually prove, among other things, the fundamental result (due to Asplund) which states that if E* is separable, then E is an Asplund space ; this clearly implies Lindenstrauss' theorem.

14

Section 1

For convex continuous functions f, it is possible (and quite useful) to characterize Frechet differentiability at x solely in terms of f, that is, without mentioning the linear functional f' (x). (This is possible since there are always candidates for the derivative lurking in af(x) . )

Proposition 1 .23. Suppose that f is continuous and convex on the nonempty open convex subset D of E. Then f is Frechet differentiable at x E D if and only if for all f > 0 there exists 6 > 0 such that f(x + ty) + f(x - ty) - 2f(x) whenever lI y ll

=

1 and 0

<

t

<

<

tf

(1. 7)

6.

Proof. H f' (x) exists, then for all f > 0 there exists b > 0 such that f(x + ty) - f(x) - (f' (X), t y)

<

(f/ 2)t ll y ll

whenever 0 < t < 6 and lI y ll = 1. Writing this down with -y in place of y and adding both expressions yields (1. 7). Conversely, suppose the stated condition holds. Choose x* E af( x); it follows that for all y and all sufficiently small t > 0 such that x ± t y E D,

(x * , ty)

=

(x* , (x + ty) - x) ::; f(x + ty) - f(x) and - (x * , ty) ::; f(x - ty) - f(x).

( 1. 8) ( 1.9)

By hypothesis, for any f > 0 there exists 6 > 0 such that (1. 7) holds for any o < t < 6 and any y with lI y ll = 1, that is,

f(x + t y) - f(x) - (x * , ty) ::; tf + f(x) - f(x - t y) - (x * , ty) . Inequality ( 1. 8) shows that the left side is greater or equal to 0 while ( 1.9) shows that the right side is less than or equal to tf, which completes the proof.

1.24. Exercise. Prove that a continuous convex function f on the open convex set D is Gateaux differentiable at xED if and only if for each yE E

t ) t ) lim t_o+ f(x + y + f(x - y t

-

2f(x)

=

.

O

The previous proposition makes it easy to formulate and prove the follow­ ing basic fact.

Proposition 1.25. Suppose that f is continuous and convex on the nonempty open convex subset D of E. Then the set G (possibly empty) of po ints x E D where f is Frechet differentiable is a G6. Proof. For each n let Gn be the set of all those x E D for which there exists > 0 such that

6

1. Convex functions on real Banach spaces.

sup

f(x + 6y) + f(x - 6y) - 2f(x) 6

<

15

1/n

the supremum being taken over all y with lIyll = 1 . Recall from Lemma 1. 2 that for fixed x and y the functions t --t r1[f(x + t( ±y)) - f(x)] are non­ increasing as t --t 0+ , hence from Prop. 1.23 we can conclude that G = nGn• Thus, it suffices to prove that each Gn is open. Suppose, then, that x E Gn. Since f is locally Lipschitzian, there exist 6 1 > 0 and M > 0 such that If(u) - f( v)1 ::; Mllu - vII whenever u, v E B(Xj 6 ), Since x E Gn, there exist 1 6> 0 and r> 0 such that for all y with Ilyll = 1 we have x ± 6y E D and

f(x + 6y) + f(x - 6y) - 2f(x) < 6

r <

1/ n.

Suppose, now, that z E B(xj 62 ) c D, where 0 < 6 2 < 61 is sufficiently small that z ± 6 2 y E D for all y with lIyll = Ij a further restriction on S2 will be indicated below. Then for any lIyll = 1 we have

S-l[J(z + Sy) + fez - Sy) - 2f(z)] ::; S-l[f(x + Sy) + f(x - Sy) - 2f(x)] + S-llf(z + Sy) - f(x + Sy)l+ +S-llf(z - Sy) - f(x - Sy)1 + 2s-1 1(J(z) - f(x)1 ::; r + 4MS-111z - xii::; r + 4MS - 1 S2•

Thus, it suffices that

S2 be chosen small enough that r + 4MS- 1 S2

<

l/n.

The usefulness of the foregoing proposition is evident: To prove generic Frechet differentiability, we need only show that the set G is densej it is auto­ matically a Go. As we noted earlier, in order to prove that a continuous convex function f is generically Ga.teaux differentiable, it suffices to show that there is a dense Go subset of points x for which af(x) is a singleton. This indicates that it is worthwhile to know more about the set-valued map x --t af(x), which has a number of special properties. Some of these have already been seen: Recall that in Prop. 1 . 1 1 we showed that the sets af(x) are nonempty, weak* compact and convex, and that the map of is itself locally bounded. We'll see further fundamental properties in the next section. We conclude this section with an easy observation relating of to optimization.

Proposition 1 .26. A continuous convex function f on a nonempty open subset D has a global minimum at x E D if and only if 0 E a f(x) . Proof. This is immediate from the definition of af(x). Remarks. The material in this introductory section is classical; it provides the background about convex functions and their derivatives needed for the remainder ·of the notes. Flett's book [Flj is frequently useful when one needs to verify fundamental questions concerning differentiability and Roberts-Varberg [R-Vj is a good source for basic el­ ementary facts about convex functions. Rademacher's theorem (stated as Theorem

16

Section 1

1 .18) has been extended in various ways to infinite dimensional spaces. The first requirement is to come up with a reasonable definition of "almost everywhere" (that is, of some notion of "measure zero") for infinite dimensional spaces, where no count­ ably additive measure with decent properties can exist. Once this is done, theorems can be proved which assert that a locally Lipschitzian map from a separable Banach space into a space with the RNP (see Sec. 5) must be Gateaux differentiable almost everywhere. It has been known for at least thirty years that no such theorem holds for Frechet differentiability; there exist simple locally Lipschitzian maps of separa­ ble Hilbert space into itself which are nowhere Frechet differentiable. There are even several examples (with subsequent citations) which purport to exhibit real-valued locally Lipschitzian fUI:lctions on Hilbert space which are nowhere Frechet differen­ tiable. In preparing our UeL notes, we found all such examples to be flawed, leaving the obvious question as to whether such functions on Hilbert space must be Frechet differentiable in a dense set, say. (Rademacher's theorem shows that Gateaux differ­ entiability must take place in a dense set, but examples on the real line show that it is too much to demand that the set be a Go.) This problem was solved beautifully by D. Preiss [Prj, as stated in Theorem 4.21. Although Mazur's theorem is both generalized and proved again later in these notes, the original proof (or a reasonable facsimile thereof, since Mazur was essen­ tially looking at Minkowski functionals) was well worth covering. This seemingly magical fact (that continuous convex functions on a separable Banach space must be Gateaux differentiable on a big set) is what originally stimulated our interest in the subject.

Section 2

2. M onot one operators, subdifferentials and Aspl und

spaces

Definition 2.1 . A set-valued map T from a Banach space E into the subsets of its dual E* is said to be a monotone operator provided (x * - y * , x - y) ;:::: 0 whenever x, y E E and x* E T(x), y* E T(y). We do not require that T(x) be nonempty. The domain (or effective domain) D(T) of T is the set of all x E E such that T( x) is nonempty.

2.2. Examples. ( a) If I is a continuous convex function on a nonempty open convex subset D of E, then T(x) = 81(x) (x E D), T(x) = 0 (x E E \ D) is a monotone operator with D(T) = D . Prool.

If

x· E 81(x), y . E 81(Y), then

(x · , y - x) ::; I ( y) - I(x) and - (y., y - x)

=

(y · , x - y) ::; I (x) - I ( y) ;

now add these two inequalities. [We will see later that the subdifferential map is a rather special monotone operator. In particular, it has certain continuity properties.]

( b ) If H is a real Hilbert space and T: H --+ H is a linear map, then T is monotone if and only if it is a positive operator: ( Tx , x) 2: 0 for all x. ( c) A function r.p: R --+ R is monotone nondecreasing in the usual sense if and only if it is monotone in the above sense: That is, r.p( tl ) ::; r.p( t 2 ) whenever t l < t 2 if and only if ( d ) Monotone derivatives necessarily arise from convex functions; more precisely, if I is a real-valued Gateaux differentiable function on E such that dl is monotone, then I is convex. Proof Given x, y E E define, for any A E R,

18

Section 2

= f( ,xx + ( 1 - ,x) y) - Vex) - ( 1 - ,x)f( y) · This function is differentiable for all ,x and 4>(0) = 0 4>(I)j we want to show that 4>( ,x )

=

4> :5 0 in the interval [0, 1]. If it were not, there would exist 0 < ,x o < 1 such that 4>( ,x o) > 0 and 4> takes its maximum value in [0, 1 ] at ,x o , hence 4>'( ,x o ) = O. Suppose ,x E ( ,x o , l ] j then 4>'( ,x ) - 4>'( ,x o) = =

(df( h + ( 1 - ,xy) , x - y) - f(x) + fey) - (df( ,x ox + ( 1 - ,xOy), x - y) + f(x) - fey) = (df(,xx + ( 1 - ,x ) y) - df(,xox + ( 1 - ,xo )y), x - y) .

Now, x -y = (A !A O ) { [,xx + ( I - ,x ) y] - [ ,x ox + ( 1 - ,x o) y] } , so by monotonicity of df (and the fact that p !A O ) > 0), we have 4>' ( ,x ) � 4>'( ,x o ) = 0, that is, 4> is nondecreasing on ( ,x o , 1 ], a contradiction. (e) This example arises in fixed-point theory. Let C be a bounded closed convex nonempty subset of Hilbert space H and let U be a (generally nonlinear) nonexpan­ sive map of C into itself: I I U(x) - U ( Y ) II :5 I l x - yll for all x, y E C. Let I denote the identity map in Hj then T = I - U is monotone, with D(T) = C. Indeed, for all x , y E C, ( T(x) - T( y), x - y) = (x - y - ( U (x) - U( y)), x - y) = 2 2 = I l x - Yl1 - ( U(x) - U( y), x - y) � I l x - Y l 1 - 11U (x) - U( y) II ' ll x - yll � O. Note that 0 is in the range of T if and only if U has a fixed point in C; this hints at the importance for applications of studying the ranges of monotone operators (something we will not do, but which is done, for instance, in [Au-Ek], [pa-Sb] and [ZeD · (f) Again, in Hilbert space, let C be a nonempty closed convex set and let P be the metric projection of H onto C defined in Example 1 .14 (d). Since (as shown there) P is the Frechet derivative of a convex continuous function on H, it is monotone. More directly, this is an immediate consequence of inequality ( 1 .6).

Definition 2.3. Let X, Y be Hausdorff topological spaces and suppose that T: X -t 2 Y is a set-valued mapping from X into the subsets of Y. If A is a subset of E , we define T(A) = U{T(x): x E A } . We say that T is upper semicontinuous at the point x E X if, for each open set V in Y containing T(x), there is an open neighborhood U of x such that T(U) C V . Upper semicontinuity on a set is defined in the obvious way. 2.4. Exercises.

(a) Prove that a set-valued map T: E -+ 2 E * is norm-to-weak* [norm-to-norm] upper semicontinuous at x E E if and only iffor every weak* open set [norm open set] V containing T(x) and every {xn} C D(T) with I Ixn - x I I -+ 0, we have T(xn ) C V for all sufficiently large n. (Equivalently, T[B(x ; o)] C V for all sufficiently small 0 > 0.) (b) Assuming that T(x) is a single point, prove that T is norm-to-norm upper semicontinuous at x if and only if lim diam T[B(x; 0)]

6 _0+

=

O.

2. Monotone operators, sub differentials and Asplund spaces

19

Proposition 2.5. If f i8 a continuOU8 convex function on the open convex 8ub8et D of E, then the 8ubdifferential map x -+ 0f( x) i8 norm-to-weak* upper 8emicontinuou8 on D . Proof. We must show that if x E D and W is any weak* open subset of E* containing of (x), then for any sequence {xn} C D with Xn -+ x, we have of(xn) c W for all sufficiently large n . If not, then there exists a subsequence ( call it {xn}) and x � E of(xn)\ W . By local boundedness of the subdifferential map, we can assume that there is a ( weak* compact ) closed ball which contains the sets of(xn) for all sufficiently large n. Let x· be a weak* cluster point of the sequence {x� }j it is easily verified that x* is in of(x) \ W , a contradiction. Lemma 2.6. Supp08e that f i8 continuou8 and convex on a nonempty open convex 8ub8et D of E and that it i8 Frechet differentiable at a point x in D. Then the 8ubdifferential map of i8 n.orm-to-norm upper 8emicontinuou8 at x. Proof. We want t o show that given any norm open neighborhood V of the functional x* = f' (x) there exists a neighborhood of x which is mapped into V by of. If this were to fail, we could choose 10 > 0, a sequence of points {x n } C D and x� E of(x n ) for each n such that Il xn - x II -+ 0 while II x� - x* 1I > 210. Consequently, there would exist Zn E E, IIz n ll = 1, such that (x� - x·, zn) > 2 10. By Frechet differentiability of f at x there would exist D > 0 such that x + y E D arid f(x + y) - f(x) - ( x*, y) :::; € lIyll whenever lI y ll :::; D. Since x� E of(xn), we have (x�, (x + y) - xn ) :::; f(x + y) - f(xn) so (x�, y) :::; f(x + y) - f(x) + (x�, Xn - x) + f(x) - f(xn) whenever Il y ll :::; D. Let Yn = DZn , so IIYn ll = D and :::; [f(x + Y n ) - f(x) - (x · , Yn )] + (x�, Xn - x) + f(x) - f(xn ) :::; :::; €D + (x�, xn - x ) + f(x) - f(xn). Since of is locally bounded and I (x�, Xn - x ) 1 :::; II x� II ' lI xn - x II , this term converges to 0, while f( x) - f( xn) -+ 0 since f is continuous. But this would yield 2€D :::; €D, an impossibility which completes the proof. Definition 2.7. A 8election

20

Section 2

Proof. Suppose that a selection r.p for af exists. Since r.p(x) E af(x), we must have (r.p(x), y - x ) � f(y) - f(x) for all y E D. For such y we also have r.p (y) E af( y ), so (r.p(y) , x - y) � f(x) - f(y). These inequalities combine to show that for all y E D, o�

f(y) - f(x) - (r.p(x), y - x) � (r.p(y) - r.p(x), y - x).

( 2. 1)

Suppose r.p is norm-to-weak* continuous at x . By taking v E E and t > 0 sufficiently small, one can make the substitution y = x + t v in ( 2. 1 ), divide by t, let t -+ 0+ and use the weak* continuity of r.p to conclude that (r.p(x), v) = d+ f(x)(v) for each v E E. That is, d+ f(x) is linear, hence f is Gateaux differentiable at x. If r.p is norm-to-norm continuous at x, then the fact the last term in ( 2. 1) is dominated by 1 I r.p(y) - r.p(x) II · lI y - x ii implies that f is Frechet differentiable at x. To prove the two converses, suppose that f is Gateaux differentiable at x, so that af( x) is a singleton. Since af is norm-to-weak* upper semicontinuous, any selection for af will necessarily be norm-to-weak* continuous. A similar argument applies if f is Frechet differentiable at x , since af is norm-to-norm upper semicontinuous, by Lemma 2.6. An interesting consequence of this result is that Frechet differentiable con­ vex functions are necessarily C1 ,

Corollary. If f is convex and Frechet differentiable on the open convex set D, then x -+ f'(x) is norm-to-norm continuous in D. We are approaching Asplund's theorem that a Banach space E with sep­ arable dual is an Asplund space. Since the sub differential a f of a continuous convex function is a monotone map and since Frechet differentiability of f can be characterized in terms of the norm-to-norm upper semicontinuity of af, we will have proved a generalization of Asplund's theorem once we have proved that monotone maps into a separable dual space are generically single-valued and norm-to-norm upper semicontinuous. We follow an approach devised by D. Preiss and L. Zajicek [Pr-Z], who have shown that the set of points of nondifferentiability is even smaller than merely being a countable union of nowhere dense closed sets. We need a couple of definitions, the first of which has been used in other contexts.

Definition 2.9. (a) If x* E E* , x* i= 0, and 0 < a < 1, define K(x * ; a ) = {x E E: a ll x ll ' ll x * 1I � ( x * , x) } . It is easily verified that K( x * ; a ) i s a closed convex cone (we call i t an a-cone) and that its nonempty interior is {x E E: a ll x ll ' lI x* 11 < (x*, x ) } .

2. Monotone operators, subdifferentials and Asplund spaces

21

Fig. 2.1 As the sketch indicates, when a becomes closer to 0, then the cone becomes wider. (b) A subset M e E is said to be a -cone meager (where 0 < a < 1) if for every x E M and f > 0 there exist z E B( x; f ) and 0 ::f x* E E* such that M n [z + int K(x*; a)] (Note that z + int K(x * ; a)

=

=

0.

{y E E: a ll x* II · ll y - z ll

<

( x* , y - z) } .)

To say that M is a-cone meager means that any point of M can be ap­ proached arbitrarily closely by vertices of cones whose interiors lie in the com­ plement of M; Figure 2 .2 below illustrates the situation (for a set M which is clearly not a-cone meager). (c) The set M is said to be angle-small if for every 0 < a < 1 it can be expressed as a countable union of a-cone meager sets.

Fig. 2.2 2.10. Exercise. Prove that if M is a-cone meager for some a > 0, then hence show that any angle-small set is of first category.

AI

ha.s empty interior,

Being a-cone meager has geometric as well as topological implications. Notice, for instance, that if M is the union in R2 of the unit circle and the origin, then it is nowhere dense, but it is not a-cone meager for any a > O. (It is, however, the union of two sets, each of which is a-cone meager for every a > 0.) Since an a-cone meager subset of R can contain at most two points,

22

Section 2

it is easily seen that a subset of R is angle-small if and only if it is countable. There exist uncountable sets of first category (such as the Cantor set).

Theorem 2.11. (Preiss-Zajicek [Pr-Z] ) . Suppose that the Banach space E has separable dual and that T: E _ 2 E • is monotone. Then there exists an angle-small set A C D(T) such that T is single-valued and norm-to-norm upper semicontinuous at each point of D(T) \ A. Proof. It suffices to show that the set A = {x E D(T): lim diam T[B(x; I5)] > O } 6-+0+

is angle small. We can obviously write A = UAn where

A n = {x E D(T): lim diam T[B(x; I5)] > l jn } . 6-+0+

Let {xn be a dense sequence in E* and suppose 0 < a < 1. Define

An,k = {x E An: dist(x i , T( x )) < a j4n}; we will show that each An,k is a-cone meager. Suppose that x E An,k and that f > O . Since x E An, there exist 0 < 15 < f and Zl , Z2 E B( x; 15) and z: E T(zj) such that li z; - z; 1I > ljn . Thus, for any x* E T(x), one of li z: - x* 1I > lj2n. Since dist(xk , T(x)) < a j4n , we can choose x * E T(x) such that Il xk - x * 11 < a j4n and therefore there exist points z E B(x; f) and z* E T( z) such that

II z * - x i II ;::: II z * - x* II - li xi - x* 1I > l j2n - a j4n > l j4n. We want to show that An,k n (z + intK(z* - x i ; a) = 0, that is,

An,k n {y E E : (z * --: xi, y - z)> a ll z * - x i ll · ll y - z ll } = 0. Now, if y E D(T) and ( z * - x i, y - z)> a ll z* - x i ll · ll y - z ll and if y* E T(y), then (y * - x i , y - z) = (y * - z * , y - z) + (z * - xi, y - z) ;:::

;::: (z * - xi, y - z) > a ll z * - x i ll · ll y - z ll ;::: (aj4n) ll y - z ll · It follows that lI y* - x i II ;::: a /4n, so y is not in An,k . This theorem, combined with Exercise 2.10, leads to the following theorem.

Theorem 2.12 ( Asplund [Asp] ) . If the dual E* of the Banach space E is separable, then E is an Asplund space. Proof. If f is continuous and convex on the open convex set D e E, then 8f is monotone, so by Theorem 2.11, it is single-valued and norm-to-norm upper semicontinuous at the points of some dense G 6 subset G of D. Thus,

2. Monotone operators, subdifferentials and Asplund spaces

23

any selection for of is continuous at the points of G, so by Proposition 2.8, f is Frechet differentiable at the points of G.

2.13. Examples. Here are some well-known examples of Banach spaces with separable duals: The sequence space Co ( but not its dual (1 ) . The spaces tP and £1'[0, 1], provided 1 < p < 00. More generally, any separable reflexive Banach space.

We will see later that separable Asplund spaces are characterized by the fact that their duals are separable. A more general fact, one of the most beautiful and fundamental results in this area, is that E is an Asplund space if and only if every separable subspace of E has a separable dual. Half of this latter result is a corollary to the next theorem.

Theorem 2.14. A Banach space E is an Asplund space if every separable closed subspace F of E is an Asplund space. Proof. Suppose that f is continuous and convex on the nonempty open convex subset D of E and suppose that the set G of all points x E D where f is Frechet differentiable is not dense in D; we will produce a separable subspace F of E such that F n D :f 0 and the points of Frechet differentiability of flF are not dense in F n D. For each n define (as in the proof of Prop. 1 .25) GnU) to be the open set of all x E D for which there exists D > 0 satisfying f(x + Dy) + f(x - Dy) - 2f(x ) < 1 /n . D lI y ll=l sup

Since D is a Baire space and since G = nGnU) , we conclude that for some m > 0, the set GmU ) is not dense in D, that is, there exists a nonempty open subset U C D \ GmU ) . We will next define by induction an increasing sequence {Fk } of separable subspaces of E. (The desired subspace F will be the closure of their union.) First, choose Xl E U. It follows (take D = Iii, j = 1 , 2, 3, . . . and use the monotonicity of the difference quotients for J) that there exists a sequence { YI ,j } with II Yl ,j l l = 1 , such that for all D > 0, SUPj

f( Xl + DYI ,j) + f(x! - DYl ,j) - 2f ( x t ) > _ 1/2m. D

Let Fl denote the closed linear span of X l and { Yl ,j } . Clearly, Fl is separable and Fl n U is nonempty (since it contains Xl ) . Given an increasing sequence of separable subspaces Fl , . . . , Fk , define FH l as follows: Let {Xk ,p } , p = 1 , 2, 3, . . . be a countable dense subset of Fk n U and for each p choose a sequence { YP ,j } with II Yp,j ll = 1 such that for all D > ° ( Xk ,p + DYp,j ) + f( Xk ,p - DYp ,j) - 2f(Xk ,p ) > 1 /2m sup). f D _

.

Let Fk+1 be the closed linear span of Fk and {Xk,p } U { YP ,j } , p, j � 1 . Let F = UFk ; then {Xk,p : k,p = 1 , 2, 3, . . . } is a dense subset of F n U . It is clear

24

Section 2

that for each k , p, the point Xk,p is not in G2m UI F) and - since U c D - the sequence {X k ,p } is in D n U, that is, {Xk,p } is a subset of (D n U) \ G2 m U I F). Since G2 mU I F) is open in F, we must have F n U c (D n U) \ G2 mU I F), which implies that f l F fails to be Frechet differentiable at each point of the relatively open nonempty subset F n U of F n D.

Corollary 2.15. A Banach space E is an Asplund space if every separable subspace of E has separable dual. It is immediate from this that every reflexive Banach space is an Asplund space.

2.16 Exercise. Show that for any set r, every separable subspace of co(r) has separable dual, so co(r) is an Asplund space. ( Recall that co(r) is the supremum-normed space of all functions x = (xl' ) on r such that for all € > 0, there exists a finite set r< c r such that I XI' I < € for every I in r\r<.) Hint: Show that if F is a separable subspace of co(r), then there exists a countable subset n C r such that F can be identified with a subspace of co(rt ).

Deflnition 2.17. By a slice of a nonempty set A we mean a (necessarily nonempty) subset of A of the form S(x * , A , a) = { x E A: (x * , x ) > o'A( x * ) - a } where x * E E*,

a

> 0 and O'A(X* ) = sup{ (x* , x) : x E A}.

Fig. 2.3 If A c E*, then we can also define a weak* slice of A to be a slice where the defining linear functional comes from E (rather than from E** ). For the moment, in fact, we will be working with weak* slices. We say that a nonempty subset A admits slices of arbitrarily small diameter if, for every e > 0 there exists a slice of A of diameter less than e. (This property is equivalent to a concept called "dentability" , so it is sometimes given the same name.)

2. Monotone operators, subdifferentials and Asplund spaces

25

The following lemma might seem a bit peculiar at first, but - as we will see later - its converse is also valid, so it contains a characterization ( a useful one) of Asplund spaces.

Lemma 2.18. If E is an Asplund space, then every nonempty bounded subset A of E* admits weak* slices of arbitrarily small diameter. Proof. Given A, nonempty and bounded in E* , define the sublinear functional p on E by p(x) = O"A(X) == sup{ (x* , x} : x * E A } . Since A is bounded, p is necessarily continuous: p(x) :::; M ll x ll if II x* 11 :::; M for every x * E A. Suppose that there exists f > 0 such that every weak* slice of A has diameter greater than f; we will show that p is nowhere Frechet differentiable. Indeed, given any x E E, for each n � 1 the weak* slice S(x, A, f/ 3n) has diameter greater than f. It follows that there exist x�, y� E S(x, A, f/ 3n) with II x � - Y� II > f, that is x �, y� E A,

(x � , x) > p( x) - f/ 3n,

(y� , x) > p(x) - f / 3n

and (x� - y�, xn) > f for some Xn E E with II xn ll

=

1. Thus

p[x + (l/n)xnl + p [x - ( l/n)xnl - 2p(x) � � ( x � , x + (lin )xn) + (y� , X - ( lin )xn ) =

�

(x � + y� , x) - 2f/ 3n

(l/n)(x � - y� , xn) - 2f/3n > fin - 2f / 3n

=

=

f /3n.

If we divide through by lin it becomes evident ( using Proposition 1.23) that p is not Frechet differentiable at x . We can now characterize separable Asplund spaces.

Theorem 2.19. A separable Banach space E is an Asplund space if and only if E* is separable. Proof. It was shown in Theorem 2.12 that if E* is separable, then E is an Asplund space. Suppose then, that E is a separable Asplund space. If E* is not separable, then its unit ball B* is not separable, hence there is an uncountable set A C B* and n � 1 such that Il x * - Y * II > lin whenever x * , y * are distinct points of A. ( Look at maximal lin-nets in B*.) Since E is separable, B* ( in its weak* topology ) is compact and metrizable, hence satisfies the second axiom of count ability. Thus, A has at most countably many points which are not weak* condensation points; we assume that these points have been deleted from A. Let S be any weak* slice of A. Since such a slice is weak* relatively open and since no point of A is weak* isolated, it must contain two distinct points of A, so its diameter is greater than lin, contradicting Lemma 2.18. We next look at some additional properties which distinguish subdifferen­ tials within the class of monotone operators, proving some basic results along the way.

26

Section 2

Definition 2.20. A set-valued map T: E -+ 2E* is said to be n-c yclically monotone provided L (Xi , Xk - Xk - l ) � 0 l $ k $n whenever n � 2 and xo , Xl , X2 , . . . xn E E, Xn = Xo , and x i E T(Xk), k = 1 , 2, 3, . . . , n . We say that T is c yclically monotone if it is n-cyclically monotone for every n. Clearly, a 2-cyclically monotone operator is monotone.

2.21 Examples. (a) The linear map in R2 defined by T(X l , X 2 ) = (X 2 , -X l ) is positive, hence monotone, but it is not 3-cyclically monotone: Look at the points ( 1 , 1), (0, 1) and ( 1 , 0). (b) Let / be a continuous convex function on an open convex set; then 8/ is cyclically monotone. (As we will see, this is almost the only example.)

Definition 2.22. A subset G of E x E* is said to be monotone provided (x* - y*, X - y) � 0 whenever (x, x*), (y, yO) E G. If T: E -+ 2E* is a monotone operator then its graph G(T)

=

{(x, x*) E E

X

E*: x* E T(x)}

is a monotone set. A monotone set is said to be maximal monotone if it is maximal in the family of monotone subsets of E x E* , ordered by inclusion. We say that a monotone operator T is maximal monotone provided its graph is a maximal monotone set. There is an obvious one-to-one correspondence between monotone sets and monotone operators. An easy application of Zorn's lemma shows that every monotone operator T can be extended to a maximal monotone operator T , in the sense that G(T) C G(T).

2.23 Exercises. (a) Prove that a monotone operator T: E -+ 2E* is maximal monotone if and only if the following condition holds: Whenever y E E, y. E E · and

(y. - x ·, y - x) ?:

°

for all x E D(T),

x· E T(x)

then it necessarily follows that y. E T( y ). (b) Prove that if T is maximal monotone, then T(x) is convex set, for every x E E. (c) Show that the operator in Example 2.21(a) is maximal monotone.

The next theorem is due to G. Minty. A much more general result has been proved by Rockafellar and will be presented in the next section ( Theorem 3.24) . We start with a simple and useful result.

2. Monotone operators, sub differentials and Asplund spaces

27

Proposition 2.24. If f is convex on D and continuous at x E D, then for all y E E, d+ f(x)(y) = sup { ( x* , y ) : x* E 8/(x)} and this supremum is attained at some point x* E 8/(x) . Proof. A s shown in Prop. 1.8, if x * E 8/(x), then (x * , y ) ::; d+ /(x)(y) for all y. On the other hand, by Corollary 1 . 7, d+ /(x) is a continuous sublinear functional, so for any y f 0 we can use the Hahn-Banach theorem to find x* E E* such that (x* , z) ::; d+f(x)(z) for all z E E ( so x* E 8f(x) by Proposition 1 .8) and (x* , y) = J+ f(x)(y), which completes the proof. Theorem 2.25. If / is continuous and convex on all of E, then its subdiffer­ ential map 8/ is maximal monotone. Proof. By Exercise 2.23( a), to show that 8f is maximal it suffices to show that whenever y E E and y* E E* are such that y* (j. 8/(y), then there exist x E E and x * E 8/(x) such that (y* - x* , y - x) < O. To simplify the proof we can replace f by the convex continuous function 9 defined by g(x) = f(x + y) - ( y* , x ) . It is easily verified that one has x* E 8g(x) if and only if x* + y* E 8/(x + y). Thus, if y* (j. 8f(y), then 0 (j. 8g(0) and if there exist x* and x with x * E 8g(x) such that (x* , x) < 0, then for z = x + y and z* = x* + y * we have z* E 8/(z) and ( y* - z*, 'y - z) = (x*, x) < O. Assume, then, that y = 0 and y* = OJ we want to produce x E E and x* E 8/(x) such that (x*, x) < O. By Prop. 1 .25, we know that 0 is not a global minimum for f, so there exists a point Xl E E such that f(O) > f(X 1 )' Consider the convex function h(t) = f(tx 1 ), O ::; t ::; 1 . Its right hand derivative at a point to E (0, 1) is clearly equal to d+ /(tOX 1 )(XI ). Suppose this quantity were nonnegative for each such toj by one form of the mean value theorem ( see, for instance, [ Fl, p. 22]) , this would imply that h(O) ::; h(1), a contradiction. Thus, it is necessarily negative for some 0 < to < 1 , so by homogeneity ( and letting x = tox! ), we have d+ f(x)(x) < O. By Prop. 2.24 above, there must exist x* E 8f(x) such that (x* , x) = d+ f(x)(x) < 0, which completes the proof. 2.26 Example. In a Banach space E define f(x) = ( 1/2)lIxIl 2 and define J = 8f; the maximal monotone map J is called the duality mapping for E. Explicitly,

J(x)

= {x· E E· : (x· , x)

=

IIx · II · lIxll and Ilx· 1I = IIxll} .

Proof. I t is readily computed that d+ f(x)(y) = IIxll . d+ llxll(y). If x = 0 , then d+ f(O)(y) = 0 for all y, hence is linear and therefore 8f(0) = {O}. Suppose, then, that x f. O. We know ( by Prop. 1.8) that x · E 8f(x) if and only if x · ::; d+ f(x), that is, if and only if IIxll- I x · ::; d + llxll, which is equivalent to y. == IIxll- I x · E 811xll, that is, if and only if (y. , y - x) ::; IIYIl - lIxll for all y E E. If, in this last inequality, we take y = x + z, II zll ::; 1 and apply the triangle inequality, we conclude that lIy· 1 I ::; 1. If we take y = 0, we conclude that IIxll ::; (y· , x) ::; lIy·II · llxll, so lIy· 1 I = 1 and (y., x) = IIxll, which is equivalent to what we want to prove. The converse

28

Section 2

I I xl l , then for all y in E, we necessarily have is easy: If " y . " = 1 and (y · , x) ( y. , y - x ) S lIyll - lI x ll, so y. E 811 x l l · If the norm of E is Gateaux differentiable, then obviously J( x) contains only one element; by a slight abuse of notation, we denote it also by J( x ) . =

We now know that subdifferential maps of continuous convex functions are maximal cyclically monotone; we will discuss later the fact that maxi­ mal cyclically monotone mappings are the sub differentials of certain convex functions. These ( and other ) considerations require a fundamental property of monotone operators which was easy to prove for sub differentials of continuous convex functions ( in Prop. 1 . 11); namely, that they are locally bounded at points in the interior of their domain. Definition 2.27.

a) A monotone operator T: E 2 E• is locally bounded at x E D(T) provided there exist M > 0 and [; > 0 such that I IY * II S M whenever y E B(x; 8) n D(T) and y* E T(y). b ) A subset ( not necessarily convex ) A c E which contains the origin is said to be absorbing if E U { AA A > O } . Equivalently, A is absorbing if for each x E E there exists t > 0 such that tx E A. A point x E A is called an absorbing point of A if the translate A - x is absorbing. It is obvious that any interior point of a set is an absorbing point. If Al is the union of the unit sphere and {O}, then Al is absorbing, even though it has empty interior. -t

=

:

Theorem 2.28. Suppose that T: E -t 2 E• is monotone and that x E D(T) . If x E int D(T), or, more generally, if x is an absorbing point of D(T), then T is locally bounded at x . Proof. By choosing any x · E T( x) and replacing T by the monotone operator y -t T(y + x) - x * , we lose no generality in assuming that x = 0 and that o E T(O) . Thus, we want to show that, under these assumptions, T is locally bounded at O. Define, for x E E, f(x) = sup { ( y * , x - y): y E D(T), lI y ll S 1 and y * E T(y)}

and let C = {x E E: f( x) S I}. As the supremum of affine continuous functions, f is convex and lower semi continuous, and hence C is closed and convex. It also contains the origin: First, since 0 E T(O), we must have f 2:: O. Second, whenever y E D(T) and y* E T(y), monotonicity implies that 0 S (y* - 0, y 0) , so f(O) S O. We claim that the closed convex symmet­ ric set A = C n (-C) is absorbing, hence, by a standard consequence of the Baire category theorem, is a neighborhood of the origin. It suffices to prove that C is absorbing, so suppose x E E. By hypothesis, D(T) is absorbing so there exists t > 0 such that T(tx) =I- 0. Choose any element u* E T(tx). If y E D(T) and y* E T(y), then by monotonicity -

2. Monotone operators, subdifferentials and Asplund spaces

29

(y * , tx - y) ::; (u * , tx - y) .

Consequently, J(t x) ::; sup{ (u * , t x - y ) : y E D(T), Il y ll ::; 1 } < (u * , t x ) + lI u* II < 00.

Choose 0 < ..\ < 1 such that ..\J (t x) < 1 . By convexity J(..\tx) ::; ..\j( t x) + (1 - ..\)J(O) = ..\j ( t x) < 1,

so ..\tx E C . Thus, A is a neighborhood of 0 and hence there exists {j > 0 such that J(x) ::; 1 whenever II x ll ::; 2 {j . Equivalently, if I I x ll ::; 2{j, then (y*, x ) ::; (y* , y) + 1 whenever y E D(T), IIYI I ::; 1 and y* E T(y ). Thus, if Y E B(Oj {j) n D(T) and y* E T(y), then 2{j IlY* II = sup{ (y* , x ) : II x II ::; 2{j} ::; Ily* II . lI y ll + 1 ::; {j IlY * II + 1,

so lI y* 1 I ::; 1/{j · Note that the foregoing result does not require that D(T) be convex. There are trivial examples which show that 0 can be an absorbing point of D(T) but not an interior point (for instance, let T be the restriction of the subdifferential of the norm to the set Al defined above). Even if D(T) is convex and T is maximal monotone, D(T) can have empty interior, as shown by the following example. (In this example, T is an unbounded linear operator, hence it is not locally bounded at any point and therefore D(T) has no absorbing points.) Example. In the Hilbert space (2 let D = {x = (x n ) E e: (2 n xn ) E (2} and define Tx = (2 n x n ) , x E D. Then D(T) D is a proper dense linear subspace of (2 and T is a positive operator, hence is monotone. We use Exercise 2.23 (a) to show that it is maximal monotone. Suppose, then, that y and y ' are in e and that for all x E D( T) =

o ::; (y '

-

Tx, Y - x )

='

(y' , y )

-

(T x, y)

we want to show that Y E D(T) and that T y and let

=

-

(y', x) + (T x , x ) ;

y ' . Fix

n

� 1,

m

(2.2)

� 1 and a E

R

Since x E D(T), we can expand the right side of (2.2) and cancel a number of terms to obtain

O « Y . , y ) - a 2 n Yn _

...,n+m Y Y k k - aYn •

- "" 1

•

+

Yn Yn + 2 n a 2 . •

Letting m -+ 00 yields a2 n (Yn - a) ::; y� (Yn - a) for all n � 1 . Since this is true for arbitrary a E R, it follows that y� = 2 n Yn for each n. Since Y' E e and Y' = (2"Yn ), we conclude that Y E D(T) and Y' = T(y).

It is conceivable that for a maximal monotone T, any absorbing point of D(T) is actually an interior point. A word of caution i� in order at this point. Our knowledge of the structure of the domain of monotone operators is incomplete, although some things are

Section 2

30

known about D(T) when T is maximal monotone. Rockafellar [Ro2] gives two special conditions under which int D(T) =I 0 for a maximal monotone T; in particular, this is the case if the convex hull of D(T) is assumed to have nonempty interior. Under this hypothesis, incidentally, the interior of D(T) is convex, and for any point x E D(T)\int D(T), the set T(x) is unbounded. (Indeed, by the separation theorem there exists Y· ::/= 0 such that (y. , x) 2: (y. , u) for every point u E D(T). Thus, for every A > 0 and any x· E T(x), u· E T(u ) , we have ( x · + A y· ) - u · , x - u) = ( x · - u * , x - u) + A (y · , X - u ) 2: 0 , by monotonicity. By maximality, x· + A y · E T(x) for every A > 0.) 2.29. Exercises. ( a) Prove that if T is maximal monotone, then T(x) is weak* compact and

convex, for all x E int D(T).

( b) Prove that if T is maximal monotone, then it is norm-to-weak* upper semi­ continuous on int D(T).

The following theorem is due to P. Kenderov [Ke2]. Note that by Lemma 2.18, the "small slices " hypothesis on E* is satisfied if E is an Asplund space. Theorem 2.30. Suppose that every nonempty bounded subset of the Banach space E* admits weak* slices of arbitrarily small diameter; then for every maximal monotone operator T: E -+ 2E• there is a dense Go subset G of W = int D(T) (which we assume nonempty) such that T is single-valued and norm-to-norm upper semicontinuous at each point of G. Proof. For each ;:::: 1 let Gn be the set of all x E W for which there is a neigh­ borhood V of x in W such that diam T(V) < lin . Clearly, T is single-valued and norm-to-norm upper semicontinuous at each point of the intersection G nGn• Since W is a Baire space, we need only show that each of the sets Gn is open and dense in W. They are open by their definition, so it remains to show that each Gn is dense. Let x E W and let U be any neighborhood of x in W. From Theorem 2.28 we know that T is locally bounded in W, so without loss of generality we can assume that T(U) is a bounded subset of E*. By hypothesis, there exist z E E and a > 0 such that the weak* slice n

=

S == S(z, T(U), a)

=

{x* E T(U): (x* , z ) > O"T(U) ( Z ) - a }

has diameter less than lin. If x * E S, then x * E T(xI ) for some point Xl E U and Xo == Xl + r z is in U for sufficiently small r > o. We claim that T(x o ) C S. Indeed, if y* E T( x o ), then we have 0 ::; (y*

" - x , xo - X l

)

=

r(y* - x" , z ) ,

which implies that y* E S. Since {x * E E·: (x· , z) > O"T(U)(Z) - a } is weak* open and since (by Exercise 2.29(b)) T is norm-to-we�* upper semicontinu­ ous, there exists /j > 0 such that B (xo ; /j ) C U and T( y ) C S for any point

2. Monotone operators, subdifferentials and Asplund spaces

31

Y E B( Xo j 6)j it follows that T[B(xo j 6)] has diameter less than lin. This says that Xo E Gn n U , which completes the proof. We can now prove the converse to Lemma 2.18, obtaining a characteri­ zation of Asplund spaces. We first prove a local extension result for convex functions, proved originally by Asplund ( although the proof given below is simpler).

Lemma 2.31. Supp08e that f i8 continuou8 and convex on the open convex 8et D e E and that Xo E D. Then tJ:ere exi8t8 a neig!!:.borhood U of Xo in D and a convex Lipschitzian function f on E 8uch that f = f in U. Proof. Given n 2:: 1, x E E, define fn to be the "inf-convolution" of f and nil 11 : fn(x) = inf{f(y) + n ll x - y ll : y E D } . ·

·

We need to know that fn > - 00 ( at least for all sufficiently large n). To this end, choose x* E 8f(xo). H n 2:: II x* lI , then for any x E E and y E D we have

f(y) - f( xo) 2:: (x* , y - xo ) 2:: - n ll y - xo ll 2:: -n ll y - x II - n ll x - xo ll , so fn(x) 2:: f(xo) - n ll x - xo ll . Thus, we will assume below that n is large enough that fn > - 00 . It follows easily from the definition that for Xl , x 2 E E, o � t � 1 and any f > 0, one has tfn (x I )+(1 - t)fn(x2 ) 2:: fn(txl + ( 1 - t)x2 ) -f, so fn is convex. Also, fn(x) � f(x) for all x E D ( take y = x in the definition ). Moreover, given u , v E E and f > 0 we can choose y E D such that

fn( u ) > f(y) + n llu - Y II - f . Since

fn(v) � f(y) + n ll v - y ll

we have

fn(v) - fn( u ) < f(y) + n ll v - y ll - [f(y) + n llu - YII - f] � n ll v - u ll + f , which shows that fn(u) - fn(v) � n llu - vII for all u , v . Interchanging u , v proves that fn has Lipschitz constant n. Since 8f is locally bounded, there exists a neighborhood U of Xo and an integer n > 0 such that 8f(U) is contained in the ball nB* . Suppose that x E U and choose any functional x* E 8f(x)j then II x * 1I � n and for all y E D

f(x) � f(y) + (x * , x - y) � f(y) + n ll x - y ll , which implies that f(x) � fn(x)j that is, f = fn in U.

Theorem 2.32. A Banach 8pace E i8 an A8plund 8pace if and only if ev­ ery nonempty bounded 8ub8et of E* admits weak* slices of arbitrarily small diameter.

32

Section 2

Proof. The necessity portion has already been proved in Lemma 2.18. Suppose, then, that f is continuous and convex on the nonempty open convex set D C E; we need only show that l' exists on a dense subset of D. Given Xo E D and a neighborhood U of Xo , we can assume that U is sufficiently small that the conclusion to Lemma 2.31 holds in U; that is, there exists a convex Lipschitz continuous function 1 on E such that 1 f on U. By Theorems 2.25 and 2.30, there is a dense G6 subset G of points in E at each of which a1 is single­ valued and norm-to-norm upper semicontinuous, hence any selection for a1 is norm-to-norm continuous. By Prop. 2.8, 1 is Frechet differentiable at each such point. But this implies that f has points of differentiability in U. =

As we will see shortly, this characterization has a number of applications. For instance, it is not obvious that a closed subspace M of an Asplund space E is itself an Asplund space; even if we could extend a continuous convex function on an open convex subset of M to one on an open convex subset of E, the (dense) set of points of differentiability of the extension could fail to intersect the (nowhere dense) set M. The Asplund property is inherited by closed subspaces; however, and Theorem 2.32 can be used to prove it. Proposition 2.33. A closed subspace M of an Asplund space E is itself an Asplund space. Proof. We need only show that any bounded nonempty subset A of M* E* /Ml.. admits weak* slices of arbitrarily small diameter. Without loss of generality, we can assume that A is weak* compact and convex. (Any weak* slice of the weak* closed convex hull of A is also a slice of A.) The quotient map Q: E* -t M* is of norm one, onto and weak*-to-weak* continuous. Suppose that f > O. Let B* be the unit ball of E* ; since Q is an open map, the set Q( B*) contains a neighborhood of the origin in M* . Since A is bounded, this implies that there exists -\ > 0 such that Q(-\B* ) = -\Q(B*) � A. Let C = -\B* n Q -l (A); clearly, C is weak* compact, convex and Q(C) = A. By Zorn's lemma there exists a minimal (under inclusion) set C with these properties; let C1 be such a minimal set. By hypothesis, there exists a weak* slice S == S( X, C1 , a) of C1 of diameter less that f; since S is relatively weak* open, the set A l = Q(C1 \ S) is a weak* compact convex set which, by the minimality of C1 , is properly contained in A . If x i , x i E A \ AI , there exist yt , yi E S such that Q( yi) = x i and

Thus, diam(A \ A I ) :::; f. By the separation theorem, there exists a weak* slice of A which misses AI , and hence has diameter at most f. By Theorem 2.32, again, we conclude that M is an Asplund space. Theorem 2.34. A Banach space is an Asplund space if and only if every separable closed subspace of E has a separable dual.

2. Monotone operators, subdifferentials and Asplund spaces

33

Proof. The sufficiency half of this theorem is precisely Corollary 2.15. The necessity follows from the previous proposition and Theorem 2.19. Problem. It is a difficult open question whether a closed subspace of a weak

Asplund space is itself a weak Asplund space.

At this point we should state the obvious: The property of being an As­ plund (or weak Asplund) space is pres erved under linear isomorphisms; that

is, replacing the norm in a Banach space by an equivalent norm will have no effect on the differentiability of functions on the space nor on a topolog­ ical property like being a dense Go . This fact has played an important role in the subject, since many results, starting with Asplund's own theorems, were obtained by showing first that particular classes of spaces admit norms which themselves have good differentiability properties, then using that fact to deduce differentiability for arbitrary continuous convex functions. The close connection between differentiable norms and Asplund spaces is illustrated by the following corollary to Theorem 2.32. Corollary 2.35. If a Banach space E is not an Asplund space, then there exists an equivalent norm on E which is nowhere Frechet differentiable.

Proof. If E is not an Asplund space, then by Theorem 2.32 there exists a bounded nonempty subset A of E* and f > 0 such that every weak* slice of A has diameter greater than f. We will use A to construct a bounded symmetric convex set U* with nonempty interior in E* such that every weak* closed slice of U* has diameter greater than f. From the proof of Lemma 2.18, it will follow that the continuous support function p for U* is nowhere Frechet differentiable. Since U* has nonempty interior, p defines an equivalent norm on E. To construct U* , note that both the convex hull C of A U -A and the sum U* = C + B*, where B* is the unit ball of E* , are bounded, symmetric and have the property that all their weak* slices have diameter greater than f. ( This last assertion for U* makes use of the fact that O"C+B* = O" c + O" B " . ) The set U* has nonempty interior, since B* does.

For one of his Gateaux differentiability theorems, Asplund used the hy­ pothesis that E could be given an equivalent norm which was itself Gateaux differentiable in a certain strong sense. We will approach this result through maximal monotone operators and another theorem of Kenderov. Definition 2.36. ( a) A norm on a Banach space E is said to be strictly convex ( or rotund) provided there are no line segments in the unit sphere; equivalently, provided Il x ll = 1 = lI y ll and x =I- y imply II AX + (1 - A)y ll < A ll x ll + (1 - A) lI y ll

whenever 0 < A < 1.

34

Section 2

(b) A norm on E is said to be smooth provided that for each x E E with II x ll = 1, there exists a unique element x* E E* such that ( x*, x) = 1 and II x* 1I = 1. This is the same, clearly, as saying that the duality map J(x) ( Example 2.26 ) is a single point for every x =f 0, which in turn is the same as saying that the norm is Gateaux differentiable at every nonzero point of E . It is common terminology to refer to a space as being smooth or strictly convex if its norm has that property.

2 .37 Exercises. ( a) P rove that if the norm in E is such that its dual norm in E· is strictly convex [smooth] , then it is itself smooth [strictly convex] . ( b ) Prove that the norm in E is strictly convex if and only if every convex subset of E has at most one point of least norm. (c ) Show that the norm in Hilbert space is both smooth and strictly convex, but that the norms in Co and £1 are neither. ( d ) Prove that the norm in E is strictly convex if and only if II x + yll < II x ll + l I y ll whenever x and y are linearly independent. It is known [ KI ] , [Tr] that there exist smooth spaces whose duals are not strictly convex; we will see in Section 4 that there also exist strictly convex spaces whose duals are not smooth. The proof of the next theorem uses the following classical fact: For any lower semi continuous real-valued function on an open subset of a complete metric space there exists a dense G6 set of points at which the function is continuous. ( See, for instance, [eh, p.ll1] . )

Theorem 2.38 ( [Ked). Suppose that E admits an equivalent norm whose dual norm is strictly convex and suppose that T: E -+ 2E• is a maximal mono­ tone map such that W = int D(T) is nonempty. Then there exists a dense G6 subset of W at every point of which T is single-valued. Proof. Assuming that the norm in E* is strictly convex, we know that for each x E W , the nonempty weak* compact set T(x) has at most one point of least norm - in fact - exactly one point of least norm, since dual norms are weak* lower semicontinuous. Define the real-valued function j3 by j3 (x)

=

min{ lI x* lI : x* E T(x)} ,

x E W.

We first show that j3 is lower semicontinuous on W ; that is, for any real number A , the set {x E W : j3( x) > A } is open. Indeed, since the norm in E* is weak* lower semicontinuous, the set V = {x* E E*: II x* 1I > A} is weak* open. From Exercise 2.29( b ) we know that maximal monotone operators are norm-to-weak* upper semicontinuous. Thus, if j3( x) > A, then T( x) C V a.nd hence there exists a neighborhood U of x in W such that T(U) C V, which implies that j3( y ) > A for all y E U.

2. Monotone operators, sub differentials and Asplund spaces

35

We next show that if (3 is continuous at x E W, then T(x) consists of a single point. Indeed, suppose that (3(x) = IIx * l I , x* E T(x), and that there exists y* E T(x), with y* t- x*j necessarily, lIy* 11 > I I x* l I . We can choose y E E, lIyl l = 1 , such that (y. , y) > I I x · 1I = (3(x). By continuity of (3 at x, there exists 8 > 0 such that B(xj 8)

c

W and (3(z) < (y* , y) whenever z E B(xj 8).

Let z x + 8y E B(x ; 8) and let z* E T(z) be such that (3(z) monotonicity, 0 :::; (z· - y* , z - x ) 8 (z· - y., y) , =

=

IIz* ll .

By

=

so (z· , y) � (y* , y) > (3( z ) IIz* lI, which is a contradiction. Finally, we complete the proof by applying the general fact about lower semicontinuous functions mentioned above. =

Corollary 2.39. (Asplund) If E admits an equivalent norm which has strictly convex dual norm, then E is a weak Asplund space. Proof. The proof is similar to that of Theorem 2.32. 2.40. Exercises. (a) Suppose that 1 · ·1 is an equivalent norm on E·, (that is, m · lI x·1 I � I x· 1 � M · lI x·1I for all x·, where 0 < m � M). Prove that 1 · · 1 is the dual of an equivalent norm on E if and only if I · ·1 is weak* lower semicontinuous. (b) Suppose that E is separable, that {xn} is a dense sequence in the unit sphere of E and, for x· E E·, define

Prove that I x · 1 = I Ix · 1 I + I I L(x·)1I 2 defines an equivalent strictly convex dual norm on E· . Hence deduce Mazur's theorem from Cor. 2.39.

We next look at a property, weaker than separability, which has implica­ tions for both Asplund spaces and weak Asplund spaces. Definition 2.41. A Banach space E is said to be weakly compactly generated (WCG ) provided there exists a weakly compact subset K of E whose linear span is dense in E. Since the closed convex hull of a weakly compact subset of a Banach space is weakly compact, one can always assume that K is convex.

For background (and proofs of some of the assertions which follow) see Diestel's lecture notes [Di].

J.

2.42. Examples. (a) If E is separable or reflexive, then it is WCG. In the first case, let {xn } be dense in E and let J( = {n-1 xn/ ll xn ll } u {O}j this is actually compact. In the second case, let K be the unit ball.

36 e..,.

Section 2 (b) The Banach space co(r), for any set r, is WCG : If, for each 'Y E r, we let denote the usual basis vector, then {e..,. : 'Y E r} U {O} is weakly compact.

(c) The Banach space [1 (r) is WCG (if and) only if it is separable, that is, if and only if r is countable (since weakly compact subsets of [l (r) are norm compact). (d) If 11 is finite, then L1 (1l) is WCG. (Since Loo(ll) C L1 (1l) and the inclusion mapping is weak* to weak continuous, the unit ball of Loo(ll) is weakly compact in Ll (Il) ' ) The [I-product of a countable family of WCG spaces is WCG, hence L1 (1l) is WCG if (and only if) 11 is O'-finite. (e) If E is WCG and there exists a bounded linear operator on E having dense range in the Banach space F, then F is WCG.

The next theorem obviously generalizes Theorem 2.12 ( that if E* is sep­ arable, then E is an Asplund space ) . Theorem 2.43. If E* is WCG, then E is an Asplund space. Proof. w� will apply Cor. 2.15, by showing that for every separable subspace F of E, its dual F* is separable. Since the latter is isometric to the quotient space E* / FJ.. , and since a continuous linear image of a wce space is wce , we can assume that F* is wce . Let K be a weakly compact subset of F* whose linear span is dense in F* ; it clearly suffices to prove that K is norm

separable. But this is immediate from the following general lemma.

Lemma 2.44. If E is a separable Banach space and if K is a weakly compact subset of E*, then K is norm separable. Proof. Since K is weakly compact, it is also weak* compact and (K, w) and (K, w*) are homeomorphic. Any weak* compact subset of the dual of a sep­ arable space is weak* metrizable, so (K, w*) is separable, and hence (K, w) is separable. Let A be a countable weakly dense subset of K and let L be

the set of all rational linear combinations of elements of A, so L is countable and hence its norm closure L is norm separable. Clearly, L contains all linear combinations of members of A, hence L is the smallest norm closed linear subspace containing A. Since L is weakly closed, it contains the weak closure of A, that is, it contains K. Thus, K is norm separable, since it is a subset of the norm separable space L. An argument ( due to P. D. Morris ) which uses some of the same ideas as above can be used to prove Kuo's theorem that if E* is isomorphic to a subspace of a WCG space, then E is an Asplund space. This may be found in R. Bourgin's lecture notes [Bou]; since H. P. Rosenthal has shown ( see [Di] ) that there exist subspaces of the wce space £ 1 [JL] ' JL finite, which are dual spaces but are not themselves wce, this is a generalization of Theorem 2.43.

2. Monotone operators, subdifferentials and Asplund spaces

37

The converse to Theorem 2.43 is false: By Exercise 2. 16, co ( r) is an As­ plund space for every set r, while its dual £ l (r) is not WCG if r is uncount­ able. We conclude this section with another of Asplund's theorems. Theorem 2.45 ( [Asp] ) . If the Banach space E is a subspace of a WeG space, then E is a weak Asplund space. Proof. By a deep renorming theorem due to Amir and Lindenstrauss (see, for instance, [Di] or [D-G-Z2]) any WCG space admits an equivalent norm having strictly convex dual. (This obviously generalizes Exercise 2.40 (b).) Moreover, the new norm inherited by any subspace is easily seen also to have strictly convex dual, so Cor. 2.39 applies.

Remarks. Theorem 2 . 1 1 was suggested by a remark in the Preiss-Zajicek paper [Pr-Z] ; they proved a sharpened form of generic differentiability for continuous convex functions (in a Banach space with separable dual) and pointed out how it could be used to obtain a generic continuity theorem for monotone mappings. (We have replaced their term "a-angle porous" by "a-cone meager" , feeling that the latter is a bit more descriptive.) They also pointed out that D. Gregory's argument could be used to give the proof of Theorem 2 .14 (that Asplund spaces are "separably determined" ) . There i s a great deal of material about monotone mappings and their applica­ tions in the two volumes by E. Zeidler [Ze] and in the book by D. Pascali and S. Sburlan [pa-SbJ, mostly for reflexive spaces. (The reader is warned that in the early sections the latter authors sometimes assume-without saying so-that their Banach spaces are reflexive.) The fundamental Theorem 2.28 (on local bounded ness of monotone operators) was first proved in slightly different form by Rockafellar [R0 2 ]. The original proof has been considerably simplified by a number of authors; there is one due to P. M. Fitzpatrick (see [pa-SbJ who have attributed to the latter author one of S. P. Fitzpatrick's early papers), and an even shorter proof, using the uniform bounded­ ness theorem, in [Dr-LJ. The one we present is a specialization to Banach spaces of a general result by J . Borwein and S. P. Fitzpatrick [Bor-F]. Recently, L. Vesely [private communication] has shown that if D(T) is convex, then a maximal mono­ tone T fails to be locally bounded at each point of bdry D(T). He first notes a Fact: If T is locally bounded at x E D(T) (definition obvious) then, using maximal monotonicity and weak* compactness, x E D(T). Assume that D(T) is convex. If x E bdry D(T) C D(T) and T is locally bounded at x, then by the Fact, x E D(T). Suppose that x were in the boundary of D(T) and that there were a neighborhood U of x with T( U) bounded. By the Bishop-Phelps theorem there would exist a point z E U n D(T) and a nonzero element w · E E· such that (w·, z ) = sup (w·, D(T)). Now, T would also be locally bounded at z, so (the Fact again) ;; E D(T) and we could choose z· E T(z). The same kind of argument used just before Exercises 2.29 would show that T(z) is unbounded, a contradiction. Since x 'f. bdry D(T), it must be in int D(T). By local bounded ness, we can choose an open set U such that x E U C int D(T) and T(U) is bounded. Thus, T is locally bounded at every point of U, which, by the Fact, implies that U C D(T) and therefore x E int D(T), another contradiction.

Section 3

3. Lower semic ontinuous c onvex functions.

Our differentiability results for convex functions made heavy and consistent use of continuity, but in both the theoretical and applied aspects of convex functions, it is sometimes desirable to weaken this hypothesis. Lower semi con­ tinuity is precisely what is needed. Uncomfortable as it may seem at first, the subject is best treated by introducing the seeming complication of admitting extended real-valued functions, that is, functions with values in R U {oo}. We adopt the conventions that r . 00 = 00 and ( -r ) . 00 = - 00 if r > 0, and ± 00 = ±oo for all r E R. r

(We won't have occasion to worry about 00 - 00 or 0 . 00 . ) Definition 3.1 Let X be a Hausdorff space and let f: X � R U {oo } . The effective domain of f is the set dom(f) = {x E X: f(x) < oo} . We say that f is lower semicontinuous provided {x E X : f ( x) :::; r} is closed in X for every r E R. This is equivalent to saying that the epigraph of f

epi(f) is closed in X

x

=

{(x, r)

E

E x R: r � f(x)}

R. Equivalently, f is lower semicontinuous provided

f(x) :::; lim inf f(xu) whenever x E X and (xu) is a net in X converging to x. We say that f is proper if dom(f) "I- 0. Our original definition of convexity for a real-valued function on a Banach space E applies without change. Note that if f is convex, then so is dom(f). Also, a function f is convex if and only if epi(f) is convex. This last fact is important; it means that

certain properties of lower semicontinuous convex functions can be deduced from properties of these (rather special) closed convex subsets of E x R. One can view this as saying that the study of lower semi continuous convex functions is a special case of the study of closed convex sets.

3. Lower semicontinuous convex functions.

39

3.2. Examples.

(a) Let C be a nonempty convex subset of Ej then the indicator function Ce , defined by Ce( x ) = 0 if x E C, = 00 otherwise, is a proper convex function which is lower semicontinuous if and only if C is closed.

[This example is one reason for introducing extended real-valued functions, since it makes it possible to deduce certain properties of a closed convex set from properties of its lower semicontinuous convex indicator function. Thus, one can use this example to assert that the study of closed convex sets is a special case of the study of lower semicontinuous convex functions. It's all a matter of which viewpoint is more convenient, the geometrical or the analyt­ ical. It is best to be able to switch readily from one to the other.] (b) Let A be any nonempty subset of E" such that the weak* closed convex hull of A is not all of E" (or, more simply, let A be a weak* closed convex proper subset of E") and let O"A ( X ) = sup { {x * , x) : x" E A}, x E Ej

then 0"A is a proper lower semicontinuous convex function, called the support function of A. (In Sec. 2 we assumed that A was bounded.)

(c) If f is a continuous convex function defined on a nonem pty closed convex set C, extend f to be 00 at the points of E \ Cj the resulting function is a proper lower semicontinuous convex function.

The next proposition uses completeness of E to show where lower semi­ continuous convex functions are necessarily continuous. Proposition 3.3. Suppose that f is a proper lower semicontinuous convex function on a Banach space E and that D = int dom(f) is nonempty; then f is continuous on D. Proof. We need only show that f is locally bounded in D , since (as observed after Prop. 1 .6) this implies that it is locally Lipschitzian in D. First, note that if f is bounded above (by M, say) in B( X; 15) c D for some 15 > 0, then it is bounded below in B(x; 15). Indeed, if y is in B(x ; 15), then so is 2x - y and 1 1 f(x) � "2 [f( Y ) + f(2x - y)] � "2 [f(y) + M]

so fey) 2': 2f(x) -M for all y E B(x; 15). Thus, to show that f is locally bounded in D , it suffices to show that it is locally bounded above in D. For each 2': 1 , let Dn {x E D : f(x) � } The sets Dn are closed and D UDn ; since D is a Baire space, for some we must have U == int Dn nonempty. We know that f is bounded above by in U; without loss of generality, we can assume that B(O; 15) C U for some 15 > O. If y is in D, with y =f 0, then there exists J-L > 1 such that z = J-Ly E D and hence (letting 0 < ,\ = J-L- 1 < 1 ) , the set n

=

n

.

=

n

n

v

=

'\z + (1 - '\)B(O; 15)

= y + (1 - '\)B(O; 15)

is a neighborhood of y in D; Fig. 3.1 below illustrates the situation.

40

Section 3

Fig. 3.1.

For any point v

=

( 1 - >.)x + >. z E V (where x E B(O; 6)) we have f(v) � ( 1 - >. )n + >.f(z),

so f is bounded above in V and the proof is complete. 3.4. Examples. (a) The function f defined by f(x) = l/x on (0, 00), f(x) = 00 on ( - 00, 0] shows that f can be continuous at a boundary point x of dom(J) where f( x) = 00 . (Recall that the neighborhoods of 00 in (-00, 00] are all the sets (a, oo], a E R.) (b) Suppose that C is nonempty closed and convex; then the lower semicontin­ uous convex indicator function Dc is continuous at x E C if and only if x E int C. Thus, if int C = 0, then Dc is not continuous at any point of C = dom( Dc ).

Definition 3.5. Recall that if E is a Banach space, then so is E x R, under any norm which restricts to the original norm on the subspace E, for instance, lI (x, r) 1I II x ll + I r l . Recall, also, that (E x R)* can be identified with E* x R, using the pairing =

( x * , r* ) , (x, r))

=

(x * , x) + r* ' r.

Remark. If a proper lower semicontinuous convex function f is continuous at some point X o E dom(f), then dom(f) has nonempty interior and epi(f) has nonempty interior in E x R. (Indeed, f(x) = 00 outside of dom(f), so Xo cannot be a boundary point of the latter. Moreover, there exists an open neighborhood U of Xo in dom(f) in which f(x) < f(xo) + 1 , so the open product set U x {r : r > f( xo ) + I } is contained in epi(f).) Definition 3.7. (a) The definition of the subdiiJerential 8f for a proper lower

semi continuous convex function f is almost the same function: If x E dom(f), define 8f(x) =

=

{x* E E* : ( x* , y - x ) � f( y )

{x* E E* : (x * , y) � f(x + y )

-

-

as

that for a continuous

f(x) for all y E E}

f(x) for all y E E},

while 8f(x) 0 if x E E \ dom(f). It may also be empty at points of dom(f), shown in the first example below. (b) If x E dom(f) we define d+ f(x) as before: =

as

3. Lower semicontinuous convex functions.

d+ f(x)( y ) = lim C 1 [f(x + ty) - f(x)], o t- +

41

y E E,

recognizing that d+ f( x )(y) = 00 if x + ty E E \ dom(f) for all t > O. (It is also possible to have d+ f(x)(y) = -00; consider, for instance, d+ f(O)(l) when f(x) = _x 1 /2 for x � 0, = 00 elsewhere.) The following important relationship is easily seen still to be valid in this more general situation: For any point x E dom(f), x * E 8f(x) if and only if (x * , y) ::; J+ f(x)(y) for all y E E.

It follows from this that for the example given above (f(x) = --IX for x � 0), it must be true that 8f(0) = 0. In the first example below, one sees that it is possible to have 8f(x) = 0 for a dense set of points x E dom(f). 3.8 Examples. {x

(a� Let C be ��e closed (in fact, compact) convex subset of £2 defined by C £ : I x n l ::; 2 , n = 1 , 2, 3, . . . } and define f on C by

=

E

Since each of the functions x -+ _(2 - n + X n )1 / 2 is continuous, convex and bounded in absolute value by 2 ( - n +1 ) / 2 , the series converges uniformly, so f is continuous and convex. We claim that af(x) = 0 for any x E C such that X n > _2- n for infinitely many n. Indeed, let e n denote the n-th unit vector in £2 . If X· E 8f(x) (so that, as noted above, x · ::; d+ f( x)), then for all n such that X n > _2- n , we have

an impossibility which implies that af(x) = 0. Note that if we make the usual extension (setting f(x) = 00 for x E e \ C), then f is lower semicontinuous, but not continuous at any point of C ( = bdry C). (b) Let C be a nonempty closed convex subset of Ej then for any x E C, the subdifferential 8I5c (x ) of the indicator function I5c is the cone with vertex 0 of all x· E E * which "support" C at x , that is, which satisfy {x * , x }

=

sup { { x * , y} : y

E

C}

== 17 c (x · ).

(Indeed, x · E 8I5c(x) if and only if { x ' , y - x } ::; I5c ( y) - l5c (x ) , while x · attains its supremum on C at x if and only if the left hand side of this latter inequality is at most 0, while the right side is always greater or equal to 0.) (c) The following is an example of a function which is continuous and convex on £2 but is not bounded on the unit ball. Define ¢>(t) for t ;::: 0 by ¢>(t) = 0 if o ::; t ::; 1/2 while ¢>(t) = 2t - 1 if t ;::: 1/2. For x = (Xk) E £2 , define 00

f(x)

=

L ¢>( Ek�n X� ). n= l

(As the supremum of a sequence of continuous convex functions, f is lower semicon­ tinuous and convex. Since the infinite sum actually has only finitely m any nonzero

Section 3

42

terms, f is finite on all of £2 , hence is continuous by Prop. 3.3. It tends to infinity on the sequence of norm-one elements of the form Xl = 7a = Xn = Xn+l while Xk = 0 if k =/: 1 , n, n + 1 . )

3.9. Exercises. (a) A proper lower semicontinuous convex function f has a global minimum on X if and only if 0 E 8f(x).

E at

(b) Suppose C is a nonempty convex subset of E and that f is a proper convex function on E with dom(t) n C =/: 0; then f i e has a minimum at the point X E C if and only if 0 E 8(t + he )(x). Suppose that f is proper convex function on E. Define its dual function (or "Fenchel transform") f' on E' by r (x ' )

= sup { ( x ' , x) - f (x) : x E E},

x · E E' .

(c) Show that f' is a proper lower semicontinuous extended real-valued convex function on E ' . (d) Show that for all x E E and x·

E

E',

( x * , x } � F (x')

+ f(x),

with equality holding if and only if x · E 8f (x) . (e) Define r' on E" in the obvious way and show that its restriction to E coincides with f if and only if the latter is lower semicontinuous.

C E"

(f) Compute r (that is, describe it as a function on E* without explict reference to f) for each of the following functions f: (i) h (x) = II x l i . (ii) h (x) = � II x ll 2 . (iii) fa = he , where C is a nonempty closed convex subset of E. (iv) h = (TA , where A is a nonempty bounded convex subset of E ' . Suppose that f and 9 are proper lower semicontinuous convex functions on E. Define their in/-convolution (or epi-sum) by (t v9 )( x )

=

inf {f( y) + g(x - y) : y E E} ,

x

E E.

(g) Show that fV9 is a proper lower semicontinuous convex function on domain dom(t) + dom(g).

E

with

(h) Given nonempty closed convex sets C) and C2 , compute bel Vhe2 . (i) Show that if C is a nonempty convex set, then

( heV i l . ' I I )(x)

=

dist( x, C).

(j) Using the definition of dual functions (above), show that (t Vg )*

=

r

+ g'.

3. Lower semicontinuous convex functions.

43

(k) Show that if the convex set dom(J) - dom(g) contains a neighborhood of the origin (in particular, if dom(J) n int dom(g) "1= 0) then (J + 9 t = rVg*. (f) Show that fV9 is continuous at

x

E E if either f or 9 is continuous at

x.

Definition 3.10. A point x of a subset X of E is said to be a support point of X provided there exists x* E E* , x* -=1= 0, such that x* attains its supremum on X at x. Any such x* is said to support X at x, or be a supporting functional of X. (This is not to be confused with the support function ax of X, but there is an obvious relationship: x* supports X at x if and only if x ( x* ) = (x*, x) .) The geometric terminology arises from the fact that a closed hyperplane is said to support X if one of its two closed half spaces contains X and if the hyperplane itself actually intersects X. If x* supports X at x, then H = {y E E: (x* , y) = aX ( x*)} contains x and is just such a hyperplane. O'

It is an easy consequence of the Hahn-Banach theorem (or separation theorem) that if a closed convex set C has nonempty interior, then every boundary point of C is a support point of C. It is not obvious that a nonempty closed convex set with empty interior has any support points. Even if it does, there is a question as to how many support functionals it admits. For instance, even though every boundary point of the unit ball B of a Banach space E is a support point of B , there may be functionals on E which do not attain their suprema (that is, their norms) on B. This obviously cannot happen if E is reflexive (since B is then weakly compact) and, in fact, a deep theorem of R. C. James says that if E is nonrefiexive, then there exists an element of E* which does not attain its supremum on B. (See Diestel's Lecture Notes [Di] for a proof.) The fact that both the support points and support functionals of C are necessarily dense (in appropriate spaces) are the Bishop-Phelps theorems. Now, it is important to know that the sub differential of a lower semicontinuous proper convex function f will be nonempty for at least some points of dom(J). By applying the Bishop-Phelps techniques in E x R to the closed convex epigraphs of lower semi continuous convex functions, A. Br�ndsted and R. T. Rockafellar showed that dom(8J) is, in fact, dense in dom(J). There are by now a number of approaches to the foregoing results and to Rockafellar 's theorem that the sub differential map is maximal monotone; our route will pass through key results by I. Ekeland and S. Simons. Definition 3.11. If 0 < A < 1 define K>. = {(x, r) E E x R: A ll x ll � - r } ;

this is easily seen to be a closed convex cone (similar to the cone used in the Preiss-Zajicek theorem) which opens downward; it is the reflection through the origin of the epigraph of the function A ll · , 11 . Since the latter is continuous, K>. has nonempty interior (containing (0, -1), for instance).

44

Section 3

The following lemma is an E x R version of a classical maximality result due to Bishop and Phelps. It simply asserts that, in the partial ordering defined by J{>. , if a closed set A satisfies a certain boundedness condition, then any point of A is dominated by a maximal point of A. Xo. ro)

Fig. 3.2. This illustrates (3. 1 ) and (3 . 2) below Lemma 3.12. Suppose that A is a closed nonempty subset of E x R, that 0 < >.. < 1 and that inf{ r: (x, r) E A } = O. For any point (xo , ro ) in A, there exists (x, r) E A such that (x, r) E A n [J{>. + (xo , ro)] and

(3.1 )

{(x, r)} = A n [J{,\ + (x , r)] .

(3 . 2)

( Figure 3.2 above illustrates properties (3. 1 ) and (3 . 2).) Proof. It is convenient to define the continuous linear functional R on E x R by R(x, r) = r. Choose (xI , rd E Ao == A n [J{>. + (xo, ro)] such that r l < inf R(Ao ) + 1 .

Continuing by induction, we can choose a sequence {(xn, rn)} such that ( X n+ l , rn +l ) E An

==

A n [J{>. + (xn, rn)] and

r n+ l < inf R(An ) + 1/(n + 1) .

We claim that for every

n,

( a) An +l C An

( b ) diam An

and

--t

O.

To see ( a) , note that J{>. + ( X n +l , rn+J ) C J{>. + [J{,\ + (xn , rn)] = J{>. + (xn, rn);

now interesect both sides with A to get the desired inclusion. To prove (b ) it suffi ces to show that if (y , 8) E A n , then l I y - xn ll + 1 8 - rn l � l i n >.. + li n . Note that 8 � infR( An ) � infR(An d > rn - li n ( the second inequality following from ( a)) and >" Il y - xn l l � rn < li n, so Il y - xn ll < li n >.. and 0 � rn - 8 < li n. Since A and J{>. are closed, completeness guarantees that nA n is a single point, which we denote by (x, r). Since (x, r) E Ao , we immediately have (3 . 1 ). To obtain (3 . 2), note that for each n, -

- 8

3. Lower semicontinuous convex functions.

K)... + (x, r)

C

45

K>. + [K>. + ( x n , rn ) ] = K>. + ( x n , rn),

so if ( y, s ) E A n [K>. + (x, r)], then ( y , s ) E A n for every have (y, s ) = (x , r).

n,

hence we must

[The proof given above goes through if E is merely a complete metric space; one simply replaces the partial ordering defined by K), by defining (x, r) ::; (y, s) if and only if >'d(x, y) ::;

s

-

r.

The Banach space version is easier to visualize geometrically, and we have no need for the more general form.]

As an immediate consequence of this result we obtain Ekeland's variational principle, which can be viewed as saying that if f(xo ) is nearly a minimum value for the lower semicontinuous function f, then a small Lipschitz contin­ uous perturbation of f attains a strict minimum at a point z relatively close to Xo. (That is, there exists a Lipschitz continuous function g, with small Lipschitz constant, such that f + 9 has a strict minimum at z .) This fact has found application in a wide variety of topics in nonlinear analysis; see Ekeland's survey in [Ek] .

Lemma 3.13 (Ekeland) . Assume that f is a proper lower semicontinuous extended real-valued function on the Banach space E which is bounded below. Suppose that f > 0 and that f(xo) ::; inf{f(x): x E E} + f. Then for any A > 0 there exists a point z E dom(f) such thai (i) and

A ll z - xo ll � f(xo ) - fez), (iii)

(ii)

A ll x - z ll + f(x) > f e z ) whenever x #- z.

Remark. Despite the fact that A appears in the denominator, the estimate in (ii) need not be large for small A; one can employ the great trick - to be used to good effect later - of taking A = -If. Proof. We assume without loss of generality that infEf = 0, so we have f(xo ) ::; f. Put the equivalent norm 2AI I . ·11 on E and apply Lemma 3.12 to the closed set A = epi(f) and the cone K1 /2 (which we denote simply by K) to obtain a point ( z, r) in E x R such that (1) ( z , r ) E A n [K + ( xo , f ( xo ))]

and (2) (( z, r) } = A n [K + ( z , r) ].

From (1) we have 0 :::; fez) :::; r < 00 and A ll z - xo ll � f(xo ) - r :::; f(xo ) - fez) :::; f(xo ) :::;

f,

which yields assertions (i) and (ii). Assertion (iii) is obvious if f(x) = 00. To see its validity in general, note first that if f( z) < r , then we must have (z, r) #- ( z , f ( z )) so from (2) it follows that ( z , f( z )) is not in K + ( z, r) , that ,

Section 3

46

is, 0 > r - fez), a contradiction. Thus, r = fez). From (2) , again, if f( x ) < 00 and x =f z, then (x, f(x)) is not in K + (z, r ), that is, ,xlix - zll > r - f(x)

= fez)

-

f(x),

which was to be shown. The Br0ndsted-Rockafellar theorem is an easy consequence of this lemma, but we first require a definition. Definition 3.14. Let f be a proper convex lower semicontinuous function and suppose x E dom(f). For any € > 0 define the €-subdifferential Bd(x) by Bd(x) = {x* : (x * , y) :5 f(x + y)

-

f( x ) + dor all y E E } .

It is clear that if 0 < €1 < €2 , then B€J(x) c B€2 f(x). We show next that this set (necessarily convex and weak* closed) is always nonempty. Proposition 3.i5. If f is a proper lower semicontinuous convex function on E, then Bd(xo ) is nonempty, for every Xo E dom(f) and every € > o. Proof. Figure 3 . 3 below shows immediately what is happening, but we must check some details.

Xo

Fig. 3.3

Since epi(f) is closed and convex and does not contain (xo, f( xo ) - €), there exists a linear functional (y*, r*) E E* X R such that ((y * , r* ), (xo , f(xo ) - f)) < ((y * , r* ) , (x, r )),

x E dom(f),

r � f(x),

that is (y * , xo) + r* [f(xo ) - €J < (y* , x) + r* · r

if x E dom(f), r � f(x).

Taking x = Xo and r = f(xo ) shows that r* > o. We can assume, in fact, that r* = 1; simply replace y* by y* jr* . If we then let x* = -y* and take r = f(x), we obtain the desired inequality. We will also need the following fact, which is of substantial interest in its own right (especially in convex optimization).

3. Lower semicontinuous convex functions.

47

Theorem 3.16. Suppose that f and 9 are convex proper lower semicontinuous functions on the Banach space E and that there is a point in dom( f ) n dom(g) where one of them, say f, is continuous. Then o(f + g) (x )

=

of(x) + o g(x ),

x E dom(f + g).

(The right side is the usual vector sum of sets.) Remark. It is immediate from the definitions that for x E dom(f + g) (which is identical to dom(f) n dom(g)), one must have of(x) + og( x )

c

o ( f + g)( x ).

This inclusion can be proper. To see this, let E = R2 , let 1 denote the indicator function Oc and g = OL , where C is the epigraph of the quadratic function y = x2 and L is the x-axis. Obviously, C and L intersect only at the origin 0 and it is easily verified that 81(0) = R-e, where e is the vector (0, 1), and 8g(0) = Re, while

8U + g)(O) = R2 f= 81(0) + 8g(0).

Proof. Suppose that X o E o(f + g)( x o ) . In order to simplify the argument, we can replace f and 9 by the functions h ex)

=

f(x + x o ) - f(xo ) - (x � , x) and gl ( X )

=

g(x + x o ) - g(xo),

x E Ej

it is readily verified from the definitions that if Xo E o(f + g)(x o ), then o ( h + gt } (O) and if 0 E Ofl (O) + Ogl ( O ) , then X o E of(xo ) + og(x o ). Without loss of generality, then, we assume that Xo = 0, X o = 0, f(O) = 0 and g(O) = O. We want to conclude that 0 is in the sum of(O) + o g( O) , under the hypothesis that 0 E o(f + g)(O). This last means that o E

(f + g)(x) � (f + g)(O)

= 0 for all x E E.

(3.3)

We now apply the separation theorem in E x R to the two closed convex sets Cl = epi(f) and C2 = {(x, r): r � - g(x)}j this is possible because f has a point of continuity in dom(f) n dom(g) and hence - recall Remark 3.6 Cl has nonempty interior. Moreover, it follows from (3.3) that C2 misses the interior of Cl = {(x, r): r > f(x)}. Since (0, 0) is common to both sets, it is contained in any separating hyperplane. Thus, there exists a functional (x*, r*) E E* x R, (0, 0) f:. (x*, r*), such that

-

(x * , x) + r* · r � 0 if r � f(x) and (x * , x) + r* · r � 0 if r � -g(x). Since 1 > f(O) = 0 we see immediately that r* � O. To see that r* f:. 0, (that is, that the separating hyperplane is not "vertical"), we argue by contradiction: If r* = 0, then we must have x * f:. OJ also (x*, x) � 0 for all x E dom(f) and (x* , x) � 0 for all x E dom(g). This says that x* separates these two sets. This is impossiblej by the continuity hypothesis, their intersection contains an

48

Section 3

interior point of dom(f). Without loss of generality, then, we can assume that r * = 1 and hence, for any x E E, (- x * , x - O) � J(x) - J(O) and ( x * , x - O) � g(x) - g(O), that is, 0 = - x* + x* E oJ(O) + og(O), which completes the proof. Theorem 3.17 (Br�ndsted-Rockafellar). Suppose that J is a convex proper lower semicontinuous function on the Banach space E. Then given any point X o E dom(f), € > 0, A > 0 and any x� E oJ (x o ) , there exist x E dom(f) and x* E E* such that

x* E oJ(x),

Il x - xo ll � € / A and I l x* - x � 11 � A.

In particular, the domain of oj is dense in dom(f). Proof. By hypothesis, (x�, x - x o) � J(x) - J(x o )

define

+ € for all x E E, so if we

g(x) = J(x) - ( x�, x ) , x E E, we see that g is proper and lower semicontinuous, with dom(g) = dom(f). Moreover, g(x o ) � infEg + €, so by Lemma 3.13 there exists z E dom(f) such that A ll z - xo ll � € and A ll x - z ll + g(x) 2: g(z) for all x E E. Letting h e x) = A ll x - z ll (x E E), this last inequality implies that 0 E o(g + h) (z ) = og(z) + oh( z) (by Theorem 3.16, since h is continuous). Thus, there exists z * E og(z) = oJ(z) - x� such that - z * E oh ( z ) = {x* E E* : IIx* 11 � A}. Let x* = z* + x� and x = Zj then x* E oJ(x), IIx* - x � 11 � A and Il x - xo ll � € / A, as required. Some important special cases of the Bishop-Phelps density theorems are easy corollaries. Theorem 3.18 (Bishop-Phelps) . Suppose that C is a nonempty closed convex subset of a Banach space E. Then (i) The support points of C are dense in the boundary bdry C of C . (ii) The support functionals of C are dense in the cone of all those functionals which are bounded above on C . Proof· (i). Suppose that Xo E bdry C and that 0 < € < 1. Let J = be be the indicator function of C. Choose X l E E \ C such that IIx o - x l ii < f and apply the separation theorem to obtain x� E E* , IIx � 1I = 1, such that O'e (x � ) < ( X�, X l ) . This implies that for all x E C

(x � , x ) < ( x � , X l ) = ( x � , x l - x o ) + ( x � , x o )

�

f + (x� , x o )

so ( x�, x - x o) � f = J(x) - J(xo ) + f, that is x� E oJ(x o ). By Theorem 3.17 (taking A = yIf) there exist x E C = dom(f) and x* E oJ(x) (which says that x* attains its supremum on C at x) such that

3. Lower semi continuous convex functions.

Il x - xo ll :::; ..fi and

49

II x * - x � 11 :::; ..fi < 1 .

The last inequality implies that x* :f 0, so it is a supporting functional of C at x. (ii). Suppose that x� E E* is such that x� :f 0 and ae(x�) < 00. Given 0 < < II x� 1 I 2 , choose Xo E C such that ( x�, xo) > ae(x* ) - E. If we again let f = fie, then €

(x � , x - xo) < E = f(x) - f(xo) + E for all x E C,

that is, x� E ad(xo) . By Theorem 3.17 (with .A = ..fi again), there exist x E dom(f) = C and x* E af(x) such that II x* - x� 1 I :::; ..fi < Il x � l I . This last inequality implies that x* :f 0, so the proof is complete. Note that assertion (ii) (above) can be reformulated as a variational prin­ ciple: If the continuous linear functional x� is bounded above on C, then there

exists a continuous linear functional y* of small norm (namely, y* = x * - x�) such that x� + y* attains its maximum on C.

The version of the Bishop-Phelps theorem most frequently applied in the theory of Banach spaces is the following special case of Theorem 3.20(ii) (above); it is what most authors mean when they refer to the "Bishop-Phelps theorem".

Let E be a Banach space. Then the set of all functionals x* in E* which attain their norms on the unit ball, that is, which satisfy

Theorem 3.19.

(x * , x ) = Il x* 11

for some x E E with

Il x ll = 1 ,

is norm dense in E* . Equivalently, the duality map J has dense range. There are much more general versions of the Bishop-Phelps theorems in the original paper [Bi-Ph], but they have not found wide application. For in­ stance, one can prove that if a functional strictly separates C from a nonempty bounded set X , then it can be approximated by a functional which supports C and strictly separates C from X . An interesting special case of this result (taking X to be a single point) has the following proposition as an immediate consequence; as we will see, it can also be proved using the present methods. Proposition 3.20. A nonempty closed convex subset C of a Banach space E is the intersection of all the closed half-spaces defined by its supporting hyperplanes.

Proof. We must show that if y E E \ C, then there is a support functional of C which separates C from y. Let d dist(y, C) and use the separation theorem to choose x� E E* of norm 1 which separates C from the ball B(y; d); this implies that ae(x�) = (x�, y) - d. =

Section 3

50

slope = d+f(x)(y)

s eparati ng hyperplane Fig.

:>

x + ty

x

x + y

3.4. The functional Xo * separates C and B(y; d)

Let f denote the indicator function Dc and choose E > 0 sufficiently small such that 0(0 + d + E) < d/ 2. Next, choose Xo E e n B(y; d + E), so (x� , Y - xo ) � lI y - xo ll � d + E. For any x E C we have (x � , x - xo) = (x � , x - y) + (x � , y - xo) �

that is, x� E od ( x o ). By Theorem 3.17 (taking ,\ 0) there exist elements x. E dom(f) == C and x; E of(x .) (that is, x ; supports C at x . ) such that II x . - xo ll � 0 and I I x; - x� 1 I � J€. It follows that for all x E C, =

(x ; , x - y) � (x ; , x . - y) = (x; - x � , x . - y) + (x � , x . - y ) �

� II x ; - x� II ' ( 1I x. - xo ll + Il xo - y ll ) + O'c ( x� ) - (x � , y) �

so

O'

c ( x; ) <

(x ; , y) .

Corollary 3.21. Suppose that f is a convex lower semicontinuous proper function on E; then f is the upper envelope of the continuous affine functions defined b y its subdifferentials, that is, for any x in dom(f), f(x) = sup{ (y * , x - y) + f ey) : y * E of(y) for some y E dom(of)}, Proof. If x E dom(f) and E > 0, apply Prop. 3.20 to obtain a closed hyperplane in E x R which supports C epi (f) (at (y, j(y)), say) and misses the point (x, j(x) - E) , as shown in Fig. 3.5 below. =

3. Lower semicontinuous convex functions.

51

Fig. 3.5

By using the same reasoning as in the proof of Prop. 3.15, this defines an element yOO E EOO which is in af(y) and which satisfies

(y OO , x - y) + f e y) > f(x) 10, -

precisely what is needed. The next two lemmas lead easily to S. Simons' proof of Rockafellar's max­ imal monotonicity theorem for subdifferentials. Lemma 3.22. Suppose that f is a lower semicontinuous proper convex func­ tion on E. If a , (3 > 0, Xo E E and f(xo ) < inf E f + a(3, then there exist x E E and xOO E af(x) such that II x - xo ll < (3 and II xOO I l < a .

Proof. Choose 10 > 0 such that f(xo) - infE f < 10 < a(3 and then choose A such that 101(3 < A < a. 1t follows that 0 E ad(xo) so by Theorem 3.17, there exist x E dom(f) and xOO E af(x) such that II xOO Il � A < a and II x - xo l l � lOl A < (3. With f as in the previous lemma, suppose that x E E (not necessarily in dom(f») and that infE f < f(x). Then there exist z E dom(f) and ZOO E af(z) such that fez ) < f(x) and (z OO , x - z) > O.

Lemma 3.23.

Proof. Fix A E R such that infE f < A < f(x) and let A - f(y) , . K = SUP y E E , y # lI y x II _

We first show that 0 < K < 00 . To that end, let F {y E E: fe y) � A } , so F is closed, nonempty and x tt. F. Since dom(f) =I- 0 , one can apply the separation theorem in E x R to find u OO E EOO and r E R such that f � u OO + r. Suppose that y E E and that y =I- x. If y E F, then =

A - f e y) � A - ( u " , y) - r � I A - ( u " , x) - r l + (u " , x - y)

52

Section 3

hence

).. - f(y) < I).. - (u*, x) - r l + * . Ilu lI lI y - x ll - dist(x, F)

If y i F, then II;�W < O . In either case, there is an upper bound for t�Si? , so K < 00. To see that K > 0, pick any y E E such that J(y) < ).. . Since ).. < f(x), we have y -:f x and K � �;{W > O. Suppose, now, that 0 < f < 1, so that (1 - f) K < K and hence, by definition of K, there exists Xo E E such that Xo -:f x and

).. - f(xo) > (1 - f)K. II xo - x II For z E E, let N(z) = K ll z - x lli we have shown that (1 - f)N(xo) + f(xo) < ).. , that is, (N + f)(xo) < ).. + fN(y). We claim that ).. ::; inf E (N + f) Indeed, if z = x, then we have ).. < J(x) = (N + f)(z), while if z -:f x, then ���Si� ::; K, from which it follows that ).. ::; (N + f)(z). Thus, we have shown that there is a point Xo E E, Xo -:f x, such that (N + f)(xo) < infE (N + f) + d{lI xo - x ii · We now apply Lemma 3.22 to N + f, with (:J = Il xo - x ii and a = fK. Thus, there exists z E dom( N + f) == dom(f) and w* E 8( N + f)( z) such that l iz - xo ll < II x - xo ll and II w* 11 < fK. It follows that l iz - xii > O. From the sum formula (Theorem 3.16), 8(N + f)(z) = 8N(z) + 8f(z), so there exist y* E 8N(z) and z* E 8J(z) such that w* = y* + z*. Since y* E 8N(z), we must have (y*, z - x) � N( z) - N(x) K ll z - x ii . Thus (z *, x - z) = ( y* , z - x) + (w *, x - z) � K ll z - x ll - ll w * II ' II x - z ll > > (1 - f)K ll z - x ii > O. Since z* E 8f(z) , we have f(x) � f(z) + (z*, x - z) > f(z), which completes =

the proof.

If f is a proper lower semicontinuous con­ vex function on a Banach space E, the its subdifferential 8f is a maximal monotone operator.

Theorem 3.24 (Rockafellar).

Proof. Suppose that x E E, that x* E E* and that x* i 8f(x). Thus, 8(f - x*)(x), which implies that inf E (f - x*) < (f - x*)(x). By Lemma 3.23 there exists z E dom(f - x*) == dom(f) and z* E 8(f - x*)(z) such that (z*, z - x) < O. Thus, there exists y* E 8f( z) such that z* = y* - x*, so that (y* - x*, z - x) < O . o i

In Section 2 we noted that the sub differential 8f of a continuous con­ vex function f is cyclically monotone, but we did not need the continuity

3. Lower semicontinuous convex functions.

53

hypothesis: If f is a convex proper lower semi continuous function on E, if xo , Xl , . . . , X n Xo are in D(8f) and if xi E 8f(Xk), k = 1 , 2, . . . , n, then =

Thus, 8f is both cyclically monotone and maximal monotone. Definition 3.25. A monotone operator T is said to maximal cyclically mono­ tone provided T = S whenever S is cyclically monotone and G(T) C G(S).

Clearly, a maximal monotone operator which is cyclically monotone is neces­ sarily maximal cyclically monotone.

It is an interesting fact that subdifferentia� are the only maximal cyclically

monotone operators.

Proposition 3.26 (Rockafellar). 1f T: E -+ 2 E * is maximal cyclically mono­ tone, with D(T) =I- 0, then there exists a proper convex lower semicontinuous function f on E such that T = 8f.

Proof. Fix Xo E D(T) and Xo E T(xo). For x E E define f(x) = sup { (x;" X - x n ) + (x� _ l ' X n - X n - l ) + . . . + (X � , X l - xo)} where the supremum is taken over all finite sets of elements Xk E D(T) and xi E T(Xk), k = 1, 2, . . . , n, n = 1 , 2, 3, . .. Since f is the pointwise supre­ .

mum of a family of continuous affine functions, it is convex and lower semi continuous and f(x) > 00 for all x. To see that f is proper, we can use monotonicity of T to show that f( xo) ::; O . Indeed, given any sum of terms of the form entering into the definition of f(xo) , let Yk X n -k and Yk x � k' k = 0, 1, . . . , n. The resulting sum is now the negative of a typical cyclic sum, hence is at most equal to O. By the cyclic maximality of T, to conclude that T = 8f we need only show that G(T) C G(8f) Suppose, then, that X E D(T) and X* E T(x). We will have (x, x*) E G(8f) if we can show that -

=

=

_

.

(x*, Y - x) ::; f(y) - A for all y E E, whenever A < f( x). ( Note that by taking y = Xo , this will imply that - A � (x*, xo - x) when­ ever A < f(x), which shows that f(x) < 00.) Now, by definition, there exist Xk E D(T) and xk E T(Xk), k 1 , 2, . . . , n, such that =

Let X n+ l = x, x�+ I

x*. For any y E E f ( y) � (X�+ l ' Y - X n+ l ) + (x�, Xn + l - X n ) + . . . + (x � , X l - XO) > (X� l ' y - X n l ) + A (X* , y - x) + A, + + =

=

which completes the proof.

>

54

Section 3

It is obvious that if f and g are proper lower semicontinuous functions which only differ by an additive constant, then af ago Rockafellar [R03] (see also [Tay]) has proved the converse; in particular, this shows that if T is a maximal cyclically monotone operator (so that T = af for an appropriate =

I), then - within an additive constant - the latter is unique.

Consider the following assertion concerning two maximal monotone oper­ ators S and T:

If D(S) n intD(T) =f 0, then S + T is maximal monotone. (3.4) (By definition, D(S + T) = D(S) n D(T)). Theorem 3.16 (that the subdiffer­

ential of the sum of two convex functions is the sum of their sub differentials ) shows that (3.4) is true whenever S = 8f and T = 8g, where f and g are proper lower semicontinuous convex functions. Indeed, if intD(T) =f f.), then [since int dome 8g) C int dom(g) and g is continuous on int dom(g)] we con­ clude that g is continous at some point of D( S) n D(T) C dom(f) n dom(g), so Theorem 3.16 implies that S + T = 8(f + g) and Theorem 3.24 implies that the latter is maximal monotone. The remark following Theorem 3.16 shows that, even in a two dimensional Banach space, the conclusion of (3.4) can fail if D(S) n intD(T) 0; in that example, the graph of af + ag is strictly contained in that of the maximal monotone operator a( f + g). Rockafellar [Ros] has shown that (3.4) is indeed true for arbitrary maximal monotone operators S and T provided E is assumed to be reflexive, a fact that has been very useful in nonlinear analysis. Its validity for nonreflexive spaces remains an interesting open question. We conclude this section by taking a brief look at a class of maximal monotone operators which do not arise from convex functions (that is, are not cyclic), but which do arise as subdifferentials of a certain class of functions. =

Definition 3.27. (a) Let E and F be Banach spaces. We say that a mapping K : E x F -+ R U {±oo } is a saddle function provided it is concave in the first

variable and convex in the second; more precisely, we require that the function x -+ -K(x, y) [resp. y K(x, y)] be a convex function for each fixed y E E [resp. each fixed x E E]. (b) The domain or effective domain dome K) of K is defined to be the set of points (x, y ) where K is finite-valued and -+

K(x', y) < 00 for all x' E E and K(x , y') > - 00 for all y' E E. We say that K is proper if its domain is nonempty. (c) If K is a proper saddle function we define its subdifferential 8K in the following way: Let a1K(x, y) c E* be the sub differential of the convex function -K(·, y) at the point (x, y) E dom(K) and let fhK(x, y) C F* be the sub differential of K (x, . ) at the same point. For any (x, y) in E x F define 8K(x, y)

=

a1K( x , y) x 82K(x, y) c E* x F*.

3. Lower semicontinuous convex functions.

55

Note that if we put any reasonable norm on E x F, then it is a Banach space ( take, for instance, II (x, y) 1 I = II x ll + Il y lI ) and its dual can be identified with E* x F* , using the pairing ((x * , y* ) , (x, y»)

= (x * , x) + (y * , y) .

3.28 Examples. ( a) Let E = F be Hilbert space and define K(x, y ) = ( 1/2) [ lIyI 1 2 - lI x l n In view of the fact that the derivative of ( 1 /2)11 ' . 11 2 at u is the functional (u, . ) , that is, can be identified with u itself, we see that 8K(x , y) = {(x, y)}.

( b ) Again, let E = F be Hilbert space and define K(x, y) = IIyl 1 2

-

(x , y).

Here we have 8d{(x, y) = { y} and 8d{(x, y) = {2 y - x}, so

8K(x, y) = {( y , 2y - x)} .

Proposition 3.29. If K : E x F --+ R U {±oo } is a proper then its subdifferential oK : E x F --+ 2 (E x F) * is monotone.

saddle function,

Proof. We must show that ((x � , yn - (x; , y; ) , (X I , YI ) - (X 2 , Y2 ») 2::

0

whenever (Xt, YI ) , (X2 , Y2 ) E E x F and that (xi, yi) E OK(Xi, Yi ), i = 1 , 2. Equivalently, given the definition of the pairing between E x F and E* x F* , we want that is,

(3.5)

Since xi E oI K (Xi, Vi ) and yi E 02K (Xi, Vi ), i = 1, 2, we have for all points (x, y) E E x F (3.6) (x � , x - Xl) � -K(x, yt } + K (X I , YI ), (3.7) (X ; , X - X 2 ) � -K(x, Y2 ) + K(X2 , Y2 ),

(3.8) (y; , y - YI ) � K (X I , Y ) - K (x t , y t } , (3.9) (Yi , Y - Y2 ) � K (X2 , y) - K (X 2 , Y2 ) ' We want to evaluate these four inequalities at X2 , X l , Y2 and YI respectively

and then add them to get (3.5) on the left side and 0 on the right side, but we must first check that the resulting cancellations on the right side do not

56

Section

3

involve adding 00 to - 00. By choosing (x, y) to be any element of dom(K), we know-by definition-that - K(x, yJ ) < 00, so (1) implies that K(Xl ' Y l ) > - 00. Also, K(Xl ' Y ) < 00, so (3.8) implies that K(Xl , yt } < 00. Similar reasoning using (3.7) and (3.9) shows that K(X2 , Y2) is also finite. Now, carry out the substitutions indicated at the beginning of this paragraph; it follows that K(X2, Yl ) and K(X l , Y2 ) are also finite valued and we can now sum all four inequalities to get the desired result. Under appropriate topological hypotheses on K (that it be proper and "closed" in a certain reasonable sense) 8K will be maximal monotone; see

[R04] .

Definition 3.30. Let E be a Banach space and suppose that K is a proper saddle function on E x E. Define TK: E --+ 2 E * by

X * E TK(X) provided (x* , x * ) E 8K(x, x ) . The graph G(TK ) of TK is readily seen to be monotone, since it is just the intersection of G( 8K) with the linear subspace D x D* of the product space (E x E) x (E* x E* ), where D [resp. D*] is the diagonal in E x E [resp. E* x E*] .

3.31 Example. Each of the saddle functions J( in Example 3.28 induces the same monotone operator TK , since TK (x) = x for all x in each case.

In recent work, E. Krauss [KrtJ (see also [Kr2] and references therein) has shown that every maximal monotone operator T: E 2 E* is of the form T = TK for some proper closed saddle function K on E x E. The previous example shows that K need not be unique, and Krauss has devoted consider­ able effort to showing how K can be chosen to have certain special properties. A different representation of general monotone operators on E (as sub differ­ entials of convex functions on E x E* ) has been investigated by S. Fitzpatrick [Fi2J, who poses a number of related open problems. --+

Remarks. The elementary properties of lower semicontinuous convex functions and their subdifferentials play a fundamental role in convex analysis (an approach to the calculus of variations and optimization which replaces differentiable functions by convex functions). Thus, any contemporary text on that subject will have some overlap with the early portions of this section (and with the first section). See, for instance, Aubin and Ekeland [Au-Ek] and Ekeland and Temam [Ek-T]. There was some reluctance on our part to abandon the original easily visualized geometric approach to the Bishop-Phelps theorems in favor of the more "analytic" approach, using Ekeland's variational principle. (M. Fabian [Fa2 ] has reversed this approach, showing how to deduce Ekeland's principle and the Br�ndsted-Rockafellar theorem (3.17) from a Bishop-Phelps lemma. ) In the first edition of these notes [Ph4] we stayed with the variational approach in part because it was a key step in a slightly

3. Lower semicontinuous convex functions.

57

cumbersome but very useful result by J. Borwein [Bor d which led to the Bri1lndsted­ Rockafellar theorem and, especially, to a new proof of Rockafellar's theorem (3 .24) on the maximal monotonicity of the subdifferential map (of a lower semi continuous proper convex function ) . The recent, dramatically simpler proof of the latter theo­ rem by S. Simons lSi) changed all that. While it still seems desirable first to prove Ekeland's variational principle ( since it is the most immediate and fundamental ex­ ample of a perturbed optimization theorem ) , it is now possible to omit a great deal of material which had been preparatory to Rockafellar's theorem. Most books which present the latter avoid giving a proof (referring the reader to Rockafellar's second of three proofs [R03 )) and at least one book reproduces Rockafellar's incorrect first proof.

Section 4

4. S mo oth variational principles, Asplund spaces, weak

A � plund spaces.

It is clear that Ekeland's variational principle (Lemma 3.13) is an extremely useful form of the "maximality points lemma" (3.12); it was a key step in a sequence of fundamental results. As shown in Ekeland's survey article [Ek] , it has found application in such diverse areas as fixed-point theorems, non­ linear semigroups, optimization, mathematical programming, control theory and global analysis. Recall the statement: IT f is lower semicontinuous on E, f > 0 and Xo is such that f( x o ) infEf + then for any >. > 0 there exists v E E such that

�

to ,

>' lI x Q

-

v II � f( x o ) - f(v) � f and f(x) + >' lI x v II > f(v) whenever x =f v. -

One drawback to this result is that, even if f be differentiable, the per­ turbed version f + >' 11 0 v II will not be differentiable at v. This objection was first overcome by J. Borwein and D. Preiss [Bor-P] (see Theorem 4.20 below). A substantially simpler version was later obtained by R. Deville, G. Godefroy and V . Zizler [D-G-Z1,3J, and that is what we present next. In order to han­ dle Gateaux and Frechet differentiability (and other kinds) simultaneously, we require the following definitions. -

Definition 4.1.

(a) A bomology on E, denoted by (3, will be any family of bounded sets S whose union is all of E, which is closed under reflection through the origin (that is, S E (3 implies - S E (3), under multiplication by positive scalars and is directed upwards (that is, the union of any two members of (3 is contained in some member of (3). There are many possibilities, but the following choices for (3 are of main interest to us: (i) Denote by (3 = G the Gateaux bornology consisting of all finite symmetric sets. (ii) By H we denote the Hadamard bomology, consisting of all compact symmetric sets. (iii) Let W denote the weak Hadamard bomology, consisting of all weakly compact symmetric sets. (iv) Finally, F denotes the Frechet bomology consisting of all bounded sym­ metric sets. It is clear that G and F are the smallest and largest possible bomologies.

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

59

(b) A real-valued function f is said to be (3-differentiable at x and x* E E* is called its (3-derivative at x, if for each S in (3,

f(x + ty) - f(x) x * , y -( ) t

-t

0 as t

-t

0+

,

uniformly for y E S. The symmetry and directed properties of a bomology imply that this right-hand limit is in fact a two-sided limit. We denote the (3derivative of f at x by \7 fd( x); it is clear that, in terms of our earlier notation, \7af(x) df(x) and \7 F f(x) f'(x) . =

=

Definition 4.2. A bump function on E is a real-valued function ¢l which is bounded and has bounded nonempty support supp(¢l) {x E E: ¢l(x) 1= O}. We will say that the Banach space E has property ( Hp) provided there exists on E a bump function b which is (3-differentiable and globally Lipschitzian. It is straightforward to verify that a (3-differentiable function f is Lipschitzian if and only if its (3-derivative x \7 pf( x) is bounded on E. =

-t

Proposition 4.3. If E admits an equivalent (3-differentiable norm (at nonzero points), then it necessarily has property (Hp). R be any C 1 function with bounded derivative and nonempty support contained in the interval [! ,. �] and define ¢l( x) = 'IjJ( I x iI ); this is (3-differentiable and vanishes if II x ll < 1/2 or Il x ll > 3/2. Since the

Proof. Let 'IjJ: R

-t

derivative of the norm (at nonzero points) always has norm one, the chain rule shows that ¢l has bounded derivative. The converse to this proposition fails dramatically: R. Haydon [Ha2] has constructed a compact Hausdorff space X with the property that there exists

a Lipschitzian Frechet differentiable bump function on C(X) but the latter does not even admit an equivalent Gateaux differentiable norm.

Definition 4.4. A function f: E ( -00, 00] attains a strong minimum at y E E if f(y) infE f and ll Yn - y ll 0 whenever Y n E E and f(Yn ) -t f(y)· IT f is bounded on E, we define Il f lloo sup { If(x)l: x E E}. -t

-t

=

=

The following general theorem of Deville, Godefroy and Zizler [DGZ3] will yield some important corollaries, which are obtained by making judicious choices of the Banach space F.

Let E be a Banach space and F a Banach space of continuous bounded real-valued functions 9 on E such that (1) IIg lioo ::; II g llF for all 9 E F. (2) For each 9 E F and x E E, the function y gx (Y) = g(x + y) is in F and II gx llF II g II F .

Theorem 4.5.

-t

=

60

Section 4

(3) For each g E F and a E R, the function y g(ay) is in F. (4) There exists a bump function in F. If f: E ( - 00, 00] is proper, lower semicontinuous and bounded below, then the set G of all g E F such that f + g attains a strong minimum on E is a dense Go subset of F. -t

-t

Proof. It is helpful to introduce a notion which is analogous to the notion of a slice used in Section 2. If g: E ( - 00, 00 ] is lower semicontinuous and bounded below, we define, for any a > 0, the closed set -t

S(gj a) = {x E E: g( x) ::; infE g + a}. Tt. is easy to verify that if a > 0 and g l , g2 are both bounded below and satisfy ( 4.1 ) gl ::; g2 + a /3 and g2 ::; g l + a/3, then S(g l j a/3) C S(g2j a). Define Un = {g E F: diam S(f + gj a) < l/n, for some a > O}. We will show that each of the sets Un is dense and open in F and that their intersection is the desired set G. To see that Un is open, suppose that g E Un , with a corresponding a > O. Then for any h E F such that Il g- h llF < a/3, we have Il g- h lloo < a/3 and hence the functions gl = f +g and g2 = f + h satisfy (4.1 ) . Thus S(f+hj a/3) C S(f+gj a) and therefore diam S(f+h; a/3) < l/n, so h E Un . To see that each Un is dense in F, suppose that g E F and e > OJ it suffices to produce h E F such that II h ll F < e and for some a > 0, diam S(f + g + h; a) < lin. By hypothesis, F contains a bump function b. Without loss of generality, Il b llF < e. By hypothesis ( 2 ) , we can assume b(O) t 0 and therefore that b(O) > O. Moreover, by hypothesis (3), we can assume that supp( b) C B(Oj 1/2n ) . Let a = b(0)/2 and choose Xo E E such that

(f + g)(xo ) < irl (f + g) + b(0)/2. Define h on E by h(x) = - b(x - xo); by hypothesis ( 2 ) , h E F and Il h l lF = II b llF < e and h(xo) = - b(O). To show that diam S(f + g + h; a) < l /n , it suffices to show that this set is contained in the ball B(xo; 1/2n ) , that is, if I I x - xull > 1/2n, then x � S(f + g + hj a), the latter being equivalent to (f + g + h)(x) > iW (f + g + h) + a.

Now, supp ( h ) C B(xoj 1/2n ) , so h(x) = 0 if I l x - xo ll > 1/2n hence

(f + g + h)(x) = (f + g)(x) � iW (f + g) > (f + g)(xo) - a = = (f + g + h)(xo ) + b(O) - b(0)/2 � iW (f + g + h) + a, as was to be shown. Suppose that g E n Un ; we want to show g E G, that is, f + g attains a strong minimum on E. First, for all n there exists an > 0

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

61

such that diamS(f + g; an ) S lin and hence there exists a unique point Xo E n S(f+g; an ) . Suppose that {yt} C E and that (f+g)(Yk) inf E (f+g) . Given n > 0 there exists ko such that (f + g)(Yk) S inf E (f + g) + an for all k 2: ko , therefore Yk E S(f + g; an ) for all k 2: ko and hence II Yk - xo ll S diam S(f + g; an ) S lin if k 2: ko . Thus, Yk Xo and therefore 9 E G. A simple proof by contradiction show that G c n Un , and completes the proof. -t

-t

The first corollary to this theorem is a version of Ekeland 's variational principle (Lemma 3.13). Whereas the latter produces a perturbation (by a small multiple of a translate of the norm) which attains a strict minimum, this corollary produces a similar perturbation which has a strong minimum. It does not, however, yield any control over the location of the minimum point. Before proceeding, we should illustrate the difference between a strong minimum and strict minimum for such perturbations. Consider, for example, any nonreflexive Banach space E; by James' theorem [Di] there exists a continuous linear functional f of norm one on E which does not attain its norm; that is f(x) < 1 whenever x E E, I I xll :::; 1. [Concrete example: f(x) = E2- n xn on E = co.] Thus, if x "I 0, then f( -x/llxll) < 1, that is f(x) + IIxll > 0, which means that f + 1 1 · 1 1 attains a strict minimum at 0. On the other hand, since IIfll = 1 , there exists a sequence {xn} C E, I Ixn ll = 1, such that f(xn) -+ - 1 and hence f(xnH llxnll = f(xnH 1 -+ 0, although Xn f+ 0.

Suppose that f: E ( - 00, 00] is proper, lower semicontinu­ ous and bounded below. Then for all € > 0 there exists Xo E E such that

Corollary 4.6.

and the perturbed function x at Xo .

-t

-t

f( x) + € II X - Xo II attains a strong minimum

Proof. Let F be the space of all bounded real-valued Lipschitz continuous functions 9 on E with II g l iF = Il g lloo + Il g ll Lip , where I g(x) - g(Y)1 Il g ll Lip = sup{ II x y ll : x, Y E E, x =f Y } . _

It is straightforward to prove that F is a Banach space which satisfies hy­ potheses (1) through (3) of Theorem 4.5. To verify hypothesis (4), one can apply the construction in Prop. 4.3 to the norm in E to produce a bounded Lipschitzian bump function. Thus, there exists 9 E F such that Il g i l F < € and f + 9 attains a strong minimum at some point Xo E E. Hence, for all x E E,

Ig(x)1 < €, I g(x) - g(xo) 1 S € II X - xo ll and (f + g)(x) 2: (f + g)(xo). It follows that for all x we have f(x) 2: f(xo)+g(xo) -g(x) 2: f(xa ) - € II X - Xo ll ; that is, x -t f(x) + € II X - xo ll attains its minimum at Xo . Also, f(x) 2: f(xo) + g(xo ) - g(x) 2: f(xo) - 2€

62

Section 4

inf E f 2: f(x o ) - 2 € . To see that X o is a strong minimum, suppose that f( Yn ) - g(x o ) + ellYn - x o ll - 0; since

SO

f( Yn ) - f(xo ) + el l Yn - x o l l

2:

(f + g)(Yn ) - (f + g)(x o ),

we must have Yn - X o . Our next application of Theorem 4.5 uses a different choice for the Banach space F.

Df3 denote the linear space of all bounded Lipschitz continuous real-valued functions g: E - R having bounded ,B-derivative 'Vf3g, provided with the norm Definition 4.7. Let

I l lg l l l = I l g li oo + II 'V f3gl l oo ==

sup{ l g(x) l : x E E} + sup{II'Vf3g(x) l I : x E E}.

It is clear that I I I . I I I makes Df3 into a normed linear space, which will be nontrivial whenever hypothesis ( H(3) is satisfied. Completeness of this space is obvious, until someone insists on the details. They are given in the following proposition. Proposition 4.8.

The space (Df3, I I I · l i D is complete.

Proof· For simplicity, if 9 E Df3, we will write g' in place of 'Vf3g. By the mean value theorem [FIl, for any f E Df3 and x, Y E E we have Ig(x) - g( Y )1 � 119'1100 ' 11x - y l I · Suppose, now, that {gn } is a I I I · I I I-Cauchy sequence in Df3. Then both {gn } and {g�} are uniformly Cauchy, so there exists g : E _ R and h: E - E* such that {gn } converges uniformly to 9 and {g� } converges uniformly to h. The boundedness of the sequence {g�} implies that there is a bound on the Lipschitz constants for {gn } , hence 9 is Lipschitzian. It remains to show that 9 is ,B-differentiable and that g' = h, that is, that for each x E E and S E ,B, the difference quotients g(x + t y) - g(x)

t

converge as t - 0+ to (h( x), y) , uniformly for y E S. To this end, fix x E E and S E ,B and let M = sup{II Y II : y E S} . Define, for t E R, t =I 0 and Y E S, gn (x + ty) - gn (x) .J.. 'l' n (t , y ) =

t

while ¢> n (O, y ) ( g�(x), y). Given > 0, the Cauchy property implies that there exists n such that I l g!,. - g� lI oo < provided m 2: n. The mean value theorem cited above shows that =

t:

t:

(4.2)

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

63

for m 2:: n, t E R and y E S. Since g n is ,a-differentiable, there is a b > 0 such that l ¢in (t, y) - ¢in (O, y) ::; f for all y E S and 0 < t < b. Combined with (4.2), this shows that if m 2:: n, 0 < t < b and Y E S, then

l ¢im (t, y) - ¢in (O, y)1 ::; (1 + M)f .

(4.3)

By hypothesis, if t =f 0, then

¢in (t, y) --. ¢i(t, y) == g(x + ty; - g(x) ,

while (g� (x), y) --. (h(x), y) . Let m 00 in (4.3) to get l ¢i(t, y) - ¢i n (O, y) l ::; (1 + M)f. Let m --. 00 for t = 0 in (4.2) to get I (g(x), y) - ¢i n (O, y) 1 ::; fM. From the last two inequalities --.

I

h(x + ty; - h(x)

_

(g(x), y)1 ::; (1 + 2M)f

for all 0 < t < b and Y E S, which completes the proof. The next corollary to Theorem 4.5 (which uses F D 13) seems startling until one recalls that the intersection of a countable collection of dense G6 subsets of a Banach space is itself a dense G6. =

Corollary 4.9. Suppose that E satisfies property (Hp) and that for each n, the function fn : E (-00, 00] is proper, lower semicontinuous and bounded below. Then for any f > 0 one can choose 9 E Dp such that II g l i oo < f, IIV'pg lloo < f and each of the functions fn + 9 attains a strong minimum on --.

E.

Suppose that the Banach space E satisfies (Hf3) and that f is a proper lower semicontinuous function on E which is bounded below. Then there exists a constant a > 0 (depending only on E) such that for all 0 < f < 1 and for any Yo E E such that f(yo ) < inf f + af 2 , there exist 9 E Df3 and Xo E E such that a) f + 9 has a strong minimum at Xo b) II g l i oo < f and lIV'f3 g lloo < f c) II xo - Yo II ::; f . Theorem 4.10. (DGZ Smooth Variational Principle)

Proof. Let b E Df3 be a bump function with supp(b) C B(Oj 1) and b(O) 2:: 1. By using an appropriate composition, we can assume that 0 ::; b ::; 1 and b(O) = 1. [Indeed, let ¢i: R --. [0, 1] be a Cl function which is monotonically nondecreasing and satisfies ¢i(0) = 0 and ¢i(1) 1. Then ¢i 0 b has bounded derivative, is 0 outside B(Oj 1) and ¢i(b(O)) 1.] Define 1 a= M = max{lIV' f3 b lloo , I } , 4M ' and =

=

64

Section 4

x - Yo x E E. h e x ) = f(x) - 2aE2 b(-- ), 10 Note that h: E -+ ( - 00 , 00) is lower semicontinuous and bounded below. Thus, by applying Theorem 4.5 to F = Dp, there exists k E Dp with Il k lloo < aE2 /2 and II Vpk lloo < 10 / 2 such that h + k attains a strong minimum on E at xo, say. We show first that II xo - Yo II :::; E. Suppose not; then II xo - Yo II > 10, that is, XO �YO ¢ B(O; 1) hence bCO�YO ) = 0 so h(xo) = f(xo) � inf E f. Also, h(yo ) = f(yo ) - 2aE2 < inf f - aE2 .

E

But (h + k)(xo) :::; (h + k)(yo ) so h(xo) :::; h(yo ) + k(yo ) - k(xo ) :::; h(yo) + 2 11 k ll00 < h(yo ) + aE2 ,

hence inf E f < h(yo ) + aE2 , a contradiction. Next, let x - Yo g(x) = - 2aE2 b( --) + k(x), 10

so 9 E Dp and f + 9

=

h + k attains its minimum at Xo . Furthermore,

II g l i oo :::; 2aE2 11 b ll00 + II k lloo :::; 2aE2 + aE2 /2 <

(since 2a

=

21 <

10

� hence a/ 2 < 1 / 8). Finally,

which completes the proof. In order to apply this result, it is extremely useful to extend the no­ tion of "subdifferential" from convex functions to arbitrary lower semi con­ tinuous extended real-valued functions; in fact we will define the notion of ",8-subdifferential" ( as well as ",8-superdifferential") for such functions. Definition 4.11. Let f: E ( ) be lower semicontinuous and suppose that f(x) is finite. We say that f is ,8-subdiiJerentiable at x and that x* E E* is a ,8-subdiiJerential of f at x if, for eacht 10 > 0 and each set S in the bomology ,8, there exists S > 0 such that for 0 < < S -+

- 00 , 00

( x * , y) < _

f(x + t Y ) - f(x) t

+ 10

( 4.4)

for all y E S. We write x* E opf( x). It follows from this definition that if ,81 C ,82 , then op2 f(x) C opJ(x). Now, we already have a definition of a subdifferential for a lower semicontinuous convex function f, so it is important to note that this new definition extends the old one. In fact, for such functions we have opf(x)

=

oaf(x) = of(x)

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

65

for all choices of 13. Indeed, by the previous remark, for any 13 we have 8pf( x) C 8af(x). On the other hand, if x*€8af(x), then for all € > 0 and y E E we have (x*, y) ::; d+ f(x)(y) + so x* ::; d+ f(x) and therefore x* E 8f(x). Finally, note that if x* E 8f( x), then the inequality (4.4) holds for all € > 0, y E E and t > 0; that is, for any 13 €,

8f(x) c 8pf(x). We define 13 -superdifferentials by reversing the inequality (4.4) and replacing by - I'.; we denote the set of all these functionals by 8P f(x). Note that €

The next proposition exhibits the relations between the foregoing notions and f3-differentiability. It also makes (in part (c)) the simple but key observa­ tion connecting the smooth variational principle with subdifferentiability.

(a) If f is lower semicontinuous, f(x) is finite and both 8pf(x) and 8P f(x) are nonempty, then f is f3-differentiable at x, and "pf(x) = df(x). (b) Suppos e that f is concave and continuous in a neighborhood of x. If 8pf(x) is nonempty, then f is f3-differentiable at x and 8pf(x) = {" pf(x)}. (c) If 9 is f3-differentiable at x and f + 9 attains a minimum at x, then f is f3-subdifferentiable at x (that is, 8pf(x) =f 0).

Proposition 4.12.

Proof. (a) Suppose that xt E 8P f(x) and x� E 8pf(x); then for any y in E and for every I'. > 0 we must have (xt , y) - (x�, y) ::; 21'., so xi = x�. Denote the common value by x*; it follows readily from the definitions that x* = "pf(x). (b) Since f is concave and continuous, the convex function - f has a nonempty sub differential and 0 =f 8( - J)( x) = 8p( - J)( x) = -8P f( x) (by the remarks following Definition 4.11). The hypothesis that 8pf( x) be nonempty therefore implies that f is f3-differentiable at x, by part (a). (c) Note first that for all u E E, (-g)(u) - (-g)(x) ::; f(u) - f(x). Let x* = "p ( -g)(x); then given S E 13 and € > 0, there exists 8 > 0 such that < 8 and y E S imply that o
-

(-g)(x + ty) - (-g)(x) t

_

(x* , y ) <- f(x + ty)t - f(x)

_

(x* , y ) ,

which completes the proof.

If E has the property (Hp) and f: E ( -00, 00] is a proper lower semicontinuous function, then {x E dom(J): f is 13 - subdifferentiable at x } is dense in dom(J).

Corollary 4.13.

-t

66

Section 4

Proof. Suppose that f(xo) < 00 and f > O. There exists 71 such that 0 < 71 < f and f( x) > f( xo) - 1 if II x - Xo II :::; 71. Define the proper lower semicontinuous function g: E R U {oo} by g(x) = f(x) if II x - xo ll :::; 71 while g(x) = 00 otherwise, so 9 is bounded below. From property (Hp) it follows that there exists ¢> E Dp such that ¢>(xo) = 0 and ¢> == 2 in E\B(xo; 71/2). Thus, we have (g + ¢» (xo) = f(xo) and (g + ¢» > f(xo) in E\B(xo ; 71/2). By Cor. 4.9, there exists h E Dp such that II h lloo < 1/4 and 9 + ¢> + h attains its minimum at some point X l . It follows that (g + ¢> + h)(x) > f(xo) + 3/4 for X ¢ B(xo; 71/2) while (g + ¢> + h)(xo) < f(xo) + 1/4. Necessarily, then, Xl E B(xO; 71/2). Now, f 9 in B(xO ; TJ/2) and 9 + (¢> + h) attains its minimum at X l ; by Prop. 4.12(c), f is ,a-sub differentiable at X l . -+

=

Suppose that E has the property (Hp), that C is a nonempty open convex subset of E and f: C R is continuous and convex; then f is ,a-differentiable on a dense subset of C. Corollary 4.14.

-+

Proof. Choose an open ball B c C such that B C C. Let 9 = - f in B while = 00 in E\B, so 9 is proper and lower semicontinuous and hence there exists a dense subset of B at each point of which 9 is ,a-subdifferentiable. In particular, there exists X E B such that 9 is ,a-sub differentiable at x. Since 9 is concave and continuous in B, by Prop. 4.12(b), 9 is ,a-differentiable at x, therefore f is ,a-differentiable at x. 9

This corollary does not say that the dense set of points of ,a-differentiability is a G6, but as we have seen (Prop. 1.25), this is automatic for Frechet differ­ entiability, hence we obtain the following corollary. Corollary 4.15. If a Banach space admits a Frechet differentiable bump func­ tion with bounded derivative, then it is an Asplund space.

This corollary is a slightly weaker form of the original theorem by I. Eke­ land and G. Lebourg [Ek-L], who proved it without assuming boundedness of the derivative. By Prop. 4.3, their result (or the one above) solved half of a long-standing open problem, whether the existence of an equivalent Frechet differentiable norm implies the Asplund property. The converse remained open for decades until R. Haydon's ingenious example [Had: There exists a scattered

compact Hausdorff space X with the property that C(X) is an Asplund space but does not even admit an equivalent Gateaux differentiable norm. It had long been known that the converse is valid for separable E; more generally, Fabian [Fal] has shown that this is true if E is assumed to be WCG. (He assumes the formally weaker hypothesis that E be "weakly count ably determined" , but then shows that these are equivalent notions in Asplund spaces.)

If a Banach space E admits an equivalent Gateaux differen­ tiable norm (or, more generally, if it satisfi es property (H G)), then for every

Corollary 4.16.

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

67

nonempty open convex subset D of E and continuous convex function f on D there exists a dense set of points in D where f is Gateaux differentiable. This corollary came close to completely solving another long-standing problem: If a Banach space admits an equivalent Gateaux differentiable norm, is it necessarily a weak Asplund space ? (As was mentioned in Section 2, there exist continuous convex functions whose set G of points of Gateaux differen­ tiability is dense but G does not contain a dense Go subset.) Spaces which sat­ isfy the conclusion of the Corollary 4.16 are called "Gateaux Differentiability Spaces"; see Section 6. The fact that the existence of a Gateaux differentiable norm does imply that the space is a weak Asplund space was proved much later by D. Preiss [P-P-N]; we will pursue this later in this section. Corollary 4.16 is sufficient to yield some negative results concerning smooth renormings.

The spaces C=, L= [0, 1] and C l (r) not admit equivalent Gateaux differentiable norms.

Proposition 4.17.

(r

uncountable) do

Cl (r) is nowhere Gateaux differentiable, and it is an interesting exercise (below) to prove the same fact about L= [O, 1]. Also, in Example 1.21 it was shown that p(x) = lim sup IX n l defines a continuous seminorm on C= which is nowhere Gateaux differentiable.

Proof. We showed in Example 1 . 4(b) that the norm in

4.18 Exercises. (a) Prove that the essential supremum norm on the Banach space LOO[O, 1] is nowhere Ga.teaux differentiable. (b) Prove that any separable Banach space admits an equivalent strictly convex (rotund) norm. (Hint: Use a dual version of the construction in Exercise 2 .40(b) . ) (c) Show that there exists a strictly convex space whose dual norm is not smooth. (Hint: Look at the separable space £1 and Prop. 4. 17) .

We mentioned at the beginning of this section that Borwein and Preiss [Bor-P] were the first to establish a smooth variational principle. We will describe their result here and refer the reader to [Bor-P] or [Ph4] for the proofs. Definition 4.19 Let e denote the family of all real-valued functions 0 on E

of the form

where EJl n = 1, Jl n � ° and {vn } converges in norm to some v E E. Thus, each 0 is a possibly infinite convex combination of squares of translates of the norm, with the translates themselves converging. It can be shown that if the norm in E is ,a-differentiable (at nonzero points), then each function in e is everywhere ,a-differentiable.

68

Section 4

Suppose that g: E ( -00, ooJ is lower semicontinuous and bounded below, that f > 0 and that ,X > O. Assume further that Xo satisfies

Theorem 4.20 (Borwein-Preiss) .

-+

g (xo) <

infE g + f. Then there exist 0 E e and v E E such that 2 9 + (2fj ,X )O attains its minimum on E at v (b)

( ) a

while Il xo

(c)

-

v ii <

,X

and (d)

Moreover, if E has a {3-smooth norm, then 0 is (3-differentiable and 8p g (v ) n (2fj 'x )B* f 0,

(e)

where B* is the dual unit ball. Figure 4.1 below illustrates ( a) , ( b) , ( c ) and ( d) , where ( b ) is written g ( x) � g ( v ) + (2fj,X 2 )[O(V) - O(x)J:

as

inf 9 + 8 E ······················· inf 9 E

•••...••.

�n

••••

g(v) + ( 2 e/A) [e(V) - e(·)]

Xo

Fig. 4.1

One can apply this theorem to prove Cor. 4.13 (using the same arguments ) . While this result is more difficult to prove than Theorem 4. 10 , it has the advantage of giving ( in conclusions ( d ) and (e )) additional information about the minimizing point of the perturbation of g. The Borwein-Preiss theorem was motivated by the following deep theorem due to Preiss [Prj . The proof is long and not easy; we content ourselves with merely giving the statement.

Any locally Lipschitzian real-valued function on an Asplund space is Frechet differentiable at the points of a dense set.

Theorem 4.21 (Preiss).

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

69

We return now to the fact that the existence of an equivalent Gateaux differentiable norm on E implies that it is a weak Asplund space; more gener­ ally, it implies that maximal monotone operators are generically single-valued on E. The technique of proof (which, as we will see later, has application to yet more general results) is due to D. Preiss [P-P-N]. It also uses a topological result called the Banach-Mazur game, which we present next. Definition 4.22. Suppose that X is a topological space and that S is any subset of X; we define a game (X, S) with players A and B, as follows. A play is a decreasing sequence UI J VI J Uz J Vz J . . . of nonempty open subsets of X which have been chosen alternately by A and B: Player A chooses UI , B chooses VI , A chooses Uz, ... etc. A strategy for B is a sequence IE = {In } of maps In where, for each n , In is defined for (U1 , VI , Uz, V2, Un) (the first 2n - 1 elements of a play) and In ( U1 , VI , U2, V2 , . . . , Un) is a nonempty open subset of Un. A play is consistent with IE = Un} provided Vn = In(U1 , VI , U2, V2 , , Un) for each n. We say that IE is a winning strategy for B if n Vn C S for every play consistent with IE. (One can similarly define a winning strategy for A, but for our purposes, it is unnecessary.) • • •

,

. • .

Recall that a set is said to be residual provided its complement is of first category. We only need the "only if" portion of the following theorem; for a proof of the other half and related results, see [Ox]. Theorem 4.23 (Banach-Mazur). Suppose that S is a subset 01 the topo­ logical space X and that A and B are players 01 the game (X, S). There exists a winning strategy fdr B if and only if S is residual in X.

Proof. Suppose there exists a winning strategy IE = Un} for B. By an I-chain of order n we mean a nested sequence of nonempty open sets such that Vi = Ii(U1 , V! , . . . Ui), i = 1, 2, . . . , n. An I-chain of order n + k is a continuation of one of order n if the first 2n terms of each are the same. We partially order the I-chains by continuation. Among all I-chains or order 1, let Fl be a maximal family with the property that any two distinct I-chains in Fl have disjoint smallest elements. This exists, by a straightforward application of Zorn's lemma. Since B has a strategy, the union WI of all the VI 'S belonging to I-chains in FI is dense in X. We now proceed by induction: Suppose that a family Tn of I-chains of order n has been defined so that the corresponding Vn's are pairwise disjoint and their union Wn is dense in X. Among the I-chains of order n + 1 which are continuations of members of Fn, use Zorn's lemma to produce a maximal family Fn+1 with the property that all the corresponding Vn+1 's are pairwise disjoint. The union Wn+l of the Vn+1 's is dense, by maximality and the existence of a strategy

70

Section 4

for B. Having found Fn for all n, let W = n Wn. For each x E W, there exists a unique sequence {en} of f-chains such that en E Fn and x is in the corresponding Vn, for all n. Now, these f-chains en are linearly ordered by continuation, so we have x E n Vn. This sequence is consistent with the strategy fB by definition of f-chain; since B has a winning strategy, x E S. Thus, we have shown that W == n Wn C S and hence X\S C U(X\ Wn) is of first category. Our first application of the Banach-Mazur game is to a proof of the fol­ lowing basic theorem due to Asplund [Asp].

If E is a weak Asplund space and T: E F is continuous, linear and onto the Banach space F, then F is a weak Asplund space.

Theorem 4.24.

-+

Proof. Suppose that D C F is open and convex and that f is a continuous convex function on D. Then DI = T- I (D) is open and convex in E and JI = f oT is continuous and convex on D1 • By hypothesis, there exists a dense Go subset G1 C DI such that dJI (x) exists for all x E G1 • Let G = T(Gd; then G is dense in D and df(x) exists for all x E G. Indeed, since T is onto, its adjoint T* is one-one, and it is simple to see that T*(8f(Tx)) C 8JI(x) for all x E D1 • In general, there is no reason to assume that G is a Go set, but the fact that it contains a dense Go set follows from the following corollary to Theorem 4.23. Let M be a complete metric space, X a Hausdorff space and X a continuous, open surjective mapping. If {Gn } is a sequence of dense open subsets of M, then the image f(G) of G = n Gn is residual in X.

Lemma 4.25.

f: M

-+

Proof. Let S = f(G) and suppose that A and B play the Banach-Mazur game (X, S). The following reasoning will show that B has a winning strategy. Suppose that A has chosen the nonempty open subset U1 of X. Since G1 is dense and open in M, there exists an open metric ball BI = B(XI , rl ) such that 0 < r l < 1 and BI C f-1 (UI) n Gl , so player B chooses VI f(BI ). Player A chooses some nonempty open subset U2 c VI . The set BI n f- I (U2) n G2 is nonempty and open, so there exists B2 = B(X2' r2 ) with 0 < r2 < 1 /2 and B2 C BI n f - I (U2) n G2 ; player B takes V2 f(B2 ). Continuing in this way, for any sequence U1 J VI J U2 J V2 J . . . J Un player B lets Vn = f(Bn), where Bn = B(xn, rn) C Bn - l , 0 < rn < lin and Bn C f - I(Un ) n Gn. These choices define a winning strategy for B, that is, n Vn C f(G). Indeed, by completeness of M, n Bn is a single point, say {xo}. If y E n Vn, then for each n there exists Zn E Bn such that y = f(zn) f(xo) and {xo } = n Bn C n Gn = G hence y = f(xo) E f(G). =

=

-+

We next consider maximal monotone operators on Gateaux smooth Ba­ nach spaces.

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

Definition 4.26. Let E be a Banach space,

on E and D

=

int D(T). Define

I7T(X, y) = sup

71

T a maximal monotone operator

{ (x*, y): x* E T(x)}, x E D, y E E.

(We will usually write I7(X, y) instead of I7T(X, y).) If A is any subset of E* and y E E, we will, for simplicity, write {(A, y)} for the set of real numbers {(x*, y): x* E A}.

We will also need a notion for monotone operators analogous t o the situation when the directional derivative df(x)(y) of f exists ( at the point x E D in the direction 0 =f y E E). Recall that this will be the case if and only if d+ f(x)( -y) - d+ f(x)(y). From Prop. 2.24 we know that d+ f(x)(y) = sup { (x*, y) : x* E af(x)}, the equality above is equivalent to =

so

sup {(af(x), y) } = inf {(af(x) , y)}, that is, this set of real numbers is actually a singleton. This motivates the following definition. Definition 4.27. For any y E E and T as above, let yT denote the set-valued mapping from E into the real line defined by

(yT)(x) O ur substitute for saying that

i8 a 8ingleton.

=

{(x*, y): x* E T(x)}.

df( x )(y) exists will be the assertion that yT( x)

Definition 4.28. The following notation will be convenient : If A C E * , y E E and Q is a real number, the assertion that (A, y) > Q means that (x*, y) > Q for each x* E A.

Let T and D be a8 above. Then (i) For each x E D, the real-valued function y I7T(X, y) i8 subadditive and p08itive homogeneou8 and, for any A > 0, I7,xT(X, y) = AI7T(X, y). (ii) For each x E D(T), sup{ l7 (x, y): lI y ll I } sup{ l7 (x, y): Il y ll ::::; I } = sup{ ll x* lI : x * E T(x)}. Proposition 4.29.

-

=

=

(iii) (yT)(x) i8 a 8ingleton if and only if I7(X, - y) = -17(X, y). (iv) If Xo E D, Y E E and (yT)(xo) i8 a 8ingleton (8ay equal to {Q}), then for all f > 0 there exi8ts a neighbohood U of Xo in D 8uch that (T(U), y} > Q - f. (v) Fixing x E D and y =f 0, letting 1 = {t E R: x + ty E D} and defining f(t) = I7(X + ty, y), t E l,

72

Section 4

the function f is monotone nondecreasing on I (and hence is continuous at all but countably many points of I). Moreover, if f is continuous at to E I, then (yT)(x + to Y ) is a singleton. Proof.

(i) and (ii) are immediate from the definitions. (iii) If (yT)(x) is a singleton, then so is (-y T)(x), with ( -y T)(x) = - O'(x , y); that is, O'(x, -y) = - O'(x , y). On the other hand, if ( yT)(x) is not a singleton, there exists xOO E T(x) such that (xOO , y) < O'(x, y) and hence O'(x, -y) 2:: (xOO, - y) 2:: - O'(x, y). (iv) Given f > 0, let W = {yOO E EOO : (yOO , y) > 0: - f l . This is a weak* open set containing T(xo ), so by the norm-to-weak* upper semicontinuity of T at Xo, there exists an open neighborhood U of Xo in D such that T(x) C W for all x E U, which was to be proved. (v) Note that we are not asserting that the open set I is an interval, but this does not affect our argument. Suppose that t l , t 2 E I with t l < t2 ; then for any xi E T(x + tiY)' i = 1, 2, we have (by monotonicity)

hence (xt , y) ::; (xi, y) . This shows that

f(h ) == sup { (xOO , y ) :

xOO E T(x + tlY)}

in particular, f(td ::; f(t2), so single-valued at x + t o Y. Then 0:

Hence if t E at to.

==

::; inf{ ( xOO , y) :

f is monotone.

xOO E T(x + t2Y)}; (4.5)

Suppose, now, that yT is not

inf { (xOO , y) : xOO E T(x + toY)} < f ( to ).

I, t < t o , then (by (4.5)), f(t) ::; 0: < f( t o), so f is not continuous

The relationship between Gateaux smoothness of the norm and �ingle­ valuedness of T is exhibited by the following lemma.

Lemma 4.30. Suppose that Xo E D and that Yo E E, II Yo II = 1, Yo T is single-valued at Xo, with value 0:. If SUP {O'T (XO , y ): lIyll =

then

I}

is such that

::; 0: ,

T(xo ) c o: ' 81 1 · · 1 1 ( yo).

In particular, if the norm is Gateaux differentiable at Yo, then singleton.

T(xo )

IS

a

Proof. Suppose that xOO E T( xo); then from Prop. 4.29 (ii), we see that 0: 2:: 0 and Il xOO II ::; 0:. Since ( Yo T)(xo) is a singleton, it follows that (x*, Yo} = 0: and so xOO E o: ' 8 1 1 · · 11 ( yo ).

4. Smooth variational principles, Asplund spaces, weak Asplund spaces.

73

Suppose that E admits an equivalent Gateaux smooth norm, that T is a maximal monotone operator on E and that D == int {x E E: T(x) -=I 0} is nonempty. Then there exists a dense G o subset G c D such that T(x) is a singleton for every x E G.

Theorem 4.31.

Proof. Let PI = 11 · 11 denote an equivalent smooth norm on E. Choose sequences of positive numbers 1 /2 > € 1 > €2 > . . . and /31 > /32 > /33 > . such that .

€k

-+

0,

E/3i < 3

and

.

Ey'€k//3k < 00.

To use the Banach-Mazur game we let D be our Hausdorff space and S the set of points in D where T is single-valued. After player A chooses a nonempty open subset U of D, player B's first choice will always be an open subset of U in which T is bounded. Thus, we may always assume that player A's first choice was an open nonempty set U1 in which T is bounded. We may also assume that sup {p; (x*): x* E T[U1 ]} > O.

[Indeed, if this supremum is 0, then B's strategy is obvious: Since T is single­ valued (equal to 0) on the entire set U1 , she need only choose Vk = Uk for each k = 1 , 2, 3 , . . . so that nVk = U1 c S. ] Now, let Sl be the unit sphere {x E E : PI (x) = I} defined by PI and define

8 1 = sup {a(x, y): (x, y) E U1

X

Sd .

4.29 ( ii ) , we can also write 8 1 sup {pr (x*): x* E T(x) and x E Ud ; by our earlier assumption, 8 1 > O. From part ( v ) of Prop. 4.29, for any y E E and x E U1 , there exist points of the form x + ty E U1 with t > 0 such that a(x, y) � a(x + ty, y) and yT is single-valued at x + ty. Thus 8 1 = sup {a(x, y): (x, y) E U1 X Sl and yT is single-valued at x}. It follows that there exists (X l , Y1 ) E U1 X Sl such that Y1 T i s single-valued at X l and such that (T(xI ), yI) > (1 - €I )8 1 . By Prop. 4.29 ( iv ) , there exists r1 , with 0 < r 1 < 1, such that B(XI, 2rd C U1 and (T[B(x 1 ; 2rI)], Y1 ) > ( 1 - €I )8 1 ' Define VI = B(x 1 , rI ). For all x E E define q1 (X) = dist (x, RxI) == inf { l i x - A xI lI : A E R} and define a new norm P2 on E by P22 - p21 + /312 q21 ' We now have 8 1 , Y1 , X l , P2 and player B's open set VI , so player A may choose any nonempty open subset U2 C VI ' Using a similar strategy to respond to From Prop.

=

_

74

Section 4

player A's choices U2 , U3 , at every step, player B chooses VI , V2 , . . . by constructing sequences of numbers S I , S2 , . . . , S k , . . . , norms PI , P2 , · · · , Pk , · · · (with dual norms pk) , spheres Sl , S2 , . . . , Sk = {x E E : Pk (X) = I } , . . . containing the vectors YI , Y2 , . . . , Yk , . . . respectively, points X l , X 2 , . . . , X k , . . . and positive radii r l , r2 , . . . , r k , . . such that Vk = B(x k , rk), • • •

.

2 2 Pk = P k-l

2 · 2 - 2 + f3k-l qk-l = P I

(where qk(X) = inf { ll x - '\x k l i : ,\ E R} ,

k-l 2 2 + '" L.J f3j qj , j= l

X E E),

S k = sup {17 ( x , y): ( x , y) E Uk X Sd = sup {pk( x * ): X * E T(Uk)}, Yk T is single-valued at X k and

Since B(x k , 2rk )

Vk-l = B(X k-l , rk-t}, we have r k ::; (lj2)rk-l ::; (lj22 )rk_2 ::; . . . ::; (lj2k-1 )rl < Ij2 k-1

as well as Vk

C

C

Uk

Uk for each k. Also, Pk � Pk-l implies that Sk

so, using Uk

C

C

C

Bk-l = {x E E : Pk-I (X) ::; I},

Uk-l and Prop. 4.29(ii),

S k ::; sup {17 ( x , y): ( x , y) E Uk-l x Bk-d = = sup {l7 ( x , y): ( x , y) E Uk-l x Sk-d = S k-l . Note that Pk (Yk- l ) = Pk-I ( Yk- l ) = 1 and X k E Uk C Uk- I , so necessarily (X k , Yk-l ) E Uk x Sk hence S k � 17(X k , Yk- d , while X k E Uk-l implies that 17(X k , Yk-l ) > ( 1 - €k-t }Sk-l . Thus, S k > ( 1 - €k- t }S k-l . Since S I > 0 and € 1 < 1 , this implies that S2 > OJ by induction, S k > 0 for all k. It follows that the decreasing sequence {s k } converges to some number Soo � O. Also, since the diameters of the sets Vk converge to zero, the intersections of their closures consists of a single point, denoted by Xooj necessarily, Xoo E Vk for all k. In order to be able to apply Lemma 4.30 and complete the proof, we will show that (i) the sequence of norms {Pk } converges to an equivalent smooth norm Poo satisfying PI ::; Poo ::; 2PI , (ii) the sequence of vectors { Yk } is convergent, with limit Yoo satisfying Poo( Yoo) = 1 and (iii) YooT is single-valued at x oo, with value Soo. Assuming that we have proved assertions (i), (ii) and (iii), we next prove the inequality in Lemma 4.30, taking, of course, Xoo and Yoo in place of Xo

4 . Smooth variational principles, Asplund spaces, weak Asplund spaces.

75

and Yo , Soo in place of a and Poo for the nonnj that is, we want to show that a(xoo , y ) :::; Soo whenever Poo( Y ) = 1 . For each k,we have Xoo E Uk and Pk ( y ) -l y E Sk and therefore a(xoo, y) = Pk ( y) . a (xoo, Pk ( y ) - l y ) :::; S k . Pk ( Y ) · This last term converges to Soo ' Poo( Y ) = Soo, so the inequality of Lemma 4 . 30 is satisfied and therefore T(xoo) c Soo ' opoo( yoo). Since poo is smooth, T(xoo) is a singletonj that is {xoo } = nVk c S. We have thus described a winning strategy for player B, so by Theorem 4.23, S contains a dense Go subset (the open set D being completely metrizable). It now remains to prove assertions (i), (ii) and (iii). (i) Since qk :::; P I for each k, we have

p� :::; p�

k- l k-l � ; ; (1 = p + L P q :::; + L P; ) . p� :::; 4 . p� , j=l j= l

hence the increasing sequence {pd of equivalent nonns converges unifonnly on bounded sets to a norm Poo satisfying P I :::; Poo :::; 2PI ' To prove that Poo is Gateaux smooth, note first that each of the functions q% is everywhere Gateaux differentiable. In fact, if q%( y ) = 0, then it is easily seen that oq%(y ) = {OJ. If . q�( y ) > 0, then for some Ao E R,

and hence, for any t > 0 and u E E, 0 :::; r l [qk( Y + tu ) + qk( y - tu ) - 2qk( y)] :::; l :::; r [pI ( y - Ao Yk + tu ) + P I ( Y - AOYk - tu ) - 2pI ( Y - AOYk)] .

(4.6)

Since Gateaux differentiability is characterized by the fact that such difference quotients converge to zero (Exercise 1 .24), Gateaux differentiability of P I at points where it is positive implies that the last term in (4.6) converges to zero, hence so does the first tenn. It now follows by standard arguments that the infinite series defining p� is everywhere Gateaux differentiable, hence Poo is Gateaux differentiable at nonzero points. (ii) To show that { Yd converges, we will show that for all k,

since the series EV€k/ Pk converges, this will suffice to show that { Yk } is a Cauchy sequence, hence is convergent to Yoo, say. To obtain the estimate above, recall first that S H I > (1 - fk )S k and f k > f k +I so

a(x HI ' YH . ) > ( 1 - f H . )S H I > (1 - f H . )( l - fk)S k > (1 - 2fk)S k > 0, while - since (X H I , Pk (Yk+I ) - I YH I ) E Uk X Sk - we have

Section 4

76

I 1 - 2 fk > O. By the definition of PHI ,

j3� q � (YHI ) = P �+I (YHI ) - p� (YkH ) < 1 - (1 - 2fk ) 2 < 4fk . Consequently, dist (YHI , Rek) = qk(YHI ) < 2Vfk/ j3k. We can write a nearest point in Rek to Yk + 1 in the form AkYk for some Ak E R; that is

YHI = AkYk + U k , where IIU k l1 = qk(YHI ) < 2 ,j€k/j3k . Now, PHI � Poo � 2PI , so PkH (Uk) � 4Vfk/j3k. Since, as noted earlier, Pk +I (Yk ) = Pk (Yk ) = 1, we have

IAk l = PHI (AkYk ) = PHI (Yk+1 - Uk) � 1 - PHI ( Uk) � 1 - 4 ,j€k/ j3k . Also,

I Ak l � Pk+ I (YHI ) + PH I ( -Uk) � 1 + 4 ,j€k/j3k . We next show that if k is large enough so that fk+ 1 + 4Vfk/j3k < 1 , then the constant Ak is positive. Indeed, since Xk+ 1 E Vk for all k, we must have o < (1 - fk )Sk < (1 - fk)Sk . (Note that by Prop. 4.29 (ii), Pk+l ( x k ) � S H 1 . ) Since (Yk+I T) ( X HI ) is a singleton, we must have ( Xk , YHI ) = ( 1 - fH t )S k+1 . It follows that (1 - fH t )S k+1 < ( x k' Akek + Uk) = A k (x k , Yk) + ( X k ' Uk) � � A k (X k , Yk ) + S k +1 . 4 ,j€k/j3k , therefore A k ( X k , Yk) > (1 - f k +1 - 4Ft/ j3k)S kH . By hypothesis, this last term is positive and since ( x k ' Yk) > 0 we conclude that Ak > 0 for all sufficiently large k . It follows that our earlier estimates on IAk l become 1 - 4Ft/13k � Ak � 1 + 4Vfk/j3k , that is, 1 1 - Ak l � 4Vfk/j3k . Consequently, using the fact that I I Yk l l = PI (Yk ) � Pk(Yk) = 1, we finally obtain the desired estimate:

IIYHI - Yk ll = II(Ak - 1)Yk + uk l l � I Ak - 1 1 · IIYk ll + I lu k l l � � IAk - 1 1 + Iluk ll � 4 ,j€k/j3k + 2 .jfkjj3k = 6 ,j€kjj3k . The fact that Poo (Yoo ) = 1 is immediate from the fact that Pk on bounded sets, since Pk(Yk ) = 1 and Yk --+ Yoo .

--+

P oo uniformly

(iii) Finally, we show that YooT is single-valued at Xoo , with value Soo. To that end, suppose that X* E T( xoo). Since Xoo E Uk for all k and since 1 = Poo (Yoo) � Pk (Yoo), the fact (proved earlier) that

Sk = sup {
Pk(Y ) � I }

implies that S k �
4.

Smooth variational principles, Asplund spaces, weak Asplund spaces.

it follows that

77

( x*, Yoo ) = Soo and the proof is complete.

The utility of "smoothability" ( the existence of an equivalent Gateaux differentiable norm) in Theorem 4.31 raises anew the question of permanence properties for such spaces. It is easy to see that smoothability is inherited by subspaces and preserved under finite products. It does not satisfy the "three­ space property" j this was shown by Talagrand [Ta2 ] when he showed that the space El = C(K), with K the compact Hausdorff "two arrow space," does not admit an equivalent smooth norm, even though it contains the separable ( hence weak Asplund) subspace C[O, I], while the quotent space EdC[O, I] is isometric to co (O, I] ( which is an Asplund space) . Theorem 4.31 provides another proof of this result: If El admitted an equivalent smooth norm, then it would be a weak Asplund space, but M. Coban and P. S. Kenderov [C -K] have shown that the set of points of Gateaux differentiability of the supremum norm in El does not contain a dense Go subset. As to permanence properties of weak Asplund spaces, the example El described above shows that weak

Asplund spaces do not satisfy the three-space property.

A natural question about monotone operators remains open: If E is a E generically

weak Asplund space, is every maximal monotone operator on single-valued ? Remarks.

The alert reader will have noticed that we have not made full use of Theo­ rem 4.10 (the D GZ smooth variational principle), since we give no applications to Hadamard or weak Hadamard differentiability. Of course, the former coincides Gateaux differentiability for locally Lipschitzian functions, while the latter coincides with Frechet differentiability in reflexive spaces. J. Borwein [Bor3] and P. Orno [Or] have shown that for continuous convex functions, weak Hadamard differentiability coincides with Frechet differentiability in any Banach space which does not contain a subspace isomorphic to £1 . J. Borwein and S. Fitzpatrick [Bor-F2 ] have shown that for a a-finite measure p , the space L 1 (p) admits an equivalent weak Hadamard dif­ ferentiable norm, hence Theorem 4.10 applies to closed subspaces of such L1 spaces. In [DGZ3] the authors apply their smooth variational principle to generalize the kinds of results on differential equations in Banach spaces proved originally by M. Crandall and P.- L. Lions in [Cr-Lh ,2 ] . I n the original version of Theorem 4.20, Borwein and Preiss utilized perturbation functions made up of sums of p-th powers of translates of the norm, 1 :$ p < 00; it seemed simpler to us to restrict attention to the case p = 2. Their paper lists several other bornologies, hence several other kinds of differentiability. Theorem 4.24 (that the weak Asplund property is preserved by continuous linear surjective maps) was proved by Asplund [Asp], but his proof seems to assume that the image of a dense G6 under such a mapping is a G 6 , something which even fails in R2 . This section has touched briefly on a number of results concerning Giiteaux or Fnkhet smooth renormings of Banach spaces and their relationship to the existence of smooth bump functions. For a comprehensive treatment of such topics, see the forthcoming book by R. Deville, G. Godefroy and V. Zizler [D-G-Z2 ]. The same authors have utilized the construction in the proof of Theorem 4.31 to generalize it to

78

Section 4

the following result [D-G-Z 1 ]: If E admits a Gateaux differentiable and Lipschitzian bump function, then E is a weak Asplund space. A different approach to proving that Gateaux smooth spaces are weak Asplund spaces (again using the construction in Theorem 4.31) has been obtained by N. Ribarska [Ria]. She avoids the Banach­ Mazur game by applying her earlier results on "fragmentability" [Rh, 2 ], a concept which plays a key role in the interplay between general topology and geometry of Banach spaces. There is a natural and straightforward definition of "/1-continuity" of a monotone operator T which reduces to /1-differentiability of f when T is the subdifferential of the convex continuous function f. Another application of the construction in Theo­ rem 4.31 has been made in [P-P-N] in order to prove that if E admits an equivalent

/1-smooth norm, then the interior of the domain of any maximal monotone operator on E contains a dense G6 set of points at which the operator is /1-continuous.

Section 5

5.

Asplund spaces, the RNP and pert urbed

optimization.

In this section we are going to look at a duality relationship between Asplund spaces and spaces with the Radon-Nikodym property (RNP). Roughly speak­ ing, a Banach space E or, more generally, a closed convex subset C of E, is said to have the Radon-Nikodym property if the classical Radon-Nikodym theorem (on the representation of absolutely continuous measures in terms of integrals) is valid for vector-valued measures whose "average range" is con­ tained in C. This property has been characterized in purely geometrical terms (which is the basis of the definition we use below). For the extraordinarily wide range of connections between the RNP and various parts of integration theory, operator theory and convexity, one should read the 1977 survey by Diestel and Uhl [Di-U] and, for more recent results (1983) the comprehensive lecture notes by Bourgin [Bou].

Definition 5.1. (a) Recall, first, the definition of a slice S(x* , A, a) of a nonempty subset A of a Banach space E: For a > 0 and x* E E*, S(x * , A, a)

=

{x E A : ( x * , x ) > O"A(X * ) - a}.

If A is a nonempty subset of E*, we define a weak* slice analogously, with the functional coming from E rather than from E** . (b) We say that a nonempty subset A of E is dentable provided it admits slices of arbitrarily small diameter; that is, for every E > 0 there exists x* E E* and a > 0 such that diam S(x* , A, a) < E. (c) If A is a nonempty subset of E*, we say that it is weak*-dentable provided it admits weak*-slices S(x, A, a) of arbitrarily small diameter. Using this terminology, Theorem 2.31 can be reformulated to say that a Banach space E is an Asplund space if and only if every nonempty bounded subset of E* is weak* dentable.

Definition 5.2. A subset A of a Banach space E is said to have the Radon­ Nikodym property (RNP) if every nonempty bounded subset of A is dent able.

80

Section 5

Since weak* dent able sets are obviously dent able, Theorem 2.31 implies that if E is an Asplund space, then E* has the RNP. One of the most striking results in this area is that the reverse implication is also valid. To that end, we need another definition.

Definition 5.3. An infinite tree in a Banach space E is a sequence {xn} such that Xn = (1j2)(X2n + X2n+ I ) for each n . An infinite tree such that I IX2n - x2n+ I I I 2: 215 for some 15 > 0 and all n is called an infinite h-tree. Ge o metri c P i c t ure

Schemati c Pi cture

"-./ V V "V

X,V XV V )( 2

)( 3

Fig. 5.1 Here is an obvious but important observation: An infinite h-tree cannot be dentable (since every slice must have diameter at least 215). What we will

do is use the characterization that E is an Asplund space ( if and) only if every separable subspace of E has separable dual ( Theorem 2.33) in order to produce a h-tree in E*. We need some preliminary results.

Lemma 5.4. Assume that E is a Banach space which contains a separa­ ble subspace F such that F* is nonseparable. Then there exists f > 0 and a nonempty subset A of the unit ball B* of E* such that every nonempty relatively weak* open subset of A has diameter greater than f.

Proof. Let B( F* ) be the unit ball of the nonseparable space F* . By the proof of Theorem 2.19, there exist f > 0 and an uncountable subset A l of B(F* ) such that each point of Al is a weak* condensation point of Al and such that I I x* - y* I I > f whenever x*, y* are distinct points of A I . It follows that any nonempty relatively weak*-open subset of the weak* closure A2 of Al contains at least two distinct points of A I . The restriction map R: E* -+ F* maps B* onto B(F* ) and is weak*-to-weak* continuous. Let A be a minimal weak* compact subset of B* such that R(A) = A2 • Thus, if U is a relatively weak* open subset of A, then the image A3 under R of the weak* compact set A \ U is a proper closed subset of A2. Since A2 \ A3 is a relatively weak* open subset of A 2 it contains at least two distinct points of Al and hence there exist distinct points x*, y* in U with II x* - y* I I > f.

5. Asplund spaces, the RNP and perturbed optimization.

81

Let {J{n } be a sequence of nonempty compact convex subsets of a topological vector space such that J{2 n U J{2 n+ l C J{n for all Then there exists an infinite tree {x n } such that X n E J{n for all

Lemma 5.5.

n.

n.

Proof. Let J{ = IIJ{n be the (compact) cartesian product x (x n ) such that X n E J{n for each Define, for each n.

=

consisting of all

n,

Each A n is closed, hence compact, and a sequence {x n } corresponding to an element x E J{ will be an infinite tree if x E nA n , that is, we need only show that this intersection is nonempty. By compactness, it suffices to prove that A l nA2 n · · ·nAk t 0 for each k. To this end, fix k and define x in J{ as follows: IT n > k, let X n be any element of J{n ' From n = k to n = 1 , use induction: Suppose that, for m = n + 1 , n + 2, . . . , a point X m E J{m has been chosen, and define X n = (lj2)(x2 n + X2 n+ l )' Clearly, X n E J{n since the latter is convex and J{2 n U J{2 n + 1 C J{n ' The resulting element x is in A l n A 2 n . . . n Ak, so the proof is complete.

Suppose that E is a Banach space and suppose that there exists a nonempty bounded set A C E* and an 10 > 0 such that diam U > 10 whenever U is a nonempty relatively weak* open subset of A. Then its weak* closed convex hull w* -cl coA contains an infinite Ej2-tree.

Proposition 5.6.

Proof. We construct a sequence {Un } of nonempty relatively weak* open sub­ sets of A and a sequence {x n } in E such that (a) II x n ll = 1 for each n , U2n U U2 n + l C Un , = 1, 2, 3, . . . , and (c) X* E U2 n and y* E U2 n+ 1 imply that (x* - y*, x n ) � for each First , let U1 = A. Suppose that, for some positive integer m, sets Uk have been defined for 1 :::; k < 2m and points X n have been defined for each 1 :::; < 2 m - 1 so that properties (a), (b) and (c) are valid for all such that 1 :::; < 2 m - I . Suppose k satisfies 2 m - 1 :::; k < 2 m . By hypothesis, we have diam Uk > 10, hence there are functionals Zo and zi in Uk such that II Zo - z i ll > E. Choose points Xk E E such that II xk l l = 1 and (zo - zi , Xk) 10 + 8 for some 8 > O. Let (b)

n

n.

10

n

n

n

=

82

Section 5

Fig. 5.2 Clearly, U2 k and U2 k H are nonempty relatively weak* open subsets of A and U2 k , U2 kH and X k satisfy ( a) , ( b) and ( c ) for n = k. Since n = 2k and n = 2k + 1 exhaust {n: 2m- l < n < 2m}, the construction is complete. We will shortly need the following elementary fact : If It , J2 are nonempty subsets of a topological vector space, then

CoJl - Coh

co(Jl - J2),

C

where co denotes the closed convex hull. [If a E Jl , then the fact that the map x -+ a - x is an affine homeomorphism shows that

a - coJ2 = Co(a - J2)

C

Co(Jl - J2 ).

For any b E coJ2 , we also have COJI b c co( Jl - J2 ), whence the result. ] Now, for each n, let Kn denote the weak* closed convex hull of Un. Each Kn is nonempty, weak* compact and convex, and ( b ) implies that K2n U K2nH C Kn . By Lemma 5.5, there is a tree {x� } in E* such that x� E Kn for each n. By the previous remark, we necessarily have -

X; n - X;n + l E K2n - K2nH

C

w * - cl CO(U2n - U2nH ),

so ( c ) implies that (X2n * - x2 nH , xn ) � f . Hence, I I X2 n - X 2 nH I I � f and {x� } is an infinite f/ 2-tree in Kl = w*-cl coA.

Theorem 5.7.

the RNP.

A Banach space E is an Asplund space if and only if E* has

Proof. As we have observed, Theorem 2.31 contains the "only if" assertion. To prove the converse, suppose that E is not an Asplund space. By Theorem 2.33 there exists a separable subspace F of E such that F* is not separable, hence Lemma 5.4 and Prop. 5.6 together show that E* contains a bounded 8-tree, for some 8 > O. Since the latter is not dent able, the space E* does not have the RNP. If C is a bounded closed convex subset of a Banach space with the RNP (or more generally, if C itself has the RNP ) one can deduce a number of

5. Asplund spaces, the RNP and perturbed optimization.

83

strong implications concerning the extremal structure of C; for instance, such a set is the closed convex hull of its extreme points, a property usually only associated (via the Krein-Milman theorem) with compact-or weakly compact­ convex sets. More generally, we will eventually show that any such set is the closed convex hull of its strongly exposed points.

Definition 5.8. A point x in a closed convex set C is said to be strongly exposed provided there exists x* ¥- 0 such that x E S( x*, C, a) for every a > 0 and these slices have diameters converging to 0 as a tends to o. Equivalently, x is strongly exposed by x* if, for {x n } C C,

(x*, x n )

--+

ac(x*) implies II x n - xii

--+

o.

A functional x* which satisfies the definition above is called a strongly exposing functional, and is said to strongly expose x. A point x E C i s called exposed if there exists x* ¥- 0 such that

ac(x*) for each y E C, y ¥- x, and x* is said to expose x. If C C E*, a point x* E C is said to be weak* strongly exposed (or weak* exposed) if analogous properties hold, with the exposing functionals coming from E. (x * , x)

= ac(x*) and

(x*, y)

<

It is obvious that exposed points are extreme points and that strongly exposed points are exposed; the following simple examples show that both inclusions can be proper.

5 .9. Examples. Let C be the closed convex hull of the orthogonal basis vectors {en } in (2 . Then is an exposed pofllt of C, but every slice of C which contains 0 has diameter at least J2, so x is not strongly exposed. o

Proof Since {en } converges weakly to 0 and C is weakly closed, we know that 0 E C. Note that if Y = ( Yn ) E C, then Yn 2': 0 for all n. This implies that if x = (xn) E (2 is such that Xn < 0 for all n, then necessarily (x, y) < 0 whenever Y E C, Y "# 0, which says that x exposes C at o . On the other hand, any slice of C which contains 0 is a relative weak neighborhood of 0, hence contains all but finitely many basis vectors, and lien - emil = J2 whenever m "# n . To see that extreme points need not be exposed, consider the following sketch of a compact convex subset C of R 2 j the point of tangency x is an extreme point, but is not exposed. I(

Fig. 5.3

84

Section 5

We next take a look at the duality relationship between Frechet differ­ entiability of the norm in E and weak* strongly exposed points of the dual ball in E*j this was first investigated by V . L. Smulyan in 1940. That such a duality is plausible is suggested by Fig. 5.3: If we assume that C is centered at the origin, it defines a unit ball in R2 . The corresponding dual unit ball ( that is, its polar ) will have the form x*

Fig. 5.4 where x* is the unique functional supporting C at x. Note that the flat ( hence very smooth) portions of C are in duality with the pointed (hence very rotund ) portions of its polar set . A slightly more general formulation of Smulyan's results can be obtained in terms of continuous nonnegative sublinear functionals ( Minkowski gauges ) and their "dual balls" . Recall some basic facts from Sec. 1 : A real-valued function p is .'lublin ear provided p( x + y) ::; p( x) + p(y) and p( tx) = t . p( x) for all x, y E E and t 2: o. It is continuous if and only if there exists M > 0 such that p(x) ::; M ' 11x ll for all x. A nonnegative continuous sublinear functional is called a Minkowski functional ( or gauge). It can be characterized in terms of its associate closed convex "ball" {x E E : p(x) ::; 1 } , but we will be more interested in its "dual ball" C(p) - or simply C - defined by

C(p) = {x * E E : (x* , x) ::; p(x) for all x E E}. This is easily seen to be weak* compact and convex. The following elementary facts are useful enough that we spell out the details.

Lemma 5.10. If p is a Minkowski gauge, then x* E op(x) if and only if x* E C and (x*, x) p(x). Moreover, p ac, the support functional for the set C. =

Proof.

=

x* E op(x) provided (x*, y - x) ::; p(y) - p(x) for all y E E.

By definition,

(5. 1 )

Assuming this, apply it to the point ry where r > 0 and y is fixed, divide by r and let r -+ 00 to get (x*, y) ::; p(y) for all y. Thus x* is in C. Moreover, setting y in (5. 1 ) equal to 0 yields (x* , x) 2: p(x), so equality holds at x. Conversely, if x* ::; p and (x* , x) = p(x), then (5.1) is immediate. It is also easy to see that

5. Asplund spaces, the RNP and perturbed optimization.

85

p(x) = O'c(x) sup {(x* , x ) : x* E C}; indeed, if x E E, then (x*, x ) :::; p(x) for all x* E C, so O'c(x) ::; p(x). Since p is continuous, there exists an element x* E op(x), so x* E C and p(x) = (x*, x ) :::; O'c(x). ==

The duality property in which we are interested is described by the fol­ lowing proposition.

Suppose that p is a Minkowski gauge on E. An element x* in C(p) is weak* strongly exposed by x E E if and only if p is Frechet differentiable at x, with p' (x) = x* .

Proposition 5.11.

Proof. Suppose that x* E C is weak* strongly exposed by x =f 0, so that (x* , x ) = O'c(x) p(x). Since p is continuous, op(y) i= 0 for all y in E. Let 'P(y) be any element in op(y) if y =f x, and let 'P(x) = x* . By Prop. 2.8, it suffices to show that 'P is norm-to-norm continuous at x. Suppose, then, that X n ---+ x. Since, for all 'P(x n ) E op(x n )' we have 'P(x n ) e C and ('P(x n ), X n ) = p(x n ). Moreover, 1('P(x n ) - x* , x ) 1 = 1('P(x n ) - x* , x - x n ) + ('P(x n ), x n ) - (x *, x n ) 1 ::; ::; 11'P(x n ) - x* II ' lI x - xn ll + Ip(x n ) - (x*, x n) l· Since p(x n ) ---+ p(x) and (x*, x n ) ---+ (x* , x ) p(x), and 11'P(x n ) - x* 1 1 IS =

n,

,

=

bounded, the right side converges to O. Therefore

(x* , x ) = p(x). Since x* is weak* strongly exposed by x, we have 11'P(x n ) - x*11 norm-to-norm continuous at x; moreover, x* = p'(x). ('P(x n ), x )

---+

---+

0,

so 'P is

To prove the converse, suppose that p is Frechet differentiable at x =f O. Let x* = p'(x); since x* E op(x) we have x* ::; p and ( x*, x ) = p( x). Suppose that there exists {x�} C C and r > 0 such that

(x�, x ) ---+ (x *, x) = p(x) but Il x� - x* 1 I There exist {Yn } C E such that ll Y n I = 1 and (x� - x*, Yn ) > r for all

>

r for all

n.

(5.2)

n.

By Frechet differentiability, given 0 < e < r / 2 there exists

0

>

0

such that

+ y) - p(x) - (x* , y ) ::; e l ly ll whenever Il y l l :::; o . Since x� E C we have (x� , x + y) ::; p(x + y) for all y E Let Zn = oYn ; then from (5.2) we have Or < (x � - x*, zn ) = (x � , x + zn ) - (x* , x ) - (x*, zn ) - (x� - x*, x ) ::; :::; [P(x + zn ) - p(x) - (x*, zn ) ] - (x� - x*, x ) ::; or / 2 - (x� - x*, x ) , 0 ::; p(x

E.

86

Section 5

which leads to a contradiction, since (x� - x*, x)

-+

o.

This proposition is the basis for the following dual characterization of Asplund spaces.

A Banach space E is an Asplund space if and only if every nonempty weak* compact convex subset C of E* is the weak* closed convex hull of its weak* strongly exposed points.

Theorem 5.12.

Proof. One direction is immediate from Theorem 2.32: If every such subset C C E* has weak* strongly exposed points, then every bounded nonempty subset A of E* is weak* dent able (since small weak* slices of the weak* closed convex hull C of A yield small weak* slices of A). To prove the converse, suppose that E is an Asplund space and that C is a nonempty weak* compact convex subset of E*. Without loss of generality, 0 E C. Let p = O"c; then p

is a Minkowski gauge and - as the definition and an easy application of the separation theorem show - C(p) C. Let D be the weak* closed convex hull of the weak* strongly exposed points of C and suppose that D � C. Then there exists x E E such that D( x) < c( x). Since C is bounded, both of these convex support functionals are continuous and so there exists a point of Frechet differentiability Xo of p contained in the open set where the strict inequality holds: D( xo) < O'c( xo). By Proposition 5.11, Xo strongly exposes C at the point x· p' C xo) and so (x·, xo) = p ( xo) O'c(xo). But D contains all the weak* strongly exposed points of C, therefore (x·, xo) � O' D(X O ) < O'c(x o ), a contradiction. =

0'

0'

0'

=

=

As another application of Proposition 5.11, we can now give the details of Example 1.14(c). Example l.14(c). There exists an equivalent norm on £ 1 which is nowhere Frechet differentiable but which is Gateaux differentiable at each nonzero point.

Proof By Exercise 2.37(a), to obtain an equivalent smooth norm on £ 1 it suffices to define an equivalent strictly convex dual norm on £00. To that end, define II . · 1 1 on £00 by Clearly, lIyl l oo � lIyll � 2 11Yll for each y E £00 , so this is an equivalent norm. Since 00 p is the supremum of a sequence of weak* continuous functions, it is weak* lower semi continuous, so II . · 11 is a dual norm. As the sum of two norms, one of which (namely, p) is strictly convex, I I 1 1 is easily verified to be strictly convex. It remains · ·

to show_that the norm 1 1 · · 11 which it induces on £1 is nowhere Frechet differentiable, and this will follow from Prop. 5.12 if we show that the unit ball it defines in £00 has no weak* strongly exposed points. Suppose, then, that x E £1 with II x ll = 1 and that y E £00, lIyll = 1, is the unique element such that ( y , x ) = 1. It suffices to produce a sequence {l} in £OCJ such that (y k , x )

-+

( y , x)

=

k 1 and lIy l l

-+

lIyll

=1

5. Asplund spaces, the RNP and perturbed optimization.

87

but lIy k - yll does not converge to zero. There are two cases to consider, depending on whether Yn --+ O. If the latter is the case, then there exists N such that IYn l < 1/4 whenever n > N. For k > N, define y k E f oo by

Yn if n :/:- k, ( y k),.

=

3/8.

(y, x ) - X k · ( Yk - 3/8)

--+

(y , x)

max { 3/8, sup IYn l } n oF "

=

IIYll oo .

( y")n Since X

k

--+

0, we have

(y" , x) Also, 1

=

=

=

1.

=

lIyll $ 2 11yll 00 so lIylloo � 1/2 . Thu s, lIy" lIoo

It follows that for k

=

--+ 00 ,

Ily k ll = IIYlloo + [ET n y� - ( y� - 9/64) . T"j l/2

--+

Ilyl i .

On the other hand, Ily" - y ll � Ily" - ylloo = 1 3/8 - y,, 1 > 1/8. Suppose, now, that {Yn} does not converge to zero. There necessarily exist £ > 0 and a subsequence {Ynk } such that IYnk I > 2£ for all k. Define y k in f oo by

(y")n Then (yk , x)

--+

(y , x)

=

Yn if n :/:- n k , (y ")nk = £. 1 and lIy"lI --+ lIyll as before, while =

lIy - y" 11 � Ily - y" lloo

=

IYnk - £ 1 > £ ,

which completes the proof.

Some of the results in this and earlier sections can be interpreted theo­ rems in infinite dimensional optimization, in that they give existence results for minimum (or maximum) points for certain types of functions. Actually, what they show is better described as "perturbed" optimization. In finite dimen­ sional optimization, existence is frequently trivial: When one has a continuous (or lower-bounded lower semicontinuous) real-valued function f on a bounded and closed set C, then, since the latter is compact, the existence of a mini­ mum point is guaranteed. In the infinite dimensional case, minimum points need not exist, and one will settle for a theorem of the form: For some Lipschitz function h, having Lipschitz norm as small as one may wish, the perturbed function f + h attains its minimum on C. Ekeland's Lemma 3.13 can be cast in this form (as noted just preceding its statement). So can the Br¢ndsted­ Rockafellar Theorem 3. 17, which has the following consequence : Suppose that f is a convex proper lower semicontinuous function which nearly attains its minimum on the Banach space E at the point x. Then there is a continuous linear functional x * of small norm such that f + x * does attain its minimum. (Moreover, it does so at a point which is close to x . ) In these results, com­ pactness was replaced by completeness and convexity. Results of this character have been obtained by C. Stegall [Ste2J (who replaces compactness by convex­ ity and the Radon-Nikodym property) and by N. Ghoussoub and B. Maurey [Gh-M2J (who look at a certain class of subsets of dual Banach spaces and as

88

Section 5

replace compactness by weak* metrizability and weak* relative compactness). We will formulate and prove Stegall's theorem in terms of maximum points for upper-bounded and upper semicontinuous functions. (The theorems of Ghous­ soub and Maurey will be stated without proof at the end of this section.) Our proof of Stegall's theorem requires a straightforward extension of the notion of "slice" , so as to include certain nonlinear functions. Definition 5.13. Suppose that C is a nonempty set and that f is an upper­ bounded real-valued function on C. For each > 0 define a

S( f, C, a) =

{x E C : f(x) > sup f - } . e

a

A point x E C is said to be a strong maximum for f if f( x ) = SUPe f and Il x - x nll 0 whenever f(x n ) f(x). (Note that we could have said that f "strongly exposes" x; some authors use this terminology.) �

�

5.14. Exercise. If C is closed and nonempty and f is a real-valued upper-bounded upper semi­ continuous function on C, then f attains a strong maximum on C if and only if diam S(f, C, o:) ..... 0 as 0: ..... 0 + .

Suppose that C c E is a nonempty closed and bounded convex set with the RNP and that f is an upper-bounded upper semi­ continuous real-valued function on C. Given € > 0, there exists x* in E* such that Il x* 1 I :::; € and f + x* attains a strong maximum on C. Theorem 5.15. (Stegall).

Remark. An obvious difference between this theorem and Ekeland's vana­

tional principle is that while both have the same hypotheses on the function to be minimized (or maximized), Stegall 's result obtains a maximum point after adding a small linear function (which is also Lipschitz, of course). This strengthening of Ekeland's variational principle comes at the cost of assuming that C has the RNP. Before we start the proof of Theorem 5.15 we need to recall the original definition of dent ability, in the form of the following exercise (which is a simple application of the separation theorem). We will also require several lemmas. 5.16. Exercise. Prove that a nonempty closed set A in a Banach space E is dentable if and only if for every € > 0 there exists x E A such that x is not in co( A \ B( Xj €».

The next lemma gives a recipe for constructing nondentable sets; we will use it when we prove Theorem 5.15 by contradiction. Lemma 5.17. Suppose that {A n } is a sequence of (eventually) nonempty subsets of a Banach space E with the following property: There exist constants € > 0 and ), > 0 such that for all x E co A n and y E E,

5. Asplund spaces, the RNP and perturbed optimization.

89

dist [x, co( An+l \ B( y ; E))) ::; >'/2n . Then the set

A = n n U {co Aj: j � n}

is nonempty and not dentable. Proof. We will show that x E co(A \ B(x; E/2)) for each x E A. First however, we show that A is nonempty and that for each n � 1 , (5.3)

where B == B(O; 1 ) . To this end, fix n sufficiently large that An is nonempty and suppose that Xo E co An . By hypothesis, there exists a point X l E co An+ l such that II xo - xI I I ::; 2>./2 n . Similarly, there exists a point X2 E co AnH such that II xI - x2 11 ::; 2>. /2n+ l . Continuing by induction, we obtain Xk E co An+k, k = 0 , 1 , 2 , . . . such that E ll xk - XkH I I < This implies that the series E(Xk - X k + l ) converges to an element y E E, of norm at most 4>'/2n . By writing y as a limit of the (collapsing) partial sums we get y Xo - Z where Z lim xk. It follows that Z E A (so A is nonempty) and Xo E A + (4)./2n )B, which proves (5.3). Suppose that x E A. Fix m such that 4>'/2m < E/2. Since we have x E {Uco Aj:j � n} for all n, for each n � m there exist j � n and Yn E co Aj such that I l x - Yn II ::; >'/2n. By hypothesis, 00.

=

=

dist [Yn , CO(AjH \ B(x; E)) ) ::; >./2j ::; >./2 n , so there exists Zn E co(Aj+1 \ B(x; E)) such that l l Yn - zn l l < 2>./2n. We can write Zn as a finite convex combination Zn

=

E>'j u j,

U j E AjH \ B(x; E).

By (5.3), for each i we have U j E A + (4)./2j )B c A + (4)./2n)B, so there exists Vj E A such that IIU j - Vj II ::; 4>./2n. Let Wn = E>'j v j; it follows that I l zn - w n ll ::; 4>./2n and hence I l x - w n ll ::; 7>./2n . Since Ilu i - x i i � E for each i, we have II Vj - x II

� E - 4>'/2 n � E - 4>./2m > E/2,

that is, Wn E coCA \ B(x; E/2)). It follows that x E coCA \ B(x; E/2)) required. The next two lemmas will allow us to reduce the proof of Theorem 5. 13 to simply showing that there are arbitrarily small perturbations f + X* of f which define small slices. The first lemma shows that we can get decreasing families of slices provided we perturb the function f slightly; the proof is a straightforward verification from the definition. Since all our slices will be in the same set C, we will write S(J, 0) in place of the usual S(J, C, 0). as

Suppose that ihe real-valued function f is bounded above on the nonempiy subset C of the unit ball B of E. For any 0 > 0 we have S(J + x*, (3) c S(J, 0) provided Il x* II < 0/2 and 0 < f3 < 0 - 2 11 x* II ·

Lemma 5.1B.

90

Section 5

Let C be a nonempty bounded closed subset of E and suppose that for every upper-bounded upper semicontinuous function f on C and every E > 0, there exist x* E E*, II x*11 < E, and 0 > 0 such that diam S(f + x*, 0) :::; 2E. Then given E > 0 there exists x* E E* such that II x*1 I < E and f + x* attains a strong maximum on C. Lemma 5.19.

Proof. We assume without loss of generality that C is contained in the unit ball of E and that 0 < E < 1. By hypothesis, there exist xt E E* , II xt I < E/2, and 0 < 01 < 1 such that diam S1 < E, where we write S1 = S(f + xr, 0 1 ) . By applying the hypotheses to f + xr and E1 01 E/22 , we obtain II xi l i < E1 so that diam S2 < 2E1 , where S2 S(f + xi + X i , 02 ) and 0 < 02 < 0 1 · Continuing by induction with 00 1 , we obtain sequences =

•

=

=

En > 0,

0 < O n < 1, x� E E* ,

Sn = S(f + Ef xi,

on)

such that

II x� II < En - 1 , En E on/2 n+ 1 and O n < 0n - 1 · Obviously, the series Ex � converges to a point x* E E* of norm at most E · Eon2-( n +1 ) < E. Note that diam Sn ---+ 0, since En ---+ O. We claim f + x* diam Sn :::; 2En - 1 ,

=

•

attains a strong maximum on C. By Exercise 5.14 it suffices to show that diam S(f + x* , 0) ---+ 0 as 0 ---+ 0+ , and for this it suffices to prove that for all there exists 0 > 0 such that S(f + x* , o) C Sn . This follows from Lemma 5.18, using the fact that f + x* = f + Eix i + w : where n

IIw� I

k :::; E�n+ 1 E Ok - 1 /2 < on/2; •

we need only choose 0 such that 0 < 0 < O n - 2 I 1w :lI .

Proof of Theorem

5. 1 5. By utilizing the previous lemma, we need only show that given any E > 0 there exist x* E E*, II x* II < E, and 0 > 0 such that diam S(f + x*, 0) :::; 2E . Proceeding by contradiction, suppose that for every II x* II < E and each 0 > 0, we have diam S(f + x* , o) > 2E . For each let n

For all sufficiently large we have E - 2 - n > 0, so that An is nonempty. Let >. = 5/2; we will show that with this choice of >., the sequence {An} satisfies the hypotheses of Lemma 5.17, the conclusion of which will contradict the hypothesis of Theorem 5.15 (that C has the RNP). Restating the main hypothesis of Lemma 5.17, we want to show that for any y E E, n,

co An c co(An+1 \ B(y; E)) + (>./2n ) . B.

(5.4)

Since the set on the right side is cunvex, it suffices to prove that it contains An. Suppose, then, that x E An but, for some y E E, it is not in the right hand side of (5.4) (which has nonempty interior). By the separation theorem there exists y* E E*, I y* I = 1, such that

5. Asplund spaces, the RNP and perturbed optimization.

91

(y* , x) � sup { (y*, u ): u E An+! \ B(y; E )} + >'/2 n . (5.5) Since x E An , there exists x* E E* with Il x* 11 ::;: E - 2- n such that x is in the slice S(f + x*, 1/4n ). Write z * = x* + (1/2 n +1 ) y*; it follows that IIz * II ::;: E - 2 - n + 2-( n+ l ) = E - 2-( n+ l ) so S(f + z*, 1/4n+! ) C An +! . Since diam S(f +z*, 1/4n + 1 ) > 2 E , this slice is not contained in B(y; E ) . This implies that there exists z E C \ B(y; E) such that f ez ) + (z * , z) > sup ( f + z *) - 1/4n+ l . (5.6) •

c

Thus, z E A n+ l \ B(y; E) and hence, from (5.5), we conclude that

(5.7) (y * , x) � (y*, z) + >'/2 n . We will prove the inclusion in (5.4) by showing that this cannot be true. Note first that since x E S(f + X* , 1/4n ) and z E C, we necessarily have f(x) + (x *, x) > sup( f + x*) - 1/4 n � fez ) + (x * , z) - 1/4n . (5.8) c

Similarly, from (5.6) we have

fez ) + (z * , z) == f ez ) + (x · , z) + ( 1 /2n+ l )(y*, z) > > sup ff + x* + (1/2 n+! )y* ] - 1/4n + 1 � c � f(x) + (x*, x) + (1/2n+ 1 )(y*, x) - 1/4n + l . Using (5.9) and (5.8) we obtain f ez ) + (x * , z) + (1/2 n+ l )(y* , z) > > f ez ) + (x * , z) - 1/4n + (1/2n+ l )(y* , x) - 1/4n + 1

(5.9)

or

(1/2 n+ 1 )(y*, x - z) < 1/4n + 1/4n+ l = 5/4n+! = 5/22 n+2 . Equivalently, (y*, x - z) < 5/2n+ 1 = (5/2)(1/2 n ) = >'/2 n , which contradicts (5.7) and completes the proof. We next illustrate the power of Theorem 5.15 by using it to give a simple

proof of the geometric fact mentioned earlier, that RNP sets are generated by their strongly exposed points.

Theorem 5.20. Suppose that C C E is a nonempty bounded closed convex set with the RNP. Then C is the closed convex hull of its strongly exposed points. Moreover, the functionals which strongly expose points of C constitute a dense G6 subset of E*.

Proof. We first prove the second assertion. To this end, define

92

Section 5

Gn

= {x* E E*: diam S(x* , C, a) < l/n for some a > O } . If y* E Gn (so that diam S(x*, C, a) < l/n for some a), then by Lemma 5.18, applied to f = y*, the a/2-neighborhood of y* is contained in Gn, hence the

latter is open. Theorem 5.15 (applied to any element f of E*) trivially implies that each Gn is dense in E*, hence by the Baire category theorem, the set G = nG n is a dense Go subset of E*. Obviously, if x* strongly exposes C, then it defines slices of arbitrarily small diameter, hence is in G. Conversely, if x * E G, then it will define a nested sequence {S(x* , C, a n ) } of slices of C with diameters converging to o. Their closures intersect in a point of C which is strongly exposed by x*. To prove the first assertion of the theorem, let D be the closed convex hull of the strongly exposed points of C. If D =f C, then by the separation theorem there exists X*E E* such that O" D ( X* ) < O"c ( x *). Since these functionals are norm continuous on E* , there exists a functional in the dense set G for which the same inequality holds, contradicting the definition of D. Since strongly exposing functionals can be used to define slices of arbi­ trarily small diameter, the foregoing theorem leads to a nice characterization. Theorem 5.21. A

Banach space E has the RNP if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points.

The following variant (and corollary) of Theorem 5.15 due to M. Fabian [Fa2] yields a strong minimum for a lower semi continuous function f defined on a space with the RNP. It replaces the restriction that f be defined only on a bounded set with a strong lower-boundedness hypothesis for f on all of E. Corollary 5.22. (Fabian) . Suppose that the Banach space E has the RNP and that f: E --? R U {oo} is a lower semicontinuous function on E for which there exist a > 0 and b E R such that

f(x) > 2all x ll + b, x E E. Then for any E > 0 there exists x* E E* such that I l x* II a strong minimum on E.

<E

and f + x* attains

Proof. Since we can replace f by f - b, we may assume that b = O. Note that if x * E E* with I l x* II < a , then for any x E E (5.10) f( x ) + (x* , x ) � 2 all x ll - all x ll = all x ll . Let r = (l/ a )[f(O) + 1] and apply Theorem 5.15 to - f restricted to the ball B == B(O; r), with 0 < E < Thus, there exist x* E E*, IIx* II < E, and a point Xo E B such that f + x* attains a strong minimum in B at X o . We need only show that this is a strong minimum for f + x * in E. If x E E is such that f( x) + (x*, x ) :::; f( xo ) + ( x * , xo ) = infB ( f + x*) :::; f(O), then from (5.10) we conclude that I l x ll :::; ( l / ) f(O) < r, so x = X o . Similarly, if {x n } C E and a.

a

5. Asplund spaces, the RNP and perturbed optimization.

(f + x*)(x n )

--t

(f + x*)(xo), then eventually f(x n ) + (x*, x n )

from (5.10) it follows that

Xn

E B and therefore X n

--t

<

93

f(O) + 1, so

Xo.

We now describe without proof some of the work of Ghoussoub and Mau­ rey on perturbed optimization; it involves the following notion. Definition 5.23. A bounded nonempty subset C c E* is said to be a strong w* -Ho set if its complement in its weak* closed convex hull D is the union of

a sequence {Kn } of weak* compact convex sets, each of which is of positive distance from C.

The reader is referred to [Gh-M2] for the motivation for this definition; we mention only that weak* compact convex subsets of E* obviously have this property, and many of the extremal properties of such sets are shown to carry over to strong w*-Ho sets (and related families of sets). Theorem 5.24. Suppose that the nonempty bounded set C w* - Ho set in its weak* closed convex hull D and that D is

C E* is a strong weak* metrizable [or that C is norm separable]; then for any lower-bounded weak* lower semi­ continuous function f on C and any f > 0, there exists x E E with I I x ll < f such that f + x attains a strict [strong] minimum on C.

Note that in this result, the function f is perturbed by a weak* continuous linear functional (which is necessarily Lipschitz). Ghoussoub and Maurey also obtain a similar result for subsets of a Banach space E which has the following property.

point of norm to weak continuity property (PCP) if every nonempty bounded subset of E admits

Definition 5.25. A Banach space is said to have the

relative weak neighborhoods of arbitrarily small norm diameter. [Equivalently, every nonempty bounded closed subset contains at least one point at which the identity map (restricted to the subset) is norm to weak continuous; this explains the terminology.] For more about the PCP, see [Ed-W] and [Gh-M1]. It is clear that the RNP implies the PCP (slices define weak neighborhoods), so the following result should be compared to Theorem 5.15. In one sense, it is more general, since it applies to nonconvex sets; on the other hand, the perturbation R need not be linear nor do we get a strong minimum; also, E is assumed to be separable. Theorem 5.26. Suppose that the separable Banach space E has the PCP and that C is a closed bounded nonempty subset of E. If f is a lower-bounded lower semicontinuous function on C and f > 0, then there exists a norm Lipschitz and weakly continuous function h, of Lipschitz constant at most f, such that f + h attains its minimum on C.

94

Section 5

Finally, N. Ghoussoub, J. Lindenstrauss and B. Maurey [G-L-M] have shown that a complex Banach space E has the "analytic" RNP if and only if for every bounded upper semi continuous real-valued function f on a closed bounded subset A of E and every E > 0, there is a plurisubharmonic function 9 on E, with supremum norm on A at most E, such that f + 9 attains a strong maximum on A. We refer to [G-L-M] for details and relevant definitions. Remarks. The fact that bounded subsets in the dual of an Asplund space admit weak* slices of arbitrarily small diameter was proved in [Na-PhJ. The converse (Theorem 5.7) was proved by C. Stegall [StelJ; the simpler proof given here is due to van D ulst and Namioka [Du-NJ. The fact that convex sets with the RNP are generated by their strongly exposed points (Theorem 5.20) has been of considerable interest. It was first proved by the author [Ph2l, using geometrical methods and assuming that the entire space had the RNP. J. Bourgain then proved the general case [BoJ, also using geometrical methods. C. A. Rogers pointed out that the original proof could easily be modified to yield the general result [La-Ph, p. 1 19J. K. Kunen and H. P. Rosenthal [Ku-RJ have proved it using vector-valued martingales, not as outlandish as it appears, if one is aware that the RNP can be characterized in terms of a martin­ gale convergence theorem (see [Bou]). A self-contained proof, using a Kenderov-like generic continuity theorem and the duality between differentiability and strongly exposed points, is given in [Ph3J. We have presented the result here, of course, as a rather easy corollary to Theorem 5.15. Our proof of the latter is due to J . Bourgain [Bo2 J, who used a modification of his proof that RNP sets can be characterized in terms of the so-called Bishop-Phelps property [BoIJ. It is interesting to note that Theorem 5.15 is an easy consequence of Corollary 5.22, at least in the special case when the entire space E is assumed to have the RNP. (This is the case applied, for instance, in [Cr-Lh,2])' Indeed, if f is lower semicontinuous and lower bounded (by m say) on the bounded closed convex set C and if the latter is contained in a ball of radius r > 0, then the function which equals f in C and +00 outside C satisfies the hypotheses of Corollary 5.22, taking a = 1/2 and b = m - r. Fabian's proof in [Fa2] of Corollary 5.22, apparently obtained independently of Stegall's (earlier) paper [Ste2l, uses the version of Theorem 5.20 found in [Ph2J, applied to certain subsets of E X R, and is much easier than the proof of Theorem 5.15 given here. It follows trivially from Theorem 5.20 that a Banach space with the RNP has the Krein-Milman Property (KMP): Every bounded closed convex subset of E is the closed convex hull of its extreme points. It remains an open question whether a Banach space with the KMP has the RNP. There have been a number of partial results; for instance, R. Huff and P. D. Morris (see [Bou, p. 91]) have shown that the answer is affirmative in any dual space, J. Bourgain and M. Talagrand (also see [Bou, p. 423]) have shown the same for any Banach lattice while V. Caselles [CaJ has given a short proof of this and of C .-H Chu's result [ChuJ that it is true in the predual of a von Neumann algebra. W. Schachermayer [SChlJ has given an affirmative answer for Banach spaces which are isomorphic to their squares, as well as for convex sets which are "strongly regular" [Sch2J. The reader has undoubtedly wondered whether the duality between Asplund spaces and the RNP goes the other way. It almost does: A Banach space E has

the RNP if and only if every continuous convex function on E* which is also weak* lower semicontinuous is Frechet differentiable at the points of a dense G6 subset of E* . This was proved by J . Collier [CoJ and generalized by S. Fitzpatrick [FiJ; see [BouJ for an exposition.

Section 6

6. Gateaux differentiability spaces.

The following class of spaces is formally larger than the class of weak Asplund spaces, but in some ways is a more natural object of study. Definition 6.1. A Banach space E is called a Gateaux Differentiability Space (GDS) provided the set G of points of Gateaux differentiability of a convex

continuous function defined on a nonempty open convex subset D e E is necessarily dense in D.

Clearly, a GDS differs from a weak Asplund space only by virtue of the fact that one does not require the set G to contain a dense Go . As shown in Prop. 1.25, the Go property is automatic for Frechet differentiability, but known examples (see e.g., [C-K] or [Tall ]) show that for Gateaux differen­ tiability (of individual convex continuous functions, at least), it is definitely an additional requirement. In this section we will examine some properties of Gateaux differentiability spaces, all but one of which remain open questions for weak Asplund spaces. Another motivation for studying Gateaux differentiability spaces is that they admit an interesting characterization which is completely analogous to Theorem 5.12 (which characterized Asplund spaces in terms of weak* strongly exposed points in their duals). We state this result now; its proof will be given below.

Banach space E is a GDS if and only if every weak* compact convex subset of E* is the weak* closed convex hull of its weak* exposed points.

Theorem 6.2. A

There is only one permanence (or stability) property known to be valid for weak Asplund spaces (they are preserved under quotient maps (Theorem 4.24)), while the Gateaux differentiability spaces have that property (even a stronger property; see Prop. 6.8 below) plus a simple but useful stability property under certain products (which is still open for weak Asplund spaces). It will help the exposition to introduce temporarily a class of spaces which is formally larger than the class of Gateaux differentiability spaces (but which will turn out to be the same).

96

Section 6

Definition 6.3. A Banach space E is said to be an M-differentiability

8pace

(MDS) provided every Minkowski gauge on E is Gateaux differentiable at the points of a dense set. Proposition 6.4. If E x R

is an MDS, then the Banach space E is a GDS.

Proof. We pair E x R with its dual E * x R by ((x* , r*), (x, r)) = (x* , x) + r * · r.

Suppose, now, that f is a continuous convex function defined on an open con­ vex subset D of E. We can assume without loss of generality that the origin is in D and that f(O) - 1. Let fl be the Minkowski functional defined. on E x R by the convex set epi(f); since the latter has nonempty interior [con­ taining, for instance, the origin], fl is necessarily continuous. If fl is Gateaux differentiable on a dense set, then by positive homogeneity it is necessarily Gateaux differentiable on a dense subset of =

G

=

((x, r) : x E D and fl(x, r) = I} .

The set G is that part of the boundary of epi(f) whose "x-coordinate" is in D, so it is the graph of f and hence it is homeomorphic with D. Thus, it suffices to show that if fl is Gateaux differentiable at the point (x, f(x)) (where x E D), then f is Gateaux differentiable at x. It is clear from Figure 6.1 below that any corners in the graph of f will be points at which epi(f) will have distinct supporting hyperplanes, hence points at which fl fails to be Gateaux differentiable.

Fig. 6.1

Specifically, suppose that xi , xi were distinct sub differentials to f at x , so that (6. 1 ) (xi , y - x) 5: f(y) - f( x ) for all y E D, i = 1, 2. Taking y 0, we see that (xi , x) - f(x) � -f(O) 1, the reciprocal S j of the quantity on the left is positive, for i 1 , 2. We claim that the two functionals s;( xi, - 1), i = 1 2 are distinct sub differentials to fl at (x, f(x)). It suffices to show that for each i, =

=

so

=

,

,

(s; (xi , - l ) , (z, r)) 5: fl (z, r) for (z, r) E E x R,

(6.2)

6. Gateaux differentiability spaces.

97

with equality at (z, r) (x, l(x)). By positive homogeneity of both sides of (6.2), it will be proved if we show that =

(S;(x i , - l ) , (z, r) ) < 1

whenever p,(z, r) < 1. The latter, of course, implies that (z, r) is interior to epi(f), so that z E D and r > I(z). Thus, by using this fact and (6.1) with y = z,

S i ( ( x i , z ) - r) < s i ( ( x i , z ) - l(z)) :S; s i ( (x i , x ) - I(x))

=

1

as was to be shown. The equality of (6.2) at (x, j(x)) is just the definition of S i , since p,(x , I(x)) 1 . Finally, the two functionals s i (x i , - 1) are distinct: if sl (xf, -1) S2 (X� , - 1), then Sl = S2 and hence xf x�. Our stability result is the following: =

=

=

Proposition 6.5.

If E is a CDS, then E x R is a CDS.

Proof. Suppose that D is an open convex subset of E x R, that I is a continuous convex function on D and that (xo , to) E D. Let U be an open neighborhood of (xo, to ); we want to find a point of Gateaux differentiability of I in U. Without loss of generality we can suppose that U contains a neighborhood of the form B x I, where B is an open ball centered at Xo and I [to - 8, to + 8] ; we can also assume that I I I is bounded in U, by M, say. Choose a differentiable extended real-valued function 9 :s; 0 on I which equals at the endpoints, is finite elsewhere and satisfies g ( to ) = O. Define =

-00

h e x)

=

sup{J(x, t) + g(t) : t E

R},

xE

B.

Since h is the supremum of a family of convex functions, it is convex; moreover, since I is bounded above by M on B, so is h. Also, for any point x E B, h( x ) ::::: I(x , to ) +g(to) ::::: - M, so h, being bounded, is continuous on B. (Recall the remark after Proposition 1 .6.) By hypothesis, h is differentiable at some point Xl E B. A compactness argument shows that h(XI ) l(x I , td + g (td for some tl E I. This implies that g (td > - 00, so that tl is an interior point of I. We will use the characterization of Gateaux differentiability as given in Exercise 1.24. For any (y, s) E E x R and all t > 0 sufficiently small, we have (Xl ± ty, tl ± ts) E B x I and =

o :s; I( Xl + ty, tl + ts) + I( Xl - ty, tl - ts) - 21( xl , tl ) :s;

h(XI + ty) + h(XI - ty) - 2h(x } ) - [ge t } + t) + g e t } - t) - 2g(td] .

Since both h a d 9 are differentiable, if we divide through these inequalities by t > 0 and let t 0, the terms on the right tend to zero, showing that I is differentiable at (Xl , tt ). n

--+

Corollary 6.6. A

Banach space E is a CDS if and only if it is an MDS.

98

Section 6

Proof. Let H be a closed hyperplane in E, so that E is isomorphic to the product H x R. IT E is an MDS, then by Prop. 6.4, H is a GDS. By Prop. 6.5, the space H x R, that is, the space E, is a GDS. 6.7. Problems. Is a closed subspace of a GDS necessarily a GDS? Is the product of two Gateaux differentiability spaces necessarily a GDS? ( It is obvious from P rop. 6.5 that, by induction, the second question has an affirmative answer for the product of a GDS wit h a finite dimensional space. )

The following permanence property for Gateaux differentiability spaces is similar to the corresponding one for weak Asplund spaces (Theorem 4.19) which required the operator T to be onto; the proof is essentially the same;

Suppose that T: E F is continuous and linear, with dense range. If the Banach space E is a GDS, then so is F.

Proposition 6.8.

---t

Proof. Suppose that D is a nonempty open convex subset of F and that I is continuous and convex on D. The function II I aT is convex and continuous on the open convex set DI = T- l (D), hence is Gateaux differentiable at the points of a dense set G 1 C DI . To see that I is Gateaux differentiable at the points of the dense set T(GJ ), it suffices to verify that T* [8I(Tx)] C 8JI(x) for all x E Dl and use the fact that density of T(E) implies that T* is one-one. =

(It follows easily from the foregoing proposition that the first question in 6.7 has an affi rmative answer for any complemented subspace M of a GDS space E; that is, for any M which is the image of a continuous linear projection on

E.)

The next proposition is competely analogous to Prop. 5.11. Proposition 6.9. Suppose that p is a Minkowski gauge on E. An element x* in C(p) = {x* E E*: (x*, x) � p(x) for all x E E} is weak* exposed by x E E if and only if p is Gateaux differentiable at x, with dp( x) = x*.

Proof. Suppose that x* E C and that x weak* exposes C at xOO; then 8p(x) {x*} . Indeed, if y * E 8p(x), then by Lemma 5.10 we have y* E C and (y*, x) p( x) = (7c ( x), hence yOO x*. Conversely, suppose that p is Gateaux differentiable at x, with xOO = dp(x). Then x* is in 8p(x), so by Lemma 5.10 again, x* E C and x attains its supremum on C at x*. Suppose there were another point y* E C such that c ( x) (yOO , x); then the other implication in Lemma 5.10 shows that y* E 8p(x), hence yOO = x*, that is, x weak* exposes C at xOO. Recall the statement of Theorem 6.2: A Banach space E is a Gateaux differentiability space if and only if every weak* compact convex subset C of E* is the weak* closed convex hull of its weak* exposed points. =

=

=

(7

=

6. Gateaux differentiability spaces.

99

Proof of necessity in Theorem 6.2. The proof that the dual of a GDS has the indicated property is identical to the proof of the analogous portion of Theo­ rem 5. 12, except for the use of Prop. 6.9 in place of Prop. 5.11 and, of course, the substitution of Gateaux differentiability for Frechet differentiability. To prove the converse, we need the following dual version of a classical "parallel hyperplane lemma" . Lemma 6.10. Suppose that x, y E E, I (x*, y) I :S 1 whenever x* E E* satisfies

with J l x J l

= 1

J l y ll , and

=

E

>

o.

If

(x *, x) 0 and Jl x * J I < 2jE, or J l x + y J l :S E. =

then either J l x - y J l :S E

The following sketch shows the reason for the name; the hypotheses require that hyperplanes defined by the functionals x and y be nearly parallel. y = 1

y = -1

Fig. 6.2

Proof. Note that, as a (weak* continuous ) linear functional on E*, y is bounded in absolute value by Ej2 on the intersection of the dual ball with the subspace H = {x* : (x*, x) O}. By the Hahn-Banach theorem, its restriction to H can be extended to a functional of norm at most Ej2 defined on all of E*. Since this extension is necessarily weak* continuous, it is defined by an element z E E. Thus, y - z = 0 on H, so y - z = a x for some a E R. Note that =

11

- Ial l IIl y Jl - Jl y - zJ l I :S J l zJ l :S Ej2. If a � 0, then J l x - y J l = J I (1 - a)x - zJ l :S 1 1 - al + J l zJl :S E. If a < 0, then J l x + y J l = J I ( 1 + a )x + z ll :S 1 1 + a l + II z ll :S E. =

Proof of sufficiency in Theorem 6. 2. By Cor. 6.6, it suffices to show that E is an MDS. To that end, suppose that p is a Minkowski gauge functional on E, so that, by Lemma 5.10, we can write p = 0; we want to approximate x by a point y where is




Section 6

100

Gateaux differentiable. Since a is positive homogeneous, we can assume that 1. Given 10 > 0 with 10 < a( x), it suffices to produce y such that II x - y ll < and y exposes C. Consider the weak* compact convex subset C1 of E* defined by C1 = co(C U N) , where a ( x)

=

10

N = {x* E E* : (x* , x)

=

0 and II x * 1 1 < 2/ € } .

Fig. 6.3

Let a1 denote the support functional of C1 . By hypothesis, C1 is the weak* closed convex hull of its weak* exposed points, so any weak* slice of C1 con­ tains a weak* exposed point. In particular, then, if 0 < Q == a(x) - then the open slice S( x, C1 , Q ) contains a weak* exposed point y* of C1 . Let y -=f 0 be a point of E which exposes C1 at y* ; we may assume that I l y l l 1. Since y* is an extreme point of C1 it is either in C or in N. The latter is impossible; since x vanishes on N we have a1 (x) a ( x ) so (y* , x) > a1( x ) - Q and the latter equals Thus, y actually exposes C at y* ; since 0 E C and y* -=f 0, this implies in particular that (y* , y) > o. 10 ,

=

=

10 .

Fig. 6.4

The sketch above suggests that the hyperplanes defined by x and y are nearly parallel (hence that y is close to x). To see that this is actually the case, note first that since (y* , z) � a(z) � II z ll for all z, we must have lI y* 11 S 1, hence (y*, y) � 1. Thus, if x * E N e C1 , then (x*, y) � a1(y) = (y* , y) � 1;

by symmetry of N this implies that l (x* , y) 1 S 1 for all x* E N. It follows from Lemma 6.10 that either II x - y ll S or II x + y l l S But the latter is impossible: Il y * II � 1 hence 10

Il x + y ll 2: (y * , x + y) > (y* , x ) > 10.

10 .

6. Gateaux differentiability spaces.

101

Remarks. Much of the material in this section originated in [La-Ph], with one major ex­ ception. Prop. 6.5 was shown to us by M. Fabian, who says the proof was motivated by work of A. Ioffe. (The function 9 introduced in the proof is what optimization specialists call a "penalty" function.) Fabian's result represents the first bit of affir­ mative progress in this topic in the last decade, although there have been a number of relevant striking examples (see [C-KJ , [Tall] and [Tah]). We are grateful for his permission to include it in these notes. The names "Asplund space" and "weak As­ plund space" were introduced in [Na-Ph]; Asplund called them "strong" and "weak differentiability spaces" . In retrospect, both definitions could easily have dropped the G6 requirement; indeed, it wouldn't have changed anything in the Frechet case, and in the Gateaux case, Asplund's name would now be attached to what appears to be a more tractable class of spaces. J. Borwein and S. Fitzpatrick [Bor-F 2 ] have sketched a way to unify the results of this section with some of those of Sec. 5. A key (and easily proved) observation is the following: An element x · of a weak* compact convex subset C C E· is weak* exposed by x E E provided O"c(x) = (x· , x) and x� --> x · in the weak* topology whenever {x�} C C and (x� , x) --> (x· , x). Now, the weak* topology is precisely the topology (call it GO) of uniform convergence on elements of the Gateaux bomology {3 = G and the norm topology in E · is the topology FO of uniform convergence on the elements of the Frechet bomology {3 = F. One is thus led to a unified definition of a weak* {3-exposed point of such a set C: replace "weak* convergence" in the observation above by "convergence in the topology {30" . One can also define a "{3-differentiability space" to be one in which every convex continuous function is {3-differentiable on a dense subset of its domain. This makes it possible to mimic the proofs of Theorems 5.12 and 6.2 to prove the general result that a Banach space E

is a {3-differentiability space if and only if every weak* compact convex subset of E · is the weak* closed convex hull of its weak* {3-exposed points.

Section 7

7.

A generalization of monotone operat ors: U se o maps.

There has been much relatively recent work showing that Kenderov's results about generic continuity of maximal monotone operators are special cases of theorems of a more topological nature. We will describe some of this, selecting the results most closely related to our primary interest in Banach spaces. Definition 7.1 . (a) Suppose that X and Y are Hausdorff topological spaces and let F : X -+ 2 Y be a map from X into the nonempty compact subsets of Y. We say that F is usco (for upper semi continuous and compact valued)

provided it is also upper semicontinuous. (Recall that upper semicontinuity means that {x E X: F( x ) c U} is open in X whenever U is open in Y.) (b) If Y is a Hausdorff linear space, then a usco map F: X 2Y is called convex usco provided F(x) is convex for each x E X. (c) We denote the graph of F by G(F) == ((x, y) E X x Y: y E F(x)}, and, as with monotone operators, we partially order these set-valued maps by the inclusion ordering on their graphs. We will usually write FI C F2 in place of G(FI ) C G(F2), and talk about one such map containing another. A usco map [convex usco map] is said to be minimal if it does not properly contain any other usco [convex usco] map. (d) If Y = E* is a dual Banach space, saying that a usco map F from X into 2Y is w*-usco means that we are taking Y to be E* in its weak* topology (so the values of F are weak* compact sets). -+

7.2. Example. Suppose that E is a Banach space, that T: E -+ 2 E* is maximal monotone and that D is a nonempty open subset of D(T). Then the restriction TD of T to D is convex w*-usco (as a set-valued map from D to E* in its weak* topology).

Proof This is a consequence of Exercise 2.29.

The kind of theorem we will consider has the form : For certain minimal usco maps F from a Baire space X into 2Y , there exists a dense Go subset D of X such that F is single-valued and continuous (for an appropriate topology in Y) at each point of D. (This is not too surprising, since we would expect minimal set-valued maps to be single-valued at many points.) In view of the

7. A generalization of monotone operators: Usco maps.

103

relationship between of Asplund spaces E and certain continuity properties of maximal monotone operators defined in E, it is also not surprising that such results would be of interest. We need some elementary facts about usco maps. Proposition 7.3. (aJ If F : X - 2Y is usco, then its graph G(F) is closed in X x Y. (bJ For every usco map [convex usco map] F there exists a minimal useo map [minimal convex usco map] contained in F.

Proof. ( a) Suppose that (xo , Yo) is a net in G( F) which converges to a point (x, y ) E X x Y , but that y is not in F( x). Since the latter is compact there

exists an open neighborhood U of F( x) whose closure U does not contain y. By upper semicontinuity, (Yo ) is eventually in U, and hence y is in U, a contradiction. ( b) This will follow from Zorn's lemma provided we can show that any decreasing chain {Fo} of [convex] usco maps contained in F has a minimal element. For x E X define Fo(x) = nFo(x); by compactness, this is nonempty, [convex] and compact. To see that Fo is upper semicontinuous, suppose that x E X, U is open in Y and Fo (x) C U. Necessarily, Fo(x) C U for some 0; indeed, if each Fo(x) \ U were nonempty, the intersection of these compact nested sets would be a nonempty subset of Fo( x) \ U. By upper semicontinuity of Fo , there exists an open set V containing x such that Fo (V) C Fo(V) C U. Example 7.2 showed that maximal monotone operators are convex usco maps; the following example shows that they need not be minimal usco maps. 7.4 Example. Define monotone operators Fo , FI and F2 from

Fo(x)

= FI(X) = F2 (x) =

R

to

2R by

sgn x if x i= 0,

while

Fo(O) = {-I}, FI (O) = { - I, I} and F2 (0) = [ - 1, 1]. Then Fo C FI C F2 , they are all distinct, F2 is maximal monotone and minimal convex usco, FI is minimal usco ( hence has closed graph ) and Fo does not have closed graph, although Fo(x) is closed for each x.

Proof Since F2 = 8f, where f(x) = I x l is continuous and convex, F2 is maximal monotone. The remaining assertions are readily verified.

The following sufficient condition for a map to be usco is quite useful. Proposition 7.5. such that G(F2 ) is

Suppose that Fl , Fz are set-valued maps from X into Y closed, F2 C Fl and Fl is useo. Then F2 is useo.

Proof. The fact that G( F2) is closed implies that F2 (x) is closed for each x E X. Since G(F2 ) C G(Fl ) implies that F2(x) C Fl (x), each F2 (x) is necessarily compact. It remains to prove that F2 is upper semicontinuous. Suppose that x E X but that F2 is not upper semicontinuous at x. Then there exist an

104

Section 7

open set U C Y containing F2 ( x ) , a net (x ) c X such that x x, and points yO' E F2(xO') \ U for each Since FI is upper semicontinuous at x, the net (YO') is eventually in every open neighborhood of FI (x). This implies that some cluster point y of (YO') is in the compact set FI(x). (Indeed, if not, then for every y E F2 (x) there would exist an open neighborhood Vy of y such that (YO') is eventually in Y \ Vy . By covering FI (x) with such neighborhoods and using compactness, we could find an open set W containing FI (x) such that (YO') is eventually outside W.) Take a subnet of (xO" YO') converging to (x, y)j since G(F2) is closed, we conclude that y E F2(x) C U. On the other hand, YO' E Y \ U for all so y E Y \ U, a contradiction which completes the proof. We want to extend the notion of "maximal monotone" in the following way: 0'

0'

---7

a.

a,

T: E 2 Eo is monotone and D e E, we say that T is maximal monotone in D provided the monotone set

Definition 7.6. H

---7

G(T) n (D x E*) == ((x, x*) E D x E*: x E D and x * E T(x)} is maximal (under set inclusion) in the family of all monotone sets contained in D x E*. We will see shortly (Cor. 7.8, below) that if T is a maximal monotone operator in E, then its restriction to any open subset D of D(T) is maximal monotone in D.

Suppose that D e E is open, that T: D 2 Eo is monotone and norm-to-weak* upper semicontinous and that T( x) is nonempty, convex and weak* closed for all X E D; then T is maximal monotone in D.

Lemma 7.7.

---7

Proof. As with the usual notion of maximal monotone, we need only show that if (y, y * ) E D x E* satisfies ( y* - x*, y - x)

2':

0 for all x E D, x* E T(x),

(7.1 )

then y * E F(y). H not, by the separation theorem there exists z E E such that F(y) c {z* E E*: (z* , z ) < (y*, z ) } W . Since W is weak* open and T is norm-to-weak* upper semi continuous, there exists a neighborhood U of y in D such that T( U ) C W . For t > 0 sufficiently small, y + t z E U and therefore T(y + t z ) C W . Applying (7.1) to any u* E T(y + tz) we get =

o �

( y * - u *, y - (y + tz)) = - t ( y * - u*, z ) ,

which implies that ( u*, z )

2':

(y*, z ) , that is, u* is not in W , a contradiction.

If T: E 2 Eo is maximal monotone and D is a nonempty open subset of D(T), then the restriction Tv of T to D is maximal monotone in D.

Corollary 7.8.

---7

105

7. A generalization of monotone operators: Useo maps.

Proof. By Example 7.2, the maximal monotonicty of T implies that the mono­ tone map TD is convex w*-usco, so the result follows from Lemma 7.7. The next theorem exhibits an interesting relationship between maximal monotone operators in an open set D and minimal usco maps in D.

If D is open in the Banach space E and if T: D 2 E* has nonempty values and is maximal monotone in D, then T is a minimal convex w* -usco map in D. Theorem 7.9.

-+

Proof. We know by Example 7.2 that T is convex and w* -usco, so the only question is whether it is minimal in this family. Suppose that F: D 2 E* is convex and w*-usco and that G(F) C G(T). By Lemma 7.7, F is maximal monotone and hence F = T. -+

It is easy to see that not every minimal convex w* -usco is monotone; consider any continuous non-monotonic function from R into itself. The application of these notions (usco maps, minimal usco maps, etc. ) to a generic continuity theorem for monotone operators requires several further lemmas.

Let X , Y be Hausdorff spaces and suppose that F: X 2Y has closed graph and is locally relatively compact, that is, every x E X has an open neighborhood V M£ch that F(V) is relatively compact in Y. Then F is a usco map. Lemma 7.10.

-+

Proof. Since G( F) is closed, every image F( x) is closed and contained in some relatively compact set, hence is compact. To see that F is upper semicontinu­ ous, it suffices to show that for each x in X and each open neighborhood V of x, the restriction of F to V is upper semicontinuous. By hypothesis, we may restrict attention to open neighborhoods V of x for which F(V) is compact. Given such a V, define FI : V 2Y by FI(x) F(V), x E V. Obviously, FI is a usco map and F l v C FI ; since G(F I �T) is closed in V x Y, Proposition 7.5 implies that F is upper semicontinuous in V. -+

=

If D is open in E and T: D 2 E* is monotone, with Iiil nonempty for each x E D, then the map T whose graph is the closure G(T) in D x (E*, w*) of G(T) is monotone and w* -usco.

Lemma 7.1 1 .

-+

Proof. Let TI : D 2 E* be a maximal monotone extension of T. By Theorem 7.9, TI is a minimal convex w*-usco map in D, and by Prop. 7.3 its graph G(TI ) is closed in D x (E*, w*). Thus, T c TI so T is monotone and-by Prop. 7.5-it is also a w* -usco map. -+

The next theorem shows how the concept of minimal usco maps can be used to prove a basic result about monotone operators.

106

Section 7

Suppose that X is a Hausdorff space and that T: X 2 E* is w* -usco. 'For x E X define coT(x) to be the weak* closed convex hull of T(x). Then the map coT is convex w* -usco.

Lemma 7.12.

-+

Proof. Since 'CoT obviously has weak* compact convex values, it suffices to prove that it is weak* upper semicontinuous. To see this, suppose x E X and that U is a weak* open subset of E* with 'CoT(x) c U. In any locally convex space, a compact convex set I< has a neighborhood base of the form I< + lV, where the closed convex sets W form a neighborhood base of o. Thus, we can assume that U is of the form U coT(x) + W, where W is a weak* closed convex neighborhood of o . By the upper semicontinuity of T there exists an open neighborhood V of x in X such that =

T(V) C coT(x) + W. It follows that coT ( V ) C coT(x) + W, so coT is upper semicontinuous. Suppose that D is an open subset of the Banach space E and that T: D 2 E* is a monotone operator, with T( x) i= 0 for all x in D. Then there is a unique maximal monotone operator M in D containing T. In fact, M can be characterized as follows: Let T be the set-valued map whose graph is the closure in D x ( E*, w* ) of G(T) and for each x E D let M(x) be the weak* closed convex hull of T(x); this defines M.

Theorem 7.13. -+

Remark. As Example 7.4 shows, one must distinguish between sets of the form T( x) and the ( possibly smaller) sets T( x ). Proof. Let TI be any maximal monotone operator containing T. By Theorem 7.9, TI is a minimal convex w*-usco map. Since it has closed graph, we must have T C T1 , and since T has closed graph, Prop. 7.5 implies that it is w*-usco. From Lemma 7.12, we conclude that the map M coT is convex w*-usco and, clearly, M C T1 • By the minimality of T1 , we have M T1 , which proves the =

=

uniqueness assertion.

There does not appear to be a unique extension theorem for monotone operators with arbitrary effective domains. If, for instance, E has dimension at least one, if (xo, Yo) E E x E* and if T is defined to be the monotone operator whose graph is {( Xo, Yo) }, then there are many maximal monotone extensions of T. The next lemma, which is purely topological in nature, has Kenderov 's Theorem 2.30 on continuity of maximal monotone mappings as an immediate corollary. The main hypothesis will seem less peculiar when we apply the lemma to maximal monotone operators.

Let F be a minimal usco map on the Baire space X with compact values in the Hausdorff space (Y, ) and let d be a metric on Y. Sup­ pose that for every nonempty open subset U of X there exist nonempty open Lemma 7.14.

r

7. A generalization of monotone operators: Usco maps.

107

subsets V of U such that F( V ) contains relatively open subsets of arbitrarily small d-diameter. Then there exists a dense Gli subset D of X such that F is single-valued and d-upper semicontinuous at each point of D . Proof. We first note that the fact that F is a minimal usco map implies that if J is a proper closed subset of G( F), then p( J) =f X, where p is the natural

projection of X x Y onto X . Indeed, if p( J) = X, then J would be the (closed) graph of a set-valued map which, by Prop. 7.5, would be a usco map properly contained in F. Next, given e > 0, let Of

=

U{G: G is an open subset of X and d - diam F(G) :s:

e}.

Clearly, Of is open in X j we will show that it is dense. Let U be a nonempty open subset of X. By hypothesis, there is a nonempty open subset V of U and a T-open subset W of Y such that F(V) n W =f 0 and d-diam (F( V ) n W) :s: e. Since G(F) n (V W) =f 0 and F is minimal (and, by Prop. 7.3 (a), G(F) is closed), we must have p[G(F) \ (V x W)] =f X. Choose Xo E X \ p[G(F) \ (V x W)]. Then p - t (xo ) n G(F) C V x W; that is, Xo E V and F(xo) C W. If G {x E X : F(x) C W} n V, then G is an open neighborhood of Xo with d-diam F(G) :s: d-diam(F(V) n W) :s: e. It follows that G C Of and therefore 0 =f V n Of C u n Of ' This proves that Of is dense in X. Now let -

=

D = n{O t /n: n

=

1 , 2, 3, ... }.

Since X is a Baire space, D is a dense G li subset of X . From the definition of D it is evident that not only is F( x) a singleton at each point x E D but that F is d-upper semicontinuous at each such point. This lemma leads to the following alternative proof of Theorem 2.30.

Let E be a Banach space such that every bounded nonempty subset of E* is weak* dentable. If T: E 2E* is maximal monotone, with X == int D(T) =f 0, then there exists a dense Gli subset D of X such that T is single-valued and norm-to-norm ttpper semicontinuous at each point of D . Theorem 7.15.

---t

Proof. We know from Example 7.2, Cor. 7.8 and Theorem 7.9 that T is a minimal convex w*-usco map from X into (E*, w*)j by Prop. 7.3 it contains a minimal w* -usco map Tt . By Theorem 2.28, T is locally bounded in X, hence the same is true of Tt . Consequently, given any nonempty open subset U of X there exists a nonempty open subset V of U such that Tt (V) is bounded. By the weak* dent ability hypothesis, Tt (V) admits nonempty relatively weak* open neighborhoods of arbitrarily small norm diameter, by Lemma 7.14 there exists a dense Gli subset D of X such that Tt is single-valued and norm­ to-norm upper semicontinuous at each point of D. Now, by Lemma 7.12, coTt is convex w*-usco, and hence by the minimality of T l x , we have coTt = T l x . so

108

Section 7

Thus, for x E X , if T1 (x) is contained in a closed ball, then so is T(x). It follows that T is also single-valued and norm-to-norm upper semicontinuous at each x E D. The following exercise exhibits a particular convex w*-usco map which is of considerable importance in optimization. 7.16. Exercises: The Clarke subdifferential. Suppose that f is a locally Lipschitzian real-valued function on a nonempty open subset D ( not necessarily convex ) of the Banach space E. For each x E D the generalized directional derivative r(x; h) of f at x in the direction h E E is defined by

r (x; h) = lim sup

( l , y )_( O + , x )

f(y + th) - f(y) . t

( a) Show that r (x; h) is finite and that for fixed x, the function h is positive homogeneous, subadditive and Lipschitzian. ( b) Show that if {xn } C D and Xn

-+

-+

r(x; h)

x E D, then for each h E E,

lim supJ" (xn ; h) :::; r (x; h). The Clarke subdifferential 8° f( x) of f at x E D is defined by

8° f(x) = {x · E E · : (x · , h} :::; r(x; h) for all h E E } .

( c ) Show that, for all x E E, the set 8° f(x) is a nonempty convex weak* compact subset of E· . ( d ) Show that the graph of 8° f is norm X weak· closed in D that 8°f is a convex weak· usco map whenever D = E.

X

E· and hence

If f is a continuous convex function, then it is locally Lipschitzian, hence one can define both its generalized directional derivative and Clarke subdifferential as above. The following proposition reassures us that these coincide with the usual notions of directional derivative and subdifferential in this case. Proposition 7.16.

If f is convex and continuous on a nonempty open convex subset D of E (hence is locally Lipschitzian), then for all x E D and h E E one has r(x; h) = d+ f(x)(h)

and hence 8° f(x) = 8f(x). Proof. It is immediate from the definitions that r( X; h) � d+f(x)(h) . To prove the reverse inequality, note that for any fixed 8 > 0, r ( x; h) = lim

sup

< - 0 + II w - " II < , 6 o < t < t:

f(y + th) - f(y )

(7.2)

7. A generalization of monotone operators: Usco maps.

109

Since (as in Lemma 1. 2 ) the difference quotient

fey + th) - f ey) t is nonincreasing as t -+ 0 + , the right side of (7.2) equals sup I·1m f - O + il y -xil < fD

fey + {h) - fey ) {

Now, there exists a neighborhood of x in which f has Lipschitz constant M say, so for all sufficiently small { > 0, if I l x - y ll < lb, then

>

0,

fey + lh) - fey ) l(x + {h) - f(x) 2bM < + { { and therefore r (x; h) ::; d+ f( x)( h) + 2b M . This being true for all b > 0, we obtain the desired inequality. The fact that the two subdifferentials coincide follows from the foregoing equality, the definition of 8° f( x) and the fact (shown in the proof of Prop. 1 .8) that 8f(x) = {x' E E' : (x' , h) ::; d+ f(x)(h) for all h E E}. Theorem 7.9 shows that if a monotone operator which takes on nonempty values is an open set D and is maximal monotone in D is necessarily a minimal convex w*-usco in D. Although the Clarke subdifferential of a Lipschitzian function f on an open set D is a convex w*-usco in D, it need not be a minimal convex w*-usco in D. Of course, it is minimal if f is convex, and J. Borwein [Bor2 ) has shown that it is minimal for certain other related classes of functions.

Remarks. Most of the results in this section were obtained independently by Drewnowski and Labuda [Dr-L) and Jokl [Jo); we have followed the exposition of [Dr-L), with some very significant improvements suggested by both S. Fitzpatrick and I. Namioka. The proof of Lemma 7.14 (which results in the new proof of one of Kenderov's theorems in Theorem 7.15) is a minor revision of one due to the highly esteemed John Rainwater [Rain), who was motivated by the work of M . E. Verona rVer). Between them, they have extended some of the generic differentiability results to Lipschitz continuous convex functions defined on a sort of quasi-interior of a closed convex set C (namely the Baire space of all non-support points of C). Some of their theorems were obtained independently by D. Noll [Nolh , 2 ). Far-reaching generalizations of these kinds of results have been obtained by J. Borwein, S. Fitzpatrick and P. Kenderov [B-F-K) and by M. E. Verona and A. Verona [V-V 1 ,2,3) . As mentioned above, the Clarke subdifferential plays an important role in opti­ mization; see, for instance, [Cl) for further properties and applications. While it is a very nice extension to locally Lipschitzian functions of the usual notion for convex functions, it has some drawbacks. For instance, it does not apply to lower semicontin­ uous convex functions which are not continuous (hence not locally Lipschitzian� , and it fails to be single-valued for some differentiable functions (such as f(x) = x sin� for x ::j:. 0, f(O) = 0; in this case it is easy to verify that 8° f(O) = [ - 1 , 1]).

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_____

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______"

INDEX

Absorbing point 28 Absorbing set 28 Affine function 1 a-cone 20 a-cone meager 21 Angle-small 2 1 Asplund space 1 3 , 22-25, 31-33, 79-82 Asplund space, weak 13, 33, 35, 95 Asplund's theorems 22, 35, 37, 70 p-derivative 59 p-differentiable 59 P-subdifferentiable 64 p-superdifferential 65 Banach-Mazur game 69 Bishop-Phelps theorems 44, 48, 49 Bomology 58 Borwein-0mo theorem 77 Borwein-Preiss theorem 68 Bounded, locally 4, 28 Bourgin's lecture notes 110 Br�ndsted-Rockafellar theorem 43, 48, 87 Bump function 59 Clarke subdifferential 108 Coban-Kenderov 77 Concave function 1 Convex function 1 Convex usco map 102 Cyclically monotone 26 Cyclically monotone, maximal 28, 53

Exposed point 83 Extended real-valued function 38 Extreme point 83 Fabian's theorems 66, 92 Farthest distance function 1 Fenchel transform 42 Frechet bomology 58 Frechet derivative 7 Frechet differential 7 Functional, gauge 1 Functional, Minkowski 1, 84 Functional, sublinear 1 Gateaux bornology 58 Gateaux derivative 3 Gateaux differentiability space 95 Gateaux differential 3 GDS 95 Generalized directional derivative 108 Generic differentiability 1 1 Ghoussoub-Maurey theorems 93 Hadamard bomology 58 Haydon's examples 59, 66 Indicator function 39 Inf-convolution 3 1 , 42 Infinite tree 80 Infinite o-tree 80 James' theorem 43, 61 John Rainwater 109 Kenderov's Theorems 30, 34, 107 Krein-Milman property 94 Kuo's theorem 36

o-tree 80 Dentability 24, 79 Dentability, weak* 79 Dick's lecture notes 110 Directional derivative 1 Directional derivative, generalized 108 Derivative, Frechet 7 Derivative, Gateaux 3 Distance function 1 Duality mapping 27

Lipschitzian extension 31 Lipschitzian, locally 4, 7 Locally bounded 4, 28 Locally bounded operator 28 Locally bounded subdifferential 7, 15 Locally Lipschitzian function 4, 16 Lower semicontinuous function 38

t-subdifferential 46 Effective domain, function 38 Effective domain, monotone operator 17 Effective domain, saddle function 54 Ekeland-Lebourg theorem 66 Ekeland's variational principle 45, 58, 6 1 , 88 Epigraph 38 Epi-sum 42

Maximal cyclically monotone 28, 53 Maximal monotone 26-27, 30, 56, 73, 104 Mazur's theorem 12 M-differentiability space 96 MDS 96 Metric projection 8, 18 Minimal usco map · 102 Minkowski gauge 1 , 84

116

Support point 43

Monotone, cyclically 26 Monotone, maximal 26, 56, 73, 104 Monotone operator 17, 26, 104 Monotone set 26

Talagrand's examples 77 Tree 80

N-cyclically monotone 26 Nearest point mapping 8, 18 Nonexpansive map 18

Upper semicontinuous map 18-19, 102 Upper semicontinuous function 88 Usco map 102

Parallel hyperplane lemma 99 PCP 93 Perturbed optimization 87 Positive operator 17 Preiss' theorem 68 Preiss-Phelps-Namioka theorem 73 Preiss-Zajicek theorem 22 Proper convex function 38 Proper saddle function 54-55 Property (H{J) 59

Variational inequality 8 Variational principle 45, 49, 58

Rademacher's theorem 1 1 Radon-Nikodym property 79 Right-hand derivative 2 Rockafellar's theorems 52, 53 Rosenthal's theorem 36 Rotund norm 33 Saddle function 54 Selection 19 Semicontinuous function, lower 38 Semicontinuous function, upper 88 Semicontinuous map, upper 18, 102 Simons' proof 51 Slice 24, 79 Slice, small diameter 24, 30, 31 Slice, weak* 24, 30-31 , 79 Smoot}> norm 34 Smootr space 34 Smooth variational principle, Borwein-Preiss 68 Smooth variational principle, DGZ 63 Stegall's theorem 88 Strictly convex norm 33 Strictly convex space 34 Strong maximum 88 Strong minimum 59 Strong w· H6 set 93 Strongly exposed point 83, 91 Subdifferential 6, 15, 19, 40 Subdifferential, Clarke 108 Subdifferential, saddle function 54 Sublinear functional 1 , 71 Support function 39 Supporting functional 43 -

WCG 13, 35, 66 Weak Asplund space 13, 35, 67 Weak Hadamard bornology 58 Weakly compactly generated 13, 35, 66 Weak* dentable 79 Weak* exposed point 83 Weak* slice 24, 79 Weak* strongly exposed point 83 Weak* usco map 102

INDEX OF SYMBOLS

B(x; r)

4

C(p)

84

D{3

62

df

3

d+ f

2

de

1

oe

39

D(T)

17

dom( f)

38

epi( f)

38

G(T )

26

1<>-

43

Pe

1

O"A

39