Topologies on Closed and Closed Convex Sets [1st ed.] 978-0-7923-2531-4;978-94-015-8149-3

Table of contents :
Front Matter ....Pages i-xi
Preliminaries (Gerald Beer)....Pages 1-33
Weak Topologies Determined by Distance Functionals (Gerald Beer)....Pages 34-77
The Attouch-Wets and Hausdorff Metric Topologies (Gerald Beer)....Pages 78-105
Gap and Excess Functionals and Weak Topologies (Gerald Beer)....Pages 106-137
The Fell Topology and Kuratowski-Painlevé Convergence (Gerald Beer)....Pages 138-182
Multifunctions: the Rudiments (Gerald Beer)....Pages 183-234
The Attouch-Wets Topology for Convex Functions (Gerald Beer)....Pages 235-269
The Slice Topology for Convex Functions (Gerald Beer)....Pages 270-305
Back Matter ....Pages 306-340

Citation preview

Topologies on Closed and Closed Convex Sets

Mathematics and Its Applications

Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 268

Topologies on Closed and Closed Convex Sets by

Gerald Beer Department of Mathematics and Computer Science, Califomia State Ulliversity, Los Allgeles, U.SA

•

"

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data Beer, Gerald Alan. Topologies on closed and closed convex sets I by Gerald Beer. p. cm. -- (Mathemat ics and its appllcatlons ; 268) Inc I udes index. ISBN 978-90-481-4333-7 ISBN 978-94-015-8149-3 (eBook) DOI 10.1007/978-94-015-8149-3

,. Topalogy. 2. Hyperspace. 3. Metrlc spaces. 4. Normed linear spaces. 1. Title. 11. Series: Mathematlcs and its applications (Kluwer Academic Publishers) ; 268. OA6" .B38 1993 514' .32--dc20 93-31538

ISBN 978-90-481-4333-7

Printed on acid-free paper

All Rights Reserved © 1993 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1993 Softcover reprint ofthe hardcover 1st edition 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanicaI, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

to Brendan, Timothy, and Kevin

Table of Contents Preface

ix

Chapter 1. Preliminaries

1

Section Section Section Section Section

1.1 1.2 1.3 1.4 1.5

Notation and Background Material Weak Topologies 8 Sernicontinuous Functions 13 Convex Sets and the Separation Theorem 20 Gap and Excess 28

Chapter 2.

Weak Topologies Determined by Distance Functionals

34

Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5

The Wijsman Topology 34 Hit-and-Miss Topologies and the Wijsman Topology 43 UC Spaces 54 The Slice Topology 60 Complete Metrizability of the Wijsman and Slice Topologies 69

Chapter 3.

The Attouch-Wets and Hausdorff Metric Topologies

Section 3.1 Section 3.2 Section 3.3 Section 3.4

The Attouch-Wets Topology 78 The Hausdorff Metric topology 85 Varying the Metrics 92 Set Convergence and Strong Convergence of Linear Functionals 100

Chapter 4.

Gap and Excess Functionals and Weak Topologies

Section 4.1 Section 4.2 Section 4.3 Section 4.4

Farnilies of Gap and Excess Functionals 106 Presentations ofthe Attouch-Wets and HausdorffMetric Topologies 113 The Scalar Topology and the Linear Topology for Convex Sets 121 Weak Topologies deterrnined by Intimal Value Functionals 128

78

106

viii

Chapter 5.

The Fell Topology and Kuratowski-Painleve Convergence

Section 5.1 Seetion 5.2 Seetion 5.3 Seetion 5.4 Section 5.5

The Fell Topology 138 Kuratowski-Painleve Convergence 145 Epi-convergence 155 Mosco Convergence and the Mosco Topology 170 Mosco Convergence versus Wijsman Convergence 178

Chapter 6.

Multifunctions:

Seetion 6.1 Seetion 6.2 Seetion 6.3 Seetion 6.4 Seetion 6.5 Seetion 6.6

Multifunctions 184 Lower and Upper Sernicontinuity for Multifunctions 192 Outer Sernicontinuity versus Upper Sernicontinuity 199 KKM Maps and their Application 208 Measurable Multifunctions 216 Two Selection Theorems 228

Chapter 7.

The Attouch-Wets Topology for Convex Functions

Seetion 7.1 Seetion 7.2 Seetion 7.3 Section 7.4 Seetion 7.5

Attouch-Wets Convergence ofEpigraphs 235 Continuity ofPolarity and the Attouch-Wets Topology 241 Regularization of Convex Functions and Attouch-Wets Convergence 250 The Sum Theorem 256 Convex Optirnization and the Attouch-Wets Topology 264

Chapter 8.

The Slice Topology for Convex Functions

Section 8.1 Seetion 8.2 Seetion 8.3 Seetion 8.4

Slice and Dual Slice Convergence of Convex Functions 270 Convex Duality and the Slice Topology 276 Subdifferentials of Convex Functions and the Slice Topology 287 Stability ofthe Geometrie Ekeland Principle 299

The Rudiments

138

183

235

270

Notes and References

306

Bibliography

315

Symbols and Notation

331

Subject Index

335

Preface In this mono graph, we give an account of the most important topologies on the closed subsets of a metric space, and on the closed convex subsets of a normed linear space. Such topologies are called hyperspace topologies. In the process, we attempt to develop a unified theory for uniformizable hyperspace topologies, based on natural set functionals. It is a remarkable fact that the most important hyperspace topologies arise as topologies induced by families of geometric set functionals. We give particular attention to the interplay between hyperspaces and geometrie functional analysis, and to the convergence of lower semicontinuous functions (especially lower semicontinuous convex functions) as associated with their epigraphs. This exposition of necessity requires some vocabulary from convex analysis, Banach space geometry, and optimization, and we make a serious attempt to fill in holes for the primarily topologically-oriented reader. On the other hand, there is a good deal of undiluted topology in the text, and we are hopeful that the reader committed to applied analysis will be willing to stretch in that direction. Although hyperspace topologies and related set convergence notions have been considered since the beginning ofthis century, the approach we take to the subject reflects decisive modem contributionsby mathematicians whose primary research interests lie outside general topology. The revival of the subject sterns from work of Robert Wijsman in the middle 1960's, and its development over the next fifteen years was to a large extent in the hands ofUmberto Mosco, Roger Wets, J.-L. Joly, Ennio De Giorgi, Hedy Attouch, and their associates. This new approach was developed for the most part in North America, France and Italy. It is impossible to overestimate the French influence on my own point of view, through the monographs of Claude Berge, Jean-Pierre Aubin, and Hedy Attouch, and frequent informal communication as weIl as formal collaboration with French colleagues. Much of the material in this book did not exist ten years ago. Still, the book is designed as a textbook as much as a research mono graph. This explains the abundance of elementary exercises, frequent digressions into background material, and the uneven use of citation and historieal commentary in the body of the text (some amends are made in Notes and References, located at the end of the text). In no case should a lack of attribution be interpreted as a claim of ownership of a result by the author. Chapter 1 includes assorted background material from topology and geometrie functional analysis. It also attempts to collect in one place basie information about weak topologies, semicontinuous functions, and geometric set functionals. Chapter 2 focuses on the Wijsman topology for the nonempty closed subsets of a metric space , which may be viewed as the topology that such sets inherit from the function space C(X,R) , equipped with the topology of pointwise convergence, under the identification A H d(- ,A), where d(-,A) is the usual distance functional for the set A. If we allow our metries to vary over all admissible metrics and take the supremum of resulting Wijsman topologies, we obtain the classieal Vietoris topology, whieh is a prototype for hit-and-miss hyperspace topologies. Suprema for less inclusive classes of metrics result in weaker topologies of the Vietoris type. Most importantly, we give several presentations of the slice topology for the nonempty closed convex subsets of a normed linear space, which is the supremum of the Wijsman topologies as determined by all norms equivalent to the initial norm. Chapter 3 contains basic information about the Hausdorff metric and Attouch-Wets topologies for the nonempty closed subsets, which are also obtained from C(X,R) under the above identification, where in the first case the function space is equipped with the ix

x

topology of uniform convergence, and in the second, the function space is equipped with the weaker topology of uniform convergence on bounded subsets. The latter topology is particularly natural from the perspective of linear analysis, since (1) strong convergence of continuous linear functionals means uniform convergence of the functionals on bounded subsets, and (2) strong convergence of continuous linear functionals corresponds to the Attouch-Wets convergence of the affine objects (graphs and level sets at fixed heights) associated with the functionals. Perhaps unexpectedly, Attouch-Wets convergence can be replaced by the weaker slice convergence in (2). Chapter 4 attempts to formulate a general approach to the study of topologies on the closed subsets of a metrizable space based on weak topologies determined by (1) geometric set functionals, and (2) infimal value functionals. For example, it is shown that the Attouch-Wets topology is the weak topology determined by gap and excess functionals with fixed c10sed and bounded left argument. Chapter 5 introduces the Kuratowski-Painleve convergence of nets of sets and the c10sely related Fell topology, distinguished by their compactness in complete generality. Convergence of lower semieontinuous functions, as identified with their epigraphs, is studied in detail. In the last two sections, we consider the Mosco topology for the weakly c10sed subsets of a normed linear space X, whieh is the supremum of the Fell topologies as deterrnined by the weak and strong topology on X. In a general Banach space, weak* (resp. Mackey) convergence of a sequence of continuous linear functionals corresponds to the Kuratowski-Painleve (resp. Mosco) convergence of the affine objects associated with the functionals. Chapter 6 presents the elements of the theory of multifunctions, focusing on continuity and measurability concepts, with an attempt to reconcile popular definitions based on the notion of the inverse image of a set under a multifunction with standard definitions in the single-valued case. An excursion is made into nonlinear analysis, where the KKM Principle for set-valued maps may be used as a unifying tool. The final section presents two basic selection theorems, one for lower sernicontinuous c10sed convex-valued multifunctions, and the other for measurable closed-valued multifunctions.

Castaing

representations are introduced as a tool to check measurability. Chapter 7 and Chapter 8 exhibit connections between topologies on c10sed convex sets and on lower sernicontinuous convex functions and convex duality in a general Banach space. For convex sets, the primary carrier of duality is the polar map, and for convex functions, it is the Young-Fenchel transform, assigning to each function its Fenchel conjugate. In the process, we carefully introduce a number of important tools of convex analysis, including variational principles, the subdifferential, and regularization by smoothing kernels. The approach we take is highly geometrie al. For example, we formulate the Ekeland Principle as a statement about the existence of cone-maximal points in an epigraph, relegating its analytic equivalent to the exercises. Similarly, following Wets and Pe not, we establish the bieontinuity of the Fenchel transform for the AttouchWets topology as a consequence of the continuity of certain geometrie operations on cones. Epigraphieal Attouch-Wets and slice convergence of convex functions is characterized in terms of the convergence of Lipschitz regularizations with respect to c1assieal function theoretic modes of convergence. Fundamental characterizations of the slice topology in terms of convex duality are also established. For a course focusing on topological results, we recommend the following sections as a syllabus: Chapter 1 : all Chapter 2 : all Chapter 3 : all

Chapter 4 : all Chapter 5 : Sections 1-3 Chapter 6 : Sections 1-3

xi

On the other hand, for a course putting more emphasis on geometrie functional analysis, a reasonable syllabus might be : Chapter 1 : all Chapter 2 : Seetions 1,2,4 Chapter 3 : Seetions 1,2,4 Chapter 4 : Seetions 1,3

Chapter 5: Chapter 6: Chapter 7: Chapter 8:

a11 Seetions 1-4 Sections 1-3 Seetions 1-3

The author is indebted to the following colleagues who read various parts of this mono graph : Andrei Verona, Roberto Lucchetti, Hedy Attouch, Camillo Costantini, Bob McCoy, Nils Mork, and especially Bob Tamaki. The author has been moved by the general level of camaradery and goodwill that exists amongresearchers in thefield,from which he has greatly profited. Over the years, I have been sustained and encouraged by so many. In addition to my reviewers, I wish to particularly acknowledge in this direction Roger Wets, Michel Thera, Sandro Levi, 10n Borwein, Yves Sonntag, Petar Kenderov, Arrigo Cellina, Stephen Simons, Som Naimpally, Alek Lechicki, 1.-P. Penot, and Wim Vervaat. I have come to realize that personal relationships are more important than the mathematics. Finally, I would like to thank my sons Tim and Kevin for their help in the production of this monograph. GERALDBEER

Los Angeles, California May, 1993

Chapter 1 Preliminaries In this text we will consider topologies on the closed subsets of ametrie space and on the closed convex subsets of a normed linear space. We shall call such topological spaces hyperspaces . Many natural hyperspace topologies are not metrizable, and some are not even Hausdorff. Minimally, we will insist that.a hyperspace topology extends the initial topology on the underlying metric space. In other words, if we restriet the topology to the singleton subsets, we want the induced subspace to agree with the initial topology on the underlying space. This property will be called admissibility. Although we will from time to time study certain topological properties of hyperspace topologies and identify coincidence for various topologies, this is not the basic purpose of this monograph. Instead, our primary goal is to expose the interplay between topologies and geometrically defined set functionals, and in the context of normed linear spaces, linear duality. We also aim to develop aspects of hyperspace topology which are applicable to unilateral analysis and one-sided optimization theory, convex analysis, geometrie functional analysis, mathematieal economics and the theory of random sets. Our focus will be on the presentation of such topologies which are convenient for such applications. This text presupposes a reasonable knowledge of general topology and, to a lesser degree, some exposure to functional analysis in the context of normed spaces. Nevertheless, there are certain aspects of these disciplines not ordinarily stressed in introductory courses and texts that playa central role in the modem theory of hyperspaces. It is the purpose of this chapter to fill in some potential gaps in the reader's background. First, we set forth some notation.

§1.1. Notation and Background Material We adopt standard notation for set operations, e.g., union, intersection, set difference, cartesian product, and complementation are represented by A U B, A n B, A\B, A x B, and AC, respectively. The set of real numbers is denotedby R and the set of positive integers by Z+. The set of all functions from a set A to a set B will be denoted by BA and the identity function for a set A will be denoted by idA. We shall for the most part develop the theory of hyperspaces in the context of metric spaces and often in the less general setting of normed linear spaces. Nevertheless, many of our results will be valid in Hausdorff topologie al spaces, and when appropriate, we may give our results in this more general setting. If be a Hausdorff space and let A be a nonempty subset of X. Then the relative topology on A is a weak topology induced by a single function, namely, the inclusion map iA: A ~ X defined by iA(a) =a. 1.2.4 Example. Let be a normed linear space with continuous dual X*. The topology a(X,x*) , called the weak topology, is the topology on X induced by X*. A local base at xo E X for a(X,x*) consists of all sets of the form {X

EX: 'V y E F, ky,x - xo>1 < E }

where F is a finite subset of X* and E > o. The elements of this local base are evidently convex sets, and the binary operations of addition and scalar multiplication

(a,x)

~

ooc.

are easily verified as continuous. A linear space equipped with a topology such that (1) at each point, there is a local base of convex neighborhoods; (2) addition and scalar multiplication are jointly continuous is called a locally convex space (l.c.s).

1.2.5 Example. Each element of a normed linear space X of course may also be regarded as a linear functional on X*. With this in mind, the weak* topology o(X*,X) is the topology on X* induced by X. This, too, is a locally convex topology. A fundamental result of functional analysis is the Banach-Alaoglu Theorem [HoI2, p. 70], which states that U* is a(X* ,X)-compact. When X**, the continuous dual of X*, is given the usual operator norm, then X sits in X* * metrically as weIl as linearly,

PRELIMINARIES

9

under the identification x H 1, where 1: X* ~ R is defined by 1(y) = (see Exercise 1.4.10). In general, this inc1usion is proper. When X =X** under tbis identification, we say that X is reflexive. The c1ass of reflexive spaces has many important characterizations among the Banach spaces to wbich we shall frequently allude in the sequel. For convenience, we provide a short list ofthese here (see [HoI2] for details): (i) the unit ball U of X is a(X,x*)-compact; (ii) X* is reflexive; (iii) for each y E

X*, there exists XE U with =lIyll*; (iv) each c10sed convex subset of X has an

element of minimal norm; (v) whenever A is c10sed and convex, then A + U is c1osed. The family of topologies on a set X forms a complete lattice - that is, given any collection of topologies on X, there is a smallest (resp. largest) topology on X containing (contained in) each member of the collection. Given a collection of topologies {'rj : j E J}, we write I\.jeJ'rj and V jeJ 7:j for their infimum and supremum with respect to tbis lattice.

Of course,

I\.je J 7:j

is just

n je J 7:j,

whereas

V je J 7:j

has

U je J 7:j as a

subbase. If each 7:j is itself a weak topology determined by a family of functions

~j

then VjeJ'rj becomes a weak topology in an obvious way.

1.2.6 Theorem. Let

{7:j: j

E J} be a family of topologies on a set X. Suppose that

each 'rj is the weak topology determined by a family of functions ~j. Then the weak topology determined by

V je J 'rj

is

U jeJ~j.

No such representation exists for the infunum of topologies, wbich in part explains the limited applicability of this construction. We also note this rather obvious fact: each supremum topology V jeJ 'rj on X is automatically a weak topology, determined by the family of functions {fj: j E J}, where Jj: X ~ Ul n U2 E 'U; (4) U E 'U => U-I E 'U; (5) 'V UI E 'U 3 U2 E 'U with U2 °U2 c:: UI. Members of 'U are called entourages , and the pair O:

Se[B]=>A}.

Exeess so defined is not symmetrie. For example, in the line with the usual metric, with and B = {5,6}, we have ed(A,B) = 5 and ed(B,A) = 1. We adopt the eonvention that if A is nonempty, then eJ{0,A) =O. The gap DJ{A,B) between nonempty subsets A and B of ametrie space is given by

A = [0,5]

Dd(A,B) = inf {d(a,B) : a E A} = inf {d(b,A) : bEB}

= inf {d(a,b) : a E A, = inf {E>O:

bEB}

An Se[B]

*0}.

Exeess and gap are illustrated in Figure 1.5.1 below. Unlike the exeess funetional, the gap funetional is finite-valued and symmetrie. Notice, that exeess and gap reduee to ordinary distanee when A is a singleton subset. Dur first result says that the exeess of A over B is deseribed by "half' of the uniform distance between the distanee funetionals for the two sets.

29

PRELIMINARIES

A

B

the excess of A over B and the gap between A and B FIGURE 1.5.1

1.5.1 Lemma. Let A and B be nonempty subsets of ametrie space . Then ed(A,B)

=

sUPxeX d(x,B) - d(x,A) .

Prooj. We work with the formula ed(A,B) = inf {e > 0: A c Se[B] }. Let us write

.Il =

sUPxeX d(x,B) - d(x,A). First, we show ed(A,B) s;.Il. If .Il = +00, there is nothing further to do. Otherwise, fix a E A and a>.Il. We have d(a,B) = d(a,B) - d(a,A) S;

.Il< a,

and it folIo ws that a E Sa[B]. Thus A c Sa[B]. This shows that inf {e > 0: A c Se[B] } s;.Il, and we obtain ed(A,B) S; .Il. We now show that ed(A,B) 2:.Il. Let e> 0 be arbitrary. Fix x EX, and choose a E A with d(x,a) < d(x,A) + el2. Now pick bEB with d(a,b) < d(a,B) + e/2 S; ed(A,B)

+ d2. We have d(x,B) S; d(x,b) S; d(x,a)

+ d(a,b) < d(x,A) + ed(A,B) + e,

so that d(x,B) - d(x,A) S; ed(A,B)

+ e.

Since x was arbitrary, we obtain sup d(x,B) - d(x,A)

S; ed(A,B)

+ e.

XEX

Since e was arbitrary, we have .Il S; e~A,B), completing the proof.

•

For convex sub sets of a normed linear space, excess has an attractive dual formulation. At the heart of our proof again is the Ascoli formula for the distance from a point x to a level set y-I(a) of a nonzero element Y E X* .

30

CHAPTER I

1.5.2 Theorem. Let be a normed linear space, and let d be the metric induced by the given norm. Then for each pair A and B of nonempty convex subsets of X, we have ed(A,B)

=

sup {s(y,A) - s(y,B): y

E

U* and y

E

dom s(·, B)}.

Let us write A = sup {s(y,A) - s(y,B) : y E U* and y E dom s(· , B) }. Since n dom sC- ,B), we have A ~ O. We first show that ed(A,B):S: A. This is obvious if ed(A,B) = O. Otherwise, let a be an arbitrary positive scalar with a< ed(A,B). By the definition of excess, there exists a E A with d(a,B) > a. Since d(a,B) = d((),a - B), by Lemma 1.4.7, there exists a norm one element y of X* with inf Proof.

()* E U*

{ : x

E

a - B}

=d((),a - B) > a.

s(y,A) - s(y,B)

~

Thus, -

SUPbeB

> a, and so

- s(y,B) > a.

Since y E dom sC- , B), we conclude that ed(A,B):S: A. The reverse inequality, ed(A,B) ~ A holds when A = O. Otherwise, fix a with o < a< A, and choose y E U* n dom sC- , B) such that s(y,A) - s(y,B) > a. By positive homogeneity of support functionals, we may assume without loss of generality that lIyll* = 1. With ß= s(y,B), we can find a E A such that > ß+ a. Noticing that y-l(ß) separates a from B, the formula for the distance from a point to the level set of a linear functional again gives ed(A,B) ~ d(a,B) ~ d(a,y -l(ß))

We conclude that

e~A,B) ~

A.

=

ky,a> lIyll*

ßI

= -

ß > u.

•

Since distance from a point to a set is a special case of excess, we have the following dual formula for distance from a point to a nonempty convex set.

1.5.3 Corollary. Let be a normed linear space, and let d be the metric induced by the given norm. Suppose x E X and B is a nonempty convex sub set of X. Then d(x,B) = sup { - s(y,B): y E U*}. Proof. If y ~ dom sC- , B), then - s(y,B)

=-00,

be disregarded in the computation of the supremum.

so that such points of U* may

•

We now give a formula for gap in the spirit of Lemma 1.5.1.

1.5.4 Lemma. Let A and B be nonempty subsets of a metric space . Then = infxex d(x,B) + d(x,A).

Dd(A,B)

Let us set A = inf {d(x,B) + d(x,A) : x EX}. The inequality A:S: Dd(A,B) follows from Dd(A,B) = inf {d(x,B) + d(x,A): XE A}. For the reverse inequality, fix x

Proof.

PRELIMINARlES E X and take a Wethenhave

E

31

A and bEB with d(x,a) ~ d(x,A) + e/2 and d(x,b) ~ d(x,B) + el2.

Dd(A,B) ~ d(a,b) ~ d(x,A) + d(x,B) + e.

Since e was arbitrary, we have Dd(A,B) Dd(A,B) ~ A follows. •

d(x,A) + d(x,B), and since x was arbitrary,

~

For nonempty subsets A and B of a normed linear space , we c1early have Dd(A,B) =d(8,A - B). For convex sets, the gap between two sets agrees with the linear separation between them, as expressed by our next result. First, a definition.

1.5.5 Definition. Let be a normed linear space and let A and B be nonempty convex sets. We say that y E x* e-separates A and B if for some a E R, y-l (a) separates A and B, and e =

IIY~I* diam (a:

y-l (a) separates A and B}.

It is not hard to show that e is the gap between those level sets of y separating A and B that are furthest apart (see Exercise 1.5.9). Clearly, convex sets A and B can be strongly separated by a continuous linear functional y if and only if y e-separates A

and B for some positive e.

1.5.6 Theorem. Let be a normed linear space, and let d be the metric induced by the given norm. Then for each pair A and B of nonempty convex subsets of X, we have Dd(A,B)

=

sup {e ~ 0: ::3 y

E

X* that e -separates A and B}.

Let A be the given supremum. We first show that Dd(A,B) ~ A. This is obvious if Dd(A,B) = O. If not, set a =Dd(A,B) > O. Since Dd(A,B) = d(8, A - B), by Lemma 1.4.7, we can separate aV from A - B by a c10sed hyperplane of the form y -l(a) where y is a norm one element of X*. This means that ~ a for each

Proof.

a E A and bEB, so that

sup + a

bEB

~

inf

aEA

As a result, the level sets of y from height s(y,B) through height s(y,B) + aseparate

A and B, and since lIyll* = 1, we conc1ude that Dd(A,B) =

y

a -separates A

and B. Thus,

a ~ A.

For the reverse inequality, suppose that y E X* e-separates A and B. We show that Dd(A,B) ~ e. If this fails, then we can find a E A and bEB such that lIa - bll < e. But then 1 lIyll* . ky,a> - 11

~ Ila -

bll < €,

32

CHAPfER I

and this contradicts the E-separation of A and B by the functional y. Dd(A,B) ~ E. Since y was arbitrary, this shows that D~A,B) ~ .1.

We conclude that

•

1.5.7 Corollary. Let A and B be nonempty convex subsets of a normed linear space . Then A and B can be strongly separated if and only ifthere is a positive gap between them. Exercise Set 1.5. 1.

2.

3.

4. 5.

6.

Let A, B and C be nonempty subsets of ametrie space . (a) Show that e~A,B) e~cl A, cl B) and D~A,B) D~cl A, cl B). (b) Establish the triangle inequality for excess: e~A,C) ~ e~A,B) + e~B,C) . Let B be a nonempty subset of ametrie space , and let {Ai: i E I} be a family of nonempty subsets of X. Establish the following formulas : (i) D~U Ai, B) =infiel D~Ai,B); (ii) ed(U Ai , B) =SUPie I e~Ai,B). Let A, B and C be nonempty subsets of a normed linear space and let .1 E [0,1]. Establish each ofthe following assertions : (a) Dd(A,B) =d(8,A - B); (b) ed(clco B, A) =e~B, A) provided A is convex; (c) ed(A,AB + (1 - A.)C) ~ M~A,B) + (1 - A.)e~A,C). (d) if A is convex, then e~AB + (1 - A.)C,A) ~ M~B,A) + (1 - A.)e~C,A); (e) if A is convex, then D~AB + (1 - A.)C,A) ~ )J)~B,A) + (1 - A.)D~C,A). Explain why sup can be replaced by max in the statement of Corollary 1.5.3. In .n. 2, let U be the closed unit ball and let B = {: Xl + X2 = I}. (a) Identifying Jl.2 with its dual, show that dom s(· ,B) = {: Xl = X2 and Xn =0 for n ~ 3 }. (b) Using Theorem 1.5.2, show that ed(U,B) = 1 + 2- 112• In the plane R2 with the usual metric, let A = {(al,a2) : a2 ~ aI} and let

=

=

= {(al,a2) : a2 ~ 2aI }. (a) Identifying R2 with its dual, explain why dom s(- ,A) =dom s(- , B) = {(O,O)} U {(al,a2): a2< O}. (b) Show that whenever a2 < 0, we have

B

and

7.

8.

(c) Using Theorem 1.5.2, compute e~A,B). Let A be a weakly (resp. weak*) closed subset of a normed linear space X (resp. of a dual normed space X*) and let K be a weakly (resp. weak*) compact subset of X (resp. X*). Prove that A and K are disjoint if and only if the gap between them is positive. Let be a normed linear space. Suppose A E C(X), y is a nonzero element of X*, and H = {x EX: ~ a}. Establish these formulas for gap andexcess:

PRELIMINARIES

9.

D(}(A,H)

1 = lIyll*' max

e(}(A,H)

=

33

{a - s(y,A) ,0 },

I lIyll*' max {s(-y,A) + a, O}.

=

Let A be a nonempty convex set and let H y -I (a) be a closed hyperplane in a normed linear space . Establish the following formulas : ed(A,H)

I = lIyll*' max {s(y,A) - a, s(-y,A) + a};

D(}(A,H)

=

I lIyll*' max {a - s(y,A), -a - s(-y,A), O}.

I Conclude that DJ(y-I(a),y-I(ß») =eJ(y-I(a),y-I(ß») = lIyll* ·Ia - ßI. 10.

Let A and B be nonempty subsets of a normed linear space with B bounded. Prove that e(}(A + B,B) sup {Ilall: a E A} (Hint : for each a E A, we can fmd y E X* with =IIall). Provide a counterexample when B is unbounded.

=

Chapter 2 Weak Topologies Determined by Distance Functionals Let be a metric space.

The Wijsman topology 'Z'wd on

CL(X) is the weak topology determined by the family {d(x,'): x EX}. Convergence in this sense for sequences of convex sets in Euclidean space was considered in a fundamental paper of R. Wijsman [Wij] that is the point of departure for the modern theory of set convergence. The main focus of this paper- convergence and convex duality- will be taken up in Chapters 7 and 8. Since each distance functional is nonnegative, a subbase for the Wijsman topology consists of all sets of the form {A

E

CL(X): d(x,A) < a}

{A E CL(X) : d(x,A)

(x EX, a> 0),

> a}

(x EX, a

> 0).

By Theorem 1.2.8, convergence of a net of closed sets s: k '/'\x - u.

Thenwhenever

(F,m)~({x},k),

k-I wehave O(F,m)r1. {WE X: p(W,x) < TqJ{x)} for

i/ cp(x)}.

each XE F, and in particular, 8(F,m) r1. {w EX: p(w,x) < k

8(F,m) can't He in the smaller ball S~X]. This completes the proof.

As a result,

•

We close this section with a result linking Wijsman convergence of level sets of continuous linear functionals at fixed heights to the convergence of the functionals themselves [So,BeI5]. This result is just one of many Hnking convergence of sequences in X* and the set convergence of the affine objects naturally associated with the terms of the sequence (see §3.4, §5.2 and §5.4).

2.1.11 Theorem. Let be a normed linear space, let y E x* and let 0, SerA] c: E. We now write :

E++ ;: {A

E

CL(X): 3 E> 0 such that SerA] c: E}.

For example, in R2 with the usual metric, {(x,a): x > 0 and xa = I} is contained in but is not strongly contained in the open set {(x,a): x > O}.

CHAPTER2

44

A nonempty closed set Amisses E provided A n E =0, which is to say that A E (EC)+. We say A strongly misses E or A isfar from E provided A E (EC)++. When E is nonempty, this means that the gap between A and E, as defined in §1.5, is positive, as illustrated in Figure 2.2.1. We remark that sets of the form E++ are not necessarily preserved if the metric d is replaced by an equivalent metric p. For example if d is the zero-one metric on Z+ and p is the metric defined by

p(ij)

E ={2i : i E Z+}, then E E ~+, but E ~ ~+ .

=I }~ - ~l land

Amisses E, but A does not strongly miss E

A E (EC) ++: A strongly misses E FIGURE

2.2.1

Metrics d and p on a set X are called uniformly equivalent if they determine the same uniformity (see §1.2).

As is weH known and easy to check, d and

p are uniformly

equivalent if and only if the identity function from to is bi-uniformly continuous. If d and

p are uniformly equivalent metrics, then for each e> 0, there

exists positive numbers al subset E of X, both

= at(e)

and and a2

=a2(e)

such that for each nonempty

and This shows that sets of the form E++ are unchanged for ametrie space if d is replaced by a uniformly equivalent metric. Suppose now that L1 is a subfamily of CL(X). By the hit-and-miss topology determined by L1, we will mean the topology having as a subbase all sets of the form Vwhere V is an open subset of X, and all sets of the form (BC)+ where BE L1. By the

WEAK TOPOLOGIES DETERMINED BY DISTANCE FUNCTIONALS

45

proximal hit-and-miss topology determined by .,1, we will mean the topology having as a subbase all sets of the form V- where V is an open subset of X, and all sets of the form (BC)++ where BE Li. Usually, we will assurne that .,1 contains the singletons. General properties of such topologies are explored in Exercises 2.2.4 - 2.2.12 as weH as in §4.4. The Wijsman topology 'rWd for CL(X) can often be presented as a proximal hitand-miss topology, which we call the ball proximal topology. For instance, this is the case when X is a normed linear space.

2.2.1 Definition. Let be ametrie space. The ball proximal topology 'rBd on CL(X) has as a subbase all sets of the form V- where V is an open subset of X, and all sets of the form (BC)++ where B is a closed ball. Evidently, we recover the same topology if we just let the open subsets V in the above definition run over the open balls of X, better justifying the name we have given to this topology (see Exercise 2.2.1). Dur next example shows that the Wijsman topology may not contain the ball proximal topology. 2.2.2 Example. Let {en: n E Z+} be the standard orthonormal base for 12, and let be the foHowing metric subspace of the sequence space : X = {e2i : i E Z+}

Let A A

E

i+ 1 ={-.-e2i-l: iE I

(BC)++.

U { i",:I 1 e2i-l : i E

Z+}. Note that for B

On the other hand, for each n

Z+}

=St[9], E

U {9}.

_'" so that we have Dd(A,B) ="'12,

Z+, the closed set An defined by An

A U {e2k: k ;::: n} meets B and thus cannot lie in (BC)++.

=

It is routine to verify that

is Wijsman convergent to A. This shows that 't'Wd may not contain 'rBd' The inclusion 'rBd ~ 'rWd is always valid [DMN]; necessary and sufficient conditions for equality, in the spirit ofTheorem 2.1.10, have been identified by Rolli and Lucchetti [HL]. These are given in our next result. 2.2.3 Theorem. Let be ametrie space. (1) The ball proximal topology 'rBd on CL(X) contains the Wijsman topology 't'Wd; (2) 'rBd

='rWd on CL(X)

if and only if each closed ball B in X is strictly

d-included in each of its open e-enlargements Se[B]. Proo.

As we noted at the beginning of this seetion, a subbase for 't'Wd consists of all

sets of the form {A E CL(X): d(x, A) < a}, where x runs over X and a> 0, plus all sets of the form {A E CL(X): d(x,A) > a}, where x runs over X and a> 0. Thus, we will show that 't'Wd

C

'rBd

by showing that each such subbasic open set lies in 'rBd'

To this end, suppose that Ao E {A E CL(X): d(x, A) < a}. Then for some a E AO we have d(x,a) < a. With e = a - d(x,a), each point in Sela] has distance less than

46

CHAPTER2

a from x, so that AO E Se[ar c: {A E CL(X): d(x, A) {A E CL(X): d(x, A)

< a}. This string shows that

< a} is open in the ball proximal topology. Next, suppose that

Ao E {A E CL(X): d(x, A)

ß="21 (a + d(x,Ao));

> a}. Let

we have

AOE {WE X: d(x,wȧ}++c: {WE X: d(x,wȧ}+ c: {A

E CL(X): d(x, A) > a}.

Since {w EX: d(x,w) > ß} is the complement of a closed ball, it follows that CL(X): d(x, A) > a} E TBd . We have now shown that 'l'Wd c: TBd'

{A E

That V- E TWd for each open V requires no assumptions whatsoever on the metric, and is immediate from Lemma 2.1.2. To verify the assertion of (2), it remains to show that we have (BC)++ E TWd for each closed ball B if and only if closed balls are strictly d-included in their enlargements. First suppose (BC)++ E

TWd

closed ball and let c> O. If Se[B] B c:

S,Ll+l[X]

c:

for each closed ball B. Let B

=X,

= S,Ll[x]

be a fixed

then

S,Ll+2[X]

c: Se[B],

and so B is strict1y d-included in Se[B]. Otherwise, write A = Se[B]C E CL(X). Since A E (BC)++, by the assumed equality of the hyperspace topologies, there exists points Xl, x2, .. .,xn in X and 8> 0 such that

Now let L == {i E {I, .. .,n} : d(Xi,A) > O}. The set L must be nonempty because (BC)++ :t= CL(X). Let a< 8 be a positive scalar with a< min { d(Xi,A) : i E L}, and write ci =d(Xi,A) - a.

We claim that B c: UieL Sei[Xi]. If not, there exists bEB

such that for each i E L, d(Xi,b)

~ ci> d(Xi,A) -

8. This means that

a contradiction. With ai =d(Xi,A) > ci for i E L, we have

This shows that B is strictly d-included in Se[B]. Conversely, suppose each closed ball is strictly d-included in each of its open enlargements. We need to prove that for each closed ball B, (BC)++ E TWd' To this end, let A E (BC)++. By definition, for some c> 0, we have Se[B] n A = 0, and by assumption, there exists a finite set ofpoints {XI,X2, ... ,xn} and positive reals cl< al, c2< a2, ... , cn < an such that

WEAK TOPOLOGIES DETERMINED BY D1STANCE FUNCTIONALS

47

Then

as required.

•

Perhaps the simplest and certainly the most well-studied hit-and-rniss hyperspace topology is the Vietoris topology, which may be defined for the closed sub sets of any Hausdorff space.

2.2.4 Definition. Let X be a Hausdorff space. The Vietoris topology 'l"v on CL(X) has as a subbase all sets of the form V-, where V is open in X, and all sets ofthe form W+, where W is open in X. For the prescribed subbase of the Vietoris topology, the hit sets consist of the open sub sets of X, and the miss sets consist of the closed sub sets of X. It should be noted immediately that in the context of metrizable spaces, the Vietoris topology is independent of the defining metric and only depends on the underlying topology. Admissibility of 'l"v is easy to check. Let lfI: X -7 CL(X) be the canonical injection ljI(x) = {x}. The injection lfI is continuous because lfI-1(V-) = lfI-I(V+) = V, and with respect to the relative topology on ljI(X), lfI is open because ljI(V) = V- n lfI(X) (see more generally Exercise 2.2.4). For arbitrary sub sets EI and E2 of X, we have (EI

n E2)+ =Ei n E!

'

whereas it is entirely possible that (EIn E2f =f. EIn E 2.. As a result, a typical basic open set determined by the prescribed subbase for

rv may be written as

where W, VI, V2, .. .,Vn are open sub sets of X (see more generally Exercise 2.2.4). A member A of a typical basic open set is illustrated in Figure 2.2.2. An alternative subbase for rv consists of all sets of the form [VI,V2, . . .,vn] == {A

E

CL(X): V i:S; n, An Vi =f. 0

and Ac UI'!:1 Vd,

where VI, V2, ... , V n is a finite family of open subsets of X. Evidently, each [VI,v2, .. ·,vn] lies in 'l"v. On the other hand, for each open V and W, we have V- = [V,X] and W + = [W]. Thus, the topology generated by all sets of the form [VI,v2, .. ·,vn] contains 'l"v. In fact, all sets of the form [VI,V2,·· .,vn] actually form a base for the topology, as Vietoris [Viel-2] first observed (see Exercise 2.2.13). For topological properties of the Vietoris topology, the reader may consult the fundamental paper of Michael [Mic1], as weIl as [Keel-2,McCl,Fle,Sm4,KT,Dor].

48

CHAPTER2

FIGURE

2.2.2

We intend to present the Vietoris topology for the closed sub sets of a metrizable space X as a weak topology, specifically, as a topology determined by a family of distance functionals obtained by varying metrics as well as points. This point of view originates from [BLLN], and the following results are established in this paper.

2.2.5 Theorem. Let X be a metrizable space, and let D denote the set of compatible metrics for X. Then the Vietoris topology 'Z"y on CL(X) is the weak topology determined by the farnily of distance functionaJs {d(x,') : x

E

X, d

E

D}.

Thus, the Vietoris topology is the supremum in the lattice of hyperspace topologies of the Wijsman topologies determined by the compatible metrics for X, and a net in CL(X) is ..y-convergent to A E CL(X) if and on1y if V x E X V d E D, we have d(x,A) = 1imll d(x,AA,).

Proof. Denote the weak topo1ogy determined by the given farni1y of distance functionaJs above by 'Z"weak. We first show that 'Z"weak c:: 'Z"y. Let d E D and x E X be fixed. For each a> 0, {B E CL(X): d(x,B) < a}

=Sa[xr

is aJready in 'Z"y. On the other hand, if A

for some ß> a, we have AnSß[x] A E

(Sß[x]C)+

c:: {B E

=0.

E

{B

E

CL(X): d(x,B) > a}, then

Then

CL(X): d(x,B) > a}.

This proves that each member of a subbase for

"Wd

be10ngs to "y, and since d was

arbitrary, we get "weak c:: ..y. For the reverse inclusion, Lemma 2.1.2 shows that each set of the form V - , where V is open in X, be10ngs to each Wijsman topology, and thus to 'Z"weak. In the

WEAK TOPOLOGIES DETERMINED BY DISTANCE FUNCTIONALS

case that W =X, we have W+

=CL(X)

E 'Z'weak, and if W

=0,

49

then W+

'Z'weak. Now let W be an arbitrary proper open subset of X, and let xo E

wc.

=0

E

Fix A E

W +; we produce a compatible metric P such that A

E

{B

E

1 CL(X) : p(xo,A) - 4" < p(xo,B)} c: W+ .

This would show that W+ contains a 'Z'weak- neighborhood of each of its points. To obtain p, let d E D be arbitrary. Since A and WC are disjoint nonempty closed sets, we can find by Urysohn's Lemma cp E C(X,[O,I]) such that CP(A) CP(WC) =1. Define p: X x X ~ [0,3/2] by p(x,y)

=min {1, d(x,y)}

°

=

and

+ Icp(x) - cp(y)1.

It is a routine exercise to verify that p as defined is ametrie equivalent to d (see Exercise 1 1.1.1.). Suppose that {B E CL(X): p(xo,A) - 4" < p(xo,B)} 0 s~ch that Sö[Gr g] c Se[Gr j]}

is a 'Z'öp-nieghborhood of f

If gE Se[Grj]++ then by the definition of excess, ep(Gr g,

Gr f) ::;; E. Since 0 and xo E X such that Se[(xo/(xo»] c V. Choose 0< el2 such that d(xo,x) < 0

WEAK TOPOLOGJES DETERMINED BY DISTANCE FUNCTIONALS

57

implies d'(j(xo),f(x» < El2. We claim that if ep(Gr JA, Gr j) < 8, then !f(xo) - JA(xo) I < e. By the definition of excess, there exists x E X such that p[(x!(x», (xo!A'(XO»] < 8. This implies that d(x,xo) < 8 and d'(j(x)!A,(xo» < 8, so that

d'(j(xo),f).(xo» < d'(j(xo)!(x» + d'(j(x)f).(xo» < e. We conclude that ep(Gr JA, Gr j) < 8 gives JA E Se[(XO,/txo»r c: V- . For (2), there exists e> 0 such that Se[Gr j] c: W. Now if ep(Gr JA,Gr j) < El2, then SEI2[Gr JA] c: Se[Gr j] c: W, which is to say that ep(Gr JA,Gr j) < El2 gives JA E W++. This establishes (2) and completes the proof ofthe lemma. • The proof of Lemma 2.3.2 shows that the p-proximal topology on qx,y) contains the topology of pointwise convergence. It also shows in effect that the pproximal topology has as a base all sets of the form {W++ : W open in X x Y}. The topology on qx,y) with base {W+ : W open in X x Y} has been called the graph topology [Nai1], and is much stronger. To reconcile proximal convergence with uniform convergence, we first state this prelirninary result, essentially observed by Naimpally [Nail].

2.3.3 Lemma. Let and 0 be arbitrary, and let f be a uniforrnly continuous function from X to Y. Choose 8< El2 such that for all x and w in X, d(x,w) < 8 implies

d'(j(x),f(w» < El2. Then if gE qx,y) and ep(Gr g,Grj) < 8, we have this estimate • for the uniform distance between fand g: SUPxeX d'(j(x), g(x» ~ 8 + El2 < e. Our final result of the section was in essence established by the author in [Be4].

2.3.4 Theorem. Let be a metric space. The following are equivalent : (1)

is a UC space;

CHAPTER2

58

(2) For each metric space

O. This means that for each nE Z+, there exists {xn,w n } c: SefX']C such that 0 < d(xn,w n ) < l/n, and since each X n and each W n is an isolated point of X, we may assume without loss of generality that the terms of XI, W!, X2, W2, . .. are distinct. Define f: X -7 [0, 1] by

fex) and for each

={ ~

nE Z+,

fn(x) =

if

X

= Xk

for some k

E

Z+

otherwise

define fn: X -7 [0, 1] by {

I - fex)

fex)

if x

=xn

or x:;;

W

n

otherwise

Each function so defined is continuous because , residing in Se[X']C, can have no cluster points. Observe that ep(Grfn, Grf) ~ d(xn,wn) < lIn , whereas for each index n, we have 1 = !f(xn) - fn(x n)I . This shows that uniform convergence fails, yet by Lemma 2.3.2 p-proximal convergence is obtained. This contradiction shows that S dX ']C is uniformly discrete for each € > 0 . It remains to show that the set X' of accumulation points of X is compact. Assume the contrary holds. To show this leads to a contradiction, we find it convenient to use the Tietze Extension Theorem [Dug, p. 149] : each continuous function with values in [0,1] defined on a closed subset of anormal topological space X can be extended to a globally defined continuous function into [0,1]. Noncompactness of X' allows us to select a sequence in X' with distinct

terms with no cluster point and a sequence of scalars such that for each n, 0 < €n < l/n, and such that the family ofballs {Sen[xn]: nE Z+} is pairwise disjoint. For each positive integer n, pick n distinct points {Xln, X2n, ... , x nn } in Sen[xn]. We next build a certain continuous function f: X -7 [0,1]. For each

XE

(U;,! Sen[xnDC, set

fex) =O. For each nE z+ and each integer i with 1 ~ i ~ n, set j(Xin) = i/no Finally, use the Tietze Theorem to extend f continuously to all of X with values in [0,1].

WEAK TOPOLOGIES DETERMINED BY DISTANCE FUNCTIONALS

59

Guided by the construction supplied in the first part of the proof, we produce a sequence in C(X,[O,ID convergent to I in the p-proximal topology but which is not uniformly convergent. By the Tietze extension theorem, for each nE Z+, we can construct In E C(X,[O,lD satisfying i

=1,2, ... , n,

and such that In(x) = fix) whenever d(x,xn) ~ En. It is routine to check that that for each nE Z+, ep(Gr In,Gr 1) ::;; 21n and !fn(xnn ) - f(x nn )I = 1. Thus, if X' is noncompact, the two function space topologies disagree, a contradiction. We may now conclude that (3) implies that is indeed a UC space. •

Exercise Set 2.3. 1. 2. 3.

4.

5.

6.

7.

Show that the restriction of the metric for a UC space to a closed subset makes the subspace a UC space. Is the product of UC spaces, equipped with the product metric, necessarily a UC space? (a) Prove that any metric space that is the union of a compact set and a uniformly discrete set must be a UC space. Prove that such aspace must be locally compact. (b) Prove that a locally compact UC space can be written as a union of a compact set and a uniformly discrete set [Wat]. Call a sequence in ametrie space pseudo -Cauchy provided that for every E> and for every nE Z+, there exist distinct integers k> n and m> n such that d(Xm,xk) < E . (a) Show each Cauchy sequence is pseudo-Cauchy. Give an example of a real pseudo-Cauchy sequence that is not a Cauchy sequence. (b) Show that ametrie space is a UC space if and only if each pseudoCauchy sequence with distinct terms has a cluster point [Toa,Be5]. Let X be a metrizable space such that X is compact. Show that there is a compatible UC metric [Nag,Rai,Mr2] (Hint [Be9]: If X'::f:. 0 and d is an initial compatible metric, let dl(X,W) =d(x,w) + max {d(X,x/), d(w,X / )} if x::f:. w). Prove that this in turn holds if and only if each closed subset of X has compact boundary. Let be ametrie space. Show that the completion of X is a UC space if and only if whenever 0, there exists ö> 0 such that d (A) < ö ==> X(A) < e. (a) Produce a sequence f, ft, fz,/3, ... in C(R,R) such that converges pointwise to I but such that limn~co ep(Grln, Grf)::F- O. (b) Produce a sequence f,/l,fz,/3, . .. such that limn~co ep(Grln, Grf) o but such that limn~co epCGrf, Gr In) ::F- O. Can I be uniformly continuous? Let fix) 2x and g(x) 2x + 1. With respect to the product metric on R2, compute ep(Gr g,Grf). Now compute the excess with respect to the Euclidean metric. Why are the function space results of this section still valid if the product metric P

=

9.

10.

=

=

on X x Y is replaced by Pl[(Xl,yt},(X2,Y2)]

=(d(Xl,x2)2 + d'(YloY2»1I2 ?

§2.4. The Slice Topology Let be a normed linear space. If we vary the norm over equivalent norms, we may weIl obtain different Wijsman topologies on CL(X) or even on C(X), the collection of c10sed convex subsets of X. In fact, Wijsman convergence for hyperplanes may not even be preserved under equivalent renorming. To see why this is so, we fIrst make the following defInition. 2.4.1 Definition. Let be a normed linear space. We say that 11·11 has the Kadec property provided weak convergence of a net to x in X along with convergence of to Ilxll together imply strong convergence of to x. In the dual space, we say that tl).e dual norm 11·11* has the weak* Kadec property provided weak* convergence of a net to y in X* along with convergence of to lIyll* together imply strong convergence of - inf = sup - 28 + ßI2, we have

20+

~

< IIb' - alil = IIb' - alio + ky,b' > - l.

We consider two cases which are exhaustive but not mutually exclusive: (i) IIb' - allo;:=: 38/2; (ii) ky,b' > - l;:=: 812 + ßI2. In case (i), since the diameter of B with

64

CHAPTER2

respect to PO is less than 8, we have dpo(a,B) ~ &2. as follows:

W= {XE X: . As we noted at the outset of this seetion, if the dual of the initial norm 11·11 fails to have the weak* Kadec property, then we can find a net of closed convex sets (in fact, hyperplanes) that is Wijsman convergent with respect to 11-11 but whose pol ars fail to Wijsman converge with respect to 11·11*, and by Theorem 2.4.6, the stronger 'l"s -convergence of polars must fail, too. In view of 'l"s -'l"s continuity of polarity, the Wijsman topology for such a norm must be properly weaker than the slice topology. To summarize: the condition 0

11·11* is weak* Kadec is necessary for the Wijsman topology on C(X) induced by 11·11 to agree with the formally stronger slice topology. This condition is also sufficient, as proved by Borwein and Vanderwerff [BoV], who showed in the process that equality of the topologies is separably sequentially determined. We state their main result without proof. 2.4.7 Theorem. Let be a Banach space. The following are equivalent :

CHAPTER2

66

(a) The dual norm on X* is weak* Kadec; (b) The Wijsman topology determined by 11·11 coincides with the slice topology on ceX); (c) IHI-Wijsman convergence and slice convergence agree for sequences in ceX) in each separable subspace of X. We c10se this section by giving a particularly practical, intuitive description of convergence of a net of c10sed convex sets with respect to the slice topology, established in [BeI9]. This points the way to a characterization of slice convergence of convex functions as identified with their epigraphs, which we shall present in §8.2.

2.4.8 Theorem. Let be a normed linear space, let Ao E ceX), and let O. Since 11 has been chosen so that Ao n int IlU"# 0, there exists ao E Ao n int IlU with > s(y, Ao n IlU) - d2. Now choose 8> 0 small enough such that ao + 8U c: IlU and 81lyll* < E/2. Again since the slice topology contains the Wijsman topology, {A E ceX): d(ao,A) < 8} is a ' - 81lylI* > s(y,Ao n j1U) - E. For all A, sufficiently large A1 n S8[ao] "# 0, and so lim inf1 s(y, AA, 1lU) ;::: s(y, Ao IlU). Evidently, lim SUP1 s(y, AA, n j1U) ~ s(y, Ao n j1U) holds when s(y, Ao n j1U) =Illlyll*, because s(y, AA, n 1lU) ~ s(y, j1U) =j1l1yll*. Otherwise, let E> 0 and choose a> 0 with

n

n

s(y,AO n 1lU) < a< rnin {Illlyll*, s(y,AO

n 1lU) + E}.

Let B = {x EX: IIxll ~ 11 and ;::: a}. Using the fact that Ao n int IlU"# 0, it is easy to check that not only Ao n IlU E (BC)++, but also that Ao E (BC)++. Again, since Ao n int IlU"# 0, by slice convergence, there exists an index An such that for all A,;::: An both AA, n int IlU"# 0, and AA, E (BC)++. As a result, for all A,;::: An, we have

n j1U) ~ a < s(y, Ao n j1U) + lim sUP1 s(y, AA, n 1lU) ~ s(y, AO n 1lU) holds. s(y, A1

and

E,

Thus, (ii) holds. For the converse, suppose (i) and (ii) both hold. Since {V- : V norm open} is contained in the Wijsman topology, (i) implies that whenever Ao n v"# 0, then for

WEAK TOPOLOGIES DETERMINED BY DISTANCE FUNCTlONALS

67

all A sufficiently large, we have A.:1, n V;f:. 0. Now suppose Ao e (BC)++ where B is closed, bounded, and convex. Choose ).L > 0 so large that Ao n int).LU;f:. 0 and B 1 c: 'i).LU. Choose by Corollary 1.5.7 y e X* and ß eRsuch that sup < ß< inf . aeAo beB By the left inequality, s(y, Ao n ).LU) < ß, so that by (ii), for all A sufficiently large, we have s(y, A.:1, n ).LU) < ß. This means that the sets A.:1, n ).LU and B can be strongly separated, so that A.:1, n ).LU e (BC)++. Since B c:

(BC)++.

~).LU,

We now obtain slice convergence of to AO.

we conclude that A.:1, e

•

It is left to the reader to formulate a dual of Theorem 2.4.8 for the dual slice topology.

Exercise Set 2.4. 1. 2.

3.

4.

5.

6.

Using Theorem 2.4.8, show that the slice topology reduces to the Wijsman topology for the nonempty closed convex subsets of n-dimensional Euclidean space Rn. Let A, Al, A2, ... be nonempty closed convex subsets of a normed linear space . Prove that A = 'rs-lim An if and only if both of the following two conditions are satisfied: (i) for each a E A, there exists a sequence strongly convergent to a such that for each n, an e An; (ii) for each y e X*, whenever is a bounded sequence such that for each n, an e An, then s(y,A) ~ lim sUPn~oo · Using Theorem 2.4.8 or Exercise 2.4.2, show that if 0, so is cl(B + aU) . (b) Prove that B is the closed unit ball for some norm equivalent to 11·110 if and only if B is closed and bounded, balanced, convex and 0 Eint B (Hint : for

=

9.

10.

11. 12.

such a set B, consider as a norm p(x) inf {a: x E aB}). (c) Prove Corradini's Theorem [Corl]: the supremum ofthe Wijsman topologies on CL(X) determined by equivalent norms has as a subbase all sets of the form V- where V is norm open, plus all sets of the form (BC)++ where B is a translate of a closed and bounded balanced convex set (Hint : use parts (a) and (b) and Theorem 2.2.3). Let be a normed linear space. (a) Let BE CB(X) be arbitrary. Show that the map A ~ cl (A + B) from C(X) to C(X) is continuous with respect to the slice topology. (b) Show that the slice topology 'rs is the weakest topology 'r on C(X) containing the Wijsman topology such that for each B E CB(X), the map A ~ cl (A + B) is 'r-continuous. (c) Show that for CE C(X), the map A ~ cl (A + C) need not be continuous with respect to the slice topology. Let be a Banach space. By a theorem of Krein and Smulian (see, e.g., [DS,HoI2]) a convex subset C of X* is weak* closed if and only if its intersection with j.lU* is weak* closed for each j.l > o. Suppose X is nonreflexive. (a) Show that there is a closed and bounded balanced convex subset of X* which is not weak* closed (Hint: let q> be a continuous linear functional on X* that is not an X-functional, and let C = {y E X*: 1q>(y)I::; I}). (b) Use (a) to produce a norm IHI1 on X* equivalent to 11·11* whose unit ball is not weak* closed (so by Exercise 1.4.8 the norm can't be a dual norm). (c) Produce a net of segments in X* which is 'rS -convergent but which fails to be Wijsman convergent with respect to IHI1 (Hint: there is a net in the unit ball as determined by IHI1 that is weak* convergent to a point outside). (d) Prove that the slice topology agrees with the dual slice topology on C*(X*) if and only if X is reflexive. Let be a nonreflexive normed linear space. Explain why C*(X*) is not a closed subset in the hyperspace is itself a Polish space. This is in some sense unexpected, in that the natural metric for the Wijsman topology displayed in §2.1 may fail to be a complete metric (see Example 2.1.6). Here, instead of thinking of the Wijsman topology as a weak topology, we find it convenient to think of CL(X) as sitting in C(X,R), equipped with the topology of pointwise convergence, under the identification A

H d( ., A).

We denote the topology of

pointwise convergence on C(X,R) by 'fp , and the closure of CL(X) in be metric spaces. By the compact-open topology

on A subfamily ".f of C(X,Y) is called equicontinuous provided for each x E X and E > 0, there exists Ö =Ö(X,E) > 0 such that for each ! E ".f, we have j{Sc5[X]) c: Se[f(x)] . For example, each equi-Lipschitzian family of functions, i.e., a family of Lipschitz functions that admits a common Lipschitz constant, is an equicontinuous family. We collect some well-known easily established facts about equicontinuous families in the following lemma. For general facts about function spaces, the reader may consult 'Z"co

C(X,Y), we mean the topology of uniform convergence on compact subsets of X.

[MNI].

2.5.2 Lemma. Let and 1. Thus, 'l's is a countable supremum of a countable farnily of Wijsman topologies, namely 'l's =sup {'l'wpy: Y E E}.

Proof. As usual, Uo will denote the closed unit ball with respect to the initial norm. Let f be the weak topology described above. By Theorem 2.4.5, f c 'l's.

For the reverse

inc1usion, eonsider a typical subbasie open set for 'l's viewed as a weak topology as presented in Theorem 2.4.5, whieh may be assumed to be ofthe form Cl = {A

E

C(X): a < py(x, A) < ß},

where x E X, Y E X*, and a and ß are sealars. Without loss of generality, we may A assume that ß> O. Let A E Cl be arbitrary. We produee Z E E and E> 0 such that A

A E {A E C(X): a

+ E< pz{x, A) < ß- E}

C

Cl .

This would show that Cl E f, and 'l'S c f would follow. We first claim the following: there exists a positive scalar 11 such that whenever Z E X* and A E C(X) satisfy pz{x, A) < ß, we have both py(x, A)

=py(x, A n (x + I1UO)) ;

74

CHAPTER2

pz(x, A)

=pz(x, A n (X + JlUO)) .

Fix z e X* and A e C(X) satisfying pz(x,A) < ß. Then for some a e A, we have IIx - allo::; pz{x,a) < ß, and we compute py(x,a)

Now set Jl

=

IIx - allo + ky,x - a>1 ::; (IIyllO + 1) IIx - allo < (IIyIlO + I)ß .

=(IIyIlO + I)ß.

The last inequality string yields pix,A) < Jl whenever

pz(x,A) < ß, so that by the definition of the metric Py we may conclude that py(x, A)

= pix, A n (x + JlUO)) .

Also, since pz(x, A) < ß::; Jl, we have pz{x, A) is established.

=pz(x, A n (x + JlUo)), A

e and z. Since A e Cl, there Now choose z e E satisfying both

We are now ready to make our specific choice of exists e> 0 with

lIy - zliO* < e/Jl

a + 2e < py(x,

A

A ) < ß - 2e .

and the claim

and Ipy(x, A ) - pz{x, A)I < e. The last inequality immediately gives A

a+

A

A

e < pz(x, A ) < ß- e .

It remains to show that whenever a + e < pz(x, A) < ß - e, then A e Cl, which is to say that a < py(x,A) < ß. Using the claim just established, we compute : Ipz

defined by qJ(w)(n)

=w

for n

= 1,2,3, ... , is

75

WEAK TOPOLOGIES DETERMINED BY DISTANCE FUNCTIONALS

an embedding. As is well-known, a countable product of metrizable (resp. second countable) topologies is metrizable (resp. second countable) [Dug, p.174 and p.19I]. Also, complete metrizability is preserved under countable products [Dug, p.295], and since the diagonal is a closed subset of the product, is a Polish space provide is a second countable metrizable space. If X is a Banach space, then with an open subset of e. Then

{A

E

C(X x R): p[(x,ß), A]

> e and p[(x,a), A] < e}

is a 'rw -neighborhood of Ao contained in the complement of Cl} . p

ß,

and

e

76

CHAPTER2

By convexity, A e (11 contains a verticalline {X} X R if and only if A is the product of some element of ceX) with R. Thus, A e 11x) if and only if A e (11 and A e (12 where (12

= {A e

ceX X R): 3 x e X and a ERsuch that (x,a) E A but

(x,a - 1)

e:

A}.

To complete the proof, we show that (12 is open in ' Fix Ao E (12, and xo E X and ao eRsuch that (xo,ao) E Ao but (xo,ao - 1) o such that p[(xo,ao - 1), Ao] > E. Then {A e ceX X R): p[(xo,ao - 1), A]

> E and

e:

Ao.

Choose E>

p[(xo,ao), A] < E}

is an open neighborhood of Ao in the Wijsman topology. We claim that it is also contained in (12. For if p[(xo,ao), A] < E, there exists (x,a) E A with IIx - xoll < E and la - aol < E. As a result, (x,a - 1) e: A, else p[(xo,ao - 1), A] ~ p[(xo,ao - 1),(x,a - 1)] < E,

which is incompatible with p[(xo,ao - 1), A] > E. lemma. •

This establishes the assertion of the

2.5.10 Theorem. Let be a separable Banach space and let p be the product metric on X X R. Then is a Polish space. If in addition, X* is strongly separable, then is completely metrizable. Unlike the Wijsman topology 1'Wd' it is not the case that is second countable.

Proof. (1)

(2). Let Ao E CL(X), let B be an arbitrary bounded subset of X, and let e> 0. Choose by (1) a finite subset F of B such that Sfl3[F] ::::> B. Then ~

Cl == {A

E

CL(X): \;:f x

E F

Id(x,A) - d(x,AO)1
0, we have liffin-7oo hausJl (A,A n) =O.

It is an unpleasant fact of life that, in general, the triangle inequality fails for hausJl (see Exercises 3.1.7 and 3.1.8).

Exercise Set 3.1. 1.

2.

Construct an example showing that Lemma 3.1.1 fails for X = (0,1), equipped with the usual metric. Let and be metric spaces. Formally, a local base for the topology 'rucb of uniform convergence on bounded sets for ceX,Y) at fE ceX,y) consists of all sets of the form

{g

E

ceX,y): 'V x

E

B d'(f(x),g(x» < E}

where B is a bounded subset of X and E> O. (a) Let xo be a fixed point of X. Show that ducb : ceX,y) x ceX,y) defined by

ducb(f,g) == :L:l 2- i . min {I,

sup d'(f(x),g(x»} d(x,xO) is a locally convex

3.

space. Why is this not the case for . (a) Let be an increasing sequence of closed subsets of a metric space such that each bounded set B is a subset of An for some n. Show that for each A E CL(X) , we have A 'rA Wd-lim n~oo A An . (b) Show that the nonempty closed and bounded subsets of a metric space are dense in the hyperspace characterizes completeness for d.

11.

Let be an unbounded metric space, and let xo E X be arbitrary. Show

12.

that the uniformity 4I for 'rAWd cannot contain the uniformity Qd [BeDC] (Hint : show that Vd[Xo;I] cannot contain Ud[xo;n] for any n). Do the uniformities coincide when is bounded? Let be a metric space. On C(X,R), let 'rAWp be the Attouch-Wets topology induced by the box metric, identifying functions with their graphs.

THE ATTOUCH-WETS AND HAUSDORFF METRIC TOPOLOGIES

85

(a) Show that Tueb ~ TAWp on C(X,R), whereas Attouch-Wets convergence of graphs need not ensure pointwise convergence. (b) Identify natural sufficient conditions on a sequence of continuous functions so that TAWp-convergence of graphs forces Tueb-convergence (see [BeDC]). (c) Obtain the following results ofHola [H02] concerning the relation between TAWp and the compact-open topology Teo on C(X,R) : (i) TAWp c: Teo provided closed and bounded subsets of X are compact; (ii) TA Wp ~ Teo provided X is locally connected. Supply limiting counterexamples.

§3.2. The Hausdorff Metric Topology For metric spaces and 0 was arbitrary, is totally bounded. Finally, ametrie spaee is eompaet if and only if it is eomplete and totally bounded [Dug, p. 298], from whieh assertion (3) follows. •

A eharaeteristie feature of the Hausdorff metrie topology is the eontinuity of exeess and gap funetionals with arbitrary fixed closed arguments. In the next result, we adopt the eonvention that "+00 - +00" is zero.

3.2.5 Proposition. Let be ametrie spaee and let B E CL(X). Then the funetionals ed(B,·): ~ [0,+00], ed(', B) : ~ [0,+00], and Dd(B, .): ~ [0,+00) are eaeh Lipsehitz eontinuous with eonstant one. Proof We verify Lipsehitz eontinuity only for the funetional ed(- ,B). Suppose AO E

CL(X), Al E CL(X), and Hd(Ao,At} is finite. By the triangle inequality for exeess (see Exereise 1.5.1), we have ed(Ao,B) ::; ed(Ao,AI)

+ ed(A},B),

ed(AI,B)::; ed(AI,Ao)

+ ed(Ao,B).

Sinee ed(AI,AO) < +00, it is clear that ed(Ao,B) =+00 if and only if edCAI,B) both are finite, then by subtraetion of real numbers, we have

=+00.

If

88

CHAPTER3

3.2.6 Theorem. Let be a metric space. Then the Hausdorff metric topology on CL(X) is the weakest topology 'r on CL(X) such that for each B E CL(X) , ed(B,·) :

~

ed(- , B) :

[0,+00],

~

Dd(B, .):

~

[0,+00], [0,+00),

are all 'r-continuous.

Proof. Let 'rweak be the weak topology induced by the collection of all such functionals (see § 1.2). Continuity of these functionals with respect to the Hausdorff metric topology follows from Proposition 3.2.5, and so 'rHd::::> 'rweak. We now show that continuity of all functionals of the form

ed(B,·) and ed(- ,B) alone is enough to show that 'rweak

contains 'rHd. Fix Ao E CL(X), and suppose AO = 'rweak-lim A,1,. By Theorem 1.2.8 with B = AO, we have lim,1, ed(Ao,A,1,) = ed(Ao,AO) = 0 and lim,1, ed(A,1,,Ao) = ed(Ao,Ao) =O. As a result,

and we obtain Hausdorff metric convergence of the net. This shows that completing the proof.

'rHd c:

'rweak,

•

Theorem 3.2.6 is a prototype: most hyperspace topologies of interest arise as weak topologies determined by families of gap and excess functionals. Often more than one description of a given topology exists. For example, we will see that the Hausdorff metric topology on CL(X) is equally weIl generated by all functionals of the form ed(B, .) and Dd(B,·) where B runs over CL(X) , in lieu of the family of functionals of the form ed(B, .). and ed(- ,B), where BE CL(X). Weak topologies determined by gap and excess functionals of course inc1ude all topologies induced by families of distance functionals that we considered in the last chapter. We shall study this general phenomenon more c10sely in Chapter 4. Theorem 1.5.2, which gives a dual formulation of excess in the context of convex sets, yields a dual formulation of the Hausdorff distance between nonempty c10sed convex subsets A and B of a normed linear space.

3.2.7 Theorem. Let be a normed linear space, let d be the metric induced by the given norm, and let A and B be nonempty c10sed convex subsets of X. If dom s(- ,A) dom s(- ,B), then

=

Hd(A,B)

If dom se· , A)

=sup {ls(y,A) - s(y,B)I:

::1=

y E U* and y E dom s(- ,A)}.

dom s(- ,B), then Hd(A,B)

= +00.

Proof If the effective domains of the support functionals agree, then the distance formula is an immediate consequence of Theorem 1.5.2. If they do not agree, then since the

THE ATTOUCH-WETS AND HAUSDORFF METRIC TOPOLOGIES

89

domain of a support functional is a cone, we may assurne without loss of generality the existence of some y E dom sC- ,B) n U* such that y ~ dom sC· ,A). Then Hd(A,B) ~ ed(A,B) ~ s(y,A) - s(y,B)

again by Theorem 1.5.2.

=+00,

•

Theorem 3.2.7 says that equality of the effective domains of support functionals is a necessary condition for the Hausdorff distance between two closed convex sets to be finite. However, this condition is far from sufficient (see Exercise 1.5.6).

3.2.8 Corollary. Let be a normed linear space, and let d be the metric induced by the given norm. Then for each pair A and B of nonempty closed and bounded convex subsets of X, we have Hd(A,B)

=sup {ls(y,A) - s(y,B)I:

y

E

U*}.

Proof. The effective domain of each support functional is the entire dual space.

•

If we equip the closed and bounded convex subsets CB(X) of a normed linear space with the operations of scalar multiplication by nonegative scalars and closed addition, defined by A Ee B == cl (A + B), it is not to hard to show directly that these operations are jointly continuous, when CB(X) is topologized by the Hausdorff metric topology (see Exercise 3.2.12). It is natural to seek a Banach space in whieh .

=

(b) Show that A

8.

9. 10.

11.

~

diam (A) is not eontinuous on

'flj.

which shows that v+ E 'fsup for each open subset V of X. Next let {Vi: i E I} be a locally finite family of nonempty open sets. We aim to show that {Vi: i E I} - E 'fsup .

To this end, suppose Ao E {Vi: i E I} -. For each i E I, choose Xi E Ao n Vi. Although i -7 Xi need not be one-to-one, it is finite-to-one, by local finiteness. Now let

B

= {Xi: i E

I}.

Fix d

E

D; by local finiteness, we can choose for each

XE

B a

number Ex> 0 such that the family of d-balls {S2ex [x] : x E B} is discrete, and moreover, such that whenever x {w EX: \Ix E B, d(w,x)

E

Vi for some index i, then S2ex [X] c: Vi. Let V

> Ex}. By Exercise 3.3.4, the set V is open so that

=

{V} U

{S2ex [X] : x E B} is a locally finite open cover of X. Let P be a metric equivalent to d

as described by Lemma 3.3.11 with respect to this cover. We claim that if Hp(A,Ao) < 1, then A

E

(Vi: i

E

Ir.

Now the condition Hp(A,AO) < 1 gives ep(B,A)::; ep(Ao,A)
- 1

11x1lS;jl

=jlllYl - nll*.

It is reasonable, then, to guess that strong convergence of continuous linear functionals in the dual X* of a normed linear space X corresponds to the Attouch-Wets convergence of the affine objects naturally associated with the functionals, namely, their graphs and their level sets at fixed heights. This conjecture is valid. On the other hand, by Theorem 2.1.11, pointwise convergence of distance functionals for these affine objects gives us something stronger than pointwise (= weak*) convergence of the functionals, as convergence of norms of the functionals ensues (see Exercise 3.4.5). In this section, we will suppress the metric subscript for the Attouch-Wets topology, writing simply 'Z'AW, whether we are working with convex subsets of X or those of X x R. What is perhaps surprising is that strong convergence of linear functionals also corresponds to the weaker slice convergence of these affine objects, as established by the author and 1. Borwein in [BB2]. Recall that slice convergence of convex sets means Wijsman convergence as determined by each equivalent norm. As we noticed in §2.4, unless the dual norm for X* has the weak* Kadec property, Wijsman convergence of level sets at fixed heights cannot guarantee strong convergence of the functionals (see Exercise 3.4.4). It turns out that the results for graphs easily fallout as corollaries of the corresponding results for level sets; so, we produce these results first. In working with level sets, it is clear that the zero functional 8* is a singularity: if is a sequence of

nonzero linear functionals convergent to Y =()*, then for each n, we have by Theorem 1.1.2 1

1

sup Id(x,y- (0» - d(x,y ~ (0»1

I!xl!::;l

= sup

Ilxll::;l

1 d(x,y ~ (0»

=

sup

Ilxll::;l

kYn.x> 1 IIYnll*

=1.

As a result, strong convergence of nonzero linear functionals to 8* cannot give AttouchWets convergence of kerneis. We also note in advance this general principle: convergence of level sets at a fixed nonzero height will usually force convergence of level sets at all heights. This fact was implicit in the statement of Theorem 2.1.11 for the particular case of Wijsman convergence. This occurs for any convergence notion defined for the convex subsets of a normed linear space X satisfying condition (1) of Lemma 2.1.2, plus translation invariance : IIx,t- xll-7 0 and A = limA,t

~

x +A = lim (x,t +A,t).

Translation invariance is a particular attribute of weak topologies determined by families of gap and excess functionals that we will establish in §4.1. As special cases, the slice and Attouch-Wets topologies exhibit translation invariance, which the reader can easily verify directly (see Exercise 3.4.1). To support our claim, suppose now that (2). We rely on the description of Attouch-Wets convergence provided in =limn~oo IIYnll* and lIylI* > 0, we can select 0> 0 such that for each n, 0< IIYn 11 * . Since strong convergence is uniform convergence on bounded sets, there exists a positive integer N such that for each n ~ N, and each x E kU we have Proposition 3.1.6, with xo = 9. Fix a E R and k E Z+. Since lIyll*

l - 1
a+ I} c {A E I: ed(BO,A) > a}.

This proves that

'l' => 'l'1.

•

We now come to a result established in [BeL3) which expresses a number of basic proximal hit-and-rniss topologies (inc1uding the slice topology and the proximal topology) as weak topologies deterrnined by a farnily of gap functionals. First, a definition.

4.1.3 Definition. Let Li be a c1ass of nonempty c10sed subsets of a metric space . We say that Li is stahle under enlargements if for each A

we have

Sa[A) E

E

Li and a> 0,

Li .

Evidently, CL(X) as weIl as CLB(X), the nonempty c10sed and bounded sets, are stable under enlargements; so is K(X) provided c10sed and bounded subsets of X are eompaet. In a normed linear spaee, the c10sed balls, the c10sed eonvex sets, the c10sed half-spaees, and the c10sed eonnected sets are stable under enlargements. If the normed linear space is reflexive, then the weakly compaet sets also have this property. The farnily of c10sed balls in ametrie space is evidently stable under enlargements provided that the c10sed ball operation is additive: for eaeh xo E X, for

°

°

eaeh a> and 11 > we have Sa[S p[XO)) =Sa+p[XO). This, of course, is true in a normed linear spaee, but it is also true for the rationals as ametrie subspace of the line. In view of condition (a) of the next proposition, it is natural to call ametrie with this property almost convex. The proof is left to the reader in Exercise 4.1.8.

4.1.4 Proposition. Let be a metric space. The following are equivalent : (a) whenever {Xl,XZ} c X and 0< a< d(xI,xz) and e> 0, there exists X3 E X such that Id(xP3) - aI < e and Id(xz,X3) - (d(xI,xz) - a)1 < e; (b) whenever {Xl,xZ} c X, a> d(xI,xz), and < ß< a, there exists W E X such that both d(xI,w) < ß and d(w,xz) < a - ß ;

°

(c) for each Xo

°and a> 0, we have °and a> 0, we have c X, 11 > °and a> 0, we have Sa[S

E

X, 11 >

Sa[Sp[XO)) = Sa+,u[XO) ;

(d) for eaeh Ac X, 11 > (e)

for each A

(f)

for each xo E X, 11 >

Sa[Sp[A))

°

=Sa+p[A) ;

p[A)) = Sa+p[A) ;

and a> 0, we have Sa[S,u[XO)) = Sa+p[XO).

4.1.5 Theorem. Let be ametrie space, and let Li be a c1ass of c10sed subsets that is stable under enlargements and that contains the singleton sub sets of X. Let I be a nonempty sub set of CL(X). Then the topology 'l' on I having as a subbase all sets of the form V- where V is open, and all sets of the form (BC)++ where B E Li, is the weak topology on I indueed by the family {Dd(B.·): B E Li}. Proof.

Let

be the weak topology so described. We first show 'l'weak => 'l'. If A E V- where V is open, we ean find a E A and e > with

'l'weak

Suppose A E I.

°

GAP AND EXCESS FUNCTIONALS AND WEAK TOPOLOGIES

109

Then {FE I:d(a,F) 0)

We now split this local presentation into two symmetrie halves. A local base for the lower

Wijsman topology 'fWd at A E CL(X) consists of all sets ofthe form {F E CL(X):

sup d(x,F) - d(x,A) < E}

(B finite, E > 0)

XEB

It is important to note and routine to verify that a subbase for

'fWd consists of all sets of the

form {V-: V open} (see Exercise 4.2.1). A local base for the upper Wijsman topology 'fWd at A E CL(X) consists of all sets of the form {F E CL(X):

sup d(x,A) - d(x,F) < E}

(B finite, E> 0).

XEB

We may play the same game for the Hausdorff metric and Attouch-Wets topologies. If B runs over CL(X) instead of the finite subsets of X, we get the lower and upper

HausdorjJmetric topologies

'fHd

and

letting B run over CLB(X), we obtain the

'flid ;

lower and upper Attouch-Wets topologies 'fÄ Wd and 'flWd • 4.2.1 Proposition. Let be ametrie space. Then: (1) 'fc5d (2)

='fWd

v 'tHd ;

'rb c5d = 7:Wd v

7:1Wd .

Proof. We prove the second assertion; the first is easier and will be left to the reader. Let us write 'f for 'fWd v 'fA+Wd. To show that 'fbc5d c: 'f, we show that for each fixed B

E

CLB(X), the map F

~

Da(B,F) is 'f-continuous.

Fix A

O. There exists a E A such that d(a,B) < Da(B,A) + ~. Now {F d(a,A) + El2} is a 'f-neighborhood of A, and Da(B,F)

semicontinuity is obvious if Da(B,A)

=0;

E

CL(X): d(a,F)
0 such that A E {F E CL(X) : p(yo,A) - 8< p(YO,F)} c: (BC)~+.

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This would show that (BC)r contains a 'l'weak-neighborhood of each of its points. Our metric p: X x X -t [0,+00) will be of the form p(x,y)

= ad(x,y) +

Id(x,B) - d(y,B)I,

where a> O. That p and d define the same uniformities follows from the d-uniform continuity of x -t d(x,B). Since p(x,y):::; (a + l)d(x,y), d-bounded sets are pbounded, and since d(x,y):::; 1 p(x,y), p-bounded sets are d-bounded. Thus, pE {2d.

a

Recalling that B consists of at least two points, the choice of a we make is: Dd(A,B)

a= 4diam B

Fix YO E Band set 8 = ad(yo,A). We intend to show that if FE CL(X) and p(yo,A) - 8< P(yo,F), then FE (BC)+;' First note that p(yo,A)

=inf aeA ad(Yo,a) + d(a,B) ~ ad(YO,A) + Dd(A,B) =

Dd(A,B) + 8. Thus, p(YO,F) > Dd(A,B). ad(x,yo) + d(x,B)

This means that for each XE F, we have

> Dd(A,B).

We consider two cases for XE F : (i) d(x,yo):::; 2diam B; (ii) d(x,yO) > 2diam B. In 1 the first case, by the choice of a, we get ad(x,yo):::; 2" Dd(A,B), so by (*) we have d(x,B) >

2"1 Dd(A,B). In the second case, since

YO E B, we have d(x,B) ~ diam B.

Thus, Dd(F,B)

~ min { ~ Dd(A,B), diam B}

and we have FE (BcYj+.

> 0,

This proves that (BC)~+ E 'l'weak for each closed and d-

bounded subset B of X, and we conclude that

'l'bi5d C

'l'weak.

•

Following the lines of the proof of Theorem 4.2.6, and invoking Proposition 4.2.7 and Theorem 3.3.3 in lieu of Theorem 2.2.7 and Theorem 3.3.2, we obtain the final result of this section. 4.2.8 Theorem. Let be a metric space. Then the Attouch-Wets topology 'l'AWd on CL(X) is the weakest topology 'l' on CL(X) such that for each BE CLB(X) and for each metric p uniformly equivalent to d that deterrnines the same bounded sets as d, the excess functional A -t ep(B,A) is 'l'-continuous.

Exercise Set 4.2. 1.

Let be a metric space.

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(a) Verify that a subbase for r Wd on CL(X) eonsists of all sets of the form {V-: V open}.

(b) Show rWd is adrnissible but is not Hausdorff, unless X is a singleton.

r;d on CL(X) is neither admissible nor Hausdorff. rWd (resp. r;) is the weakest topology r on CL(X) such that

(e) Show in general that (d) Show that 2.

V x E X, d(x,·) is r-upper (resp. r-lower) semicontinuous. Let be ametrie spaee and let A E CL(X). (a) Show that a loeal base for {F

E

rHd at A eonsists of all sets of the form

CL(X): Ac Se[F]}, where e> O.

(b) Showthataloealbasefor rAWd at A eonsists ofall sets ofthe form {FE

3. 4.

5.

6.

CL(X) : A n B c Se[F]}, where BE CLB(X) and e> O. (e) Formulate analagous presentations for the upper halves of these topologies, and use these to give other loeal presentations of the proximal topology, the bounded proximal topology, and their duals. Complete the proofs of Proposition 4.2.1 and Theorem 4.2.2. Let 1: be a nonempty farnily of closed subsets of ametrie spaee , and let Al and Az be (possibly empty) subsets of CL(X). Let r be the weak topology on 1: indueed by 9\ = {Dd(B,'): BE Ad U {ed(', B): BE Az}. (a) Show using Exereise 1.5.2 that if is a sequenee in 1: such that An c An+l for n = 1,2,3, ... and cl (U An) E 1:, then cl (U An) = r-lim An. Provide eounterexamples if exeess funetionals of the form ed(B,') are among the family of funetionals 9\ (Hint; eonsider the bounded dual proximal or the Attouch-Wets topologies). (b) Suppose 1: contains the finite subsets of X, and E is a dense subset of X. Show that the finite subsets of E are dense in 0 for alilarge A, which is to say that a> s(y,A,ü for such A. Finally, suppose that a< s(y,A). Take ß between a and s(y,A). With H = {x : :2! ß}, we have D(H,A) = O. By 'fweakconvergence, there exists Ao such that for all A:2! Ao, we have D(H,A;J < ß- a. For all such A, there exists aÄ E AÄ with d(aÄ,H) < ß - a, and since lIyll* = 1, we get s(y,AÄ) :2! a . • The scalar topology need not contain the Wijsman topology, even restricted to the closed and bounded convex sets CB(X). For example, in 12, the weak convergence of to the origin implies that {e} ='fsc-lim ren}. This example also shows that sets of the form V- for V norm open need not be open sets in the scalar topology. That the Wijsman topology need not contain the scalar topology is also easy to see, for in the plane, the horizontal axis is the Wijsman limit of the sequence of lines . n One is thus led to consider the supremum of the Wijsman and scalar topologies, perhaps first considered by C. Hess [Hesi]. We call this topology the linear topology 'fL. Theorem 4.3.3 below, obtained in [Be13], indicates the close relations hip between the linear topology and the weaker slice topology, which is anticipated by Theorem 2.4.8. First, we give apreparatory lemma.

4.3.2 Lemma. Let be a normed linear space, and let B E C(X). Then the set (BC)++ is open in 'fsc . (BC)++, i.e., D(A,B) > O.

By Corollary 1.5.7, there exists y E X* and a ERsuch that SUPaeA < a< infbeB . With H = {x EX: :2! a}, we have Proof Suppose A is a convex set with A

A E {F E C(X) : D(F,H)

E

> O} O}

By Theorem 4.3.1, this proves that (BC)++ E 'fsc.

=(BC)++ .

•

4.3.3 Theorem. Let be a normed linear space. Then a subbase for the linear topology 'fL = 1W v 'fsc on C(X) consists of all sets of the form V-, where V is norm open in X, and all sets of the form (BC)++, where BE C(X). Thus, the linear topology is induced by the family of gap functionals {D(B,·): BE C(X)}. Proof Let 'f be the topology on C(X) with these sets as a subbase. Since 'fW 0).

It is natural to seek an admissible Hausdorff topology on C(X) with respect to which these operations are jointly continuous. Although these operations are jointly continuous with respect to the scalar topology, this topology is not admissible (see Exercise 4.3.1). But the linear topology fits the bill perfectly. In fact we know of no other hyperspace topology on C(X) fulfilling this modest set of requirements. Before we give positive results, we point out some shortcomings of topologies we have already considered with respect to joint continuity. The Vietoris topology is unsatisfactory because even translation by a fixed vector is not continuous. We next present exarnples showing that (i) scalar multiplication fails to be jointly continuous with respect to the Hausdorff metric topology and the proximal topology; (ii) the operations of closed addition and closed convex hull are not jointly continuous with respect to the Wijsman topology, the slice topology, the Attouch-Wets topology, and the bounded proximal topology, which all agree in finite dimensions.

4.3.5 Example. In the plane, let A be the epigraph of x ~ x 2 . Then for each a > 0, aA is the epigraph of x ~ tri x2 . In particular, for each a> 1, the excess of aA over A is infinity (see Exercise 1.5.6), and so aA fails to converge to 1·A in the proximal topology as a ~ 1. Thus, convergence also fails in the stronger Hausdorff metric topology. 4.3.6 Example. the ray {(x, -

~x)

1

In the plane, let An be the ray {(x, -x): x;::: O} and let B n be n : x.,:; O}.

Then rw-converges to the nonnegative horizontal

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axis A, and "w-converges to the nonpositive horizontal axis B, and A EB B clco (A U B) = the horizontal axis. On the other hand, for each n An EB Bn

=clco (An U Bn) = {(x,y): y ~

=

1

I -n X I },

and so = 0), is jointly continuous.

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125

Proof. In view of Theorem 1.2.8, continuity of these operations may be checked by evaluating support functionals along nets of convex sets. To this end, suppose that A = Tsc-lim A,t and B = Tsc-lim B,t and o. Let y E X* be fixed. By the definition of scalar convergence, we have lim,ts(y,A,t) =s(y,A) and lim,ts(y,B,t) = s(y,B). Using Exercise 3.2.13, we now compute limv(y,A,t E9 B,t)

= lim,t [s(y,A,t) + s(y,B,t)] = s(y,A) + s(y,B) = s(y,A E9 B);

lim,ts(y, clco(A,t U B,t)

= lim,tmax {s(y,A,t), s(y,B,t)} =max

{s(y,A), s(y,B)}

= s(y, clco(A U B); lim,t s(y,a.v\,t)

=lim,t aA,S(Y,A,t) =as(y,A) =s(y,aA).

•

Putting together the two preceding propositions, we obtain

4.3.9 Theorem. Let be a normed linear space, and suppose C(X) is equipped with the linear topology TL. Then each of the operations (A,B) ~ A E9 B, (A,B) ~ clco (A U B), and (a,A) ~ aA (a > 0), is jointly continuous. Intersection is not really an operation on C(X), in that the intersection of two closed convex sets may be empty. As one might expect, positive results regarding continuity of intersection for convex sets with nonempty intersection require a significant overlap of the limit sets AO and Bo, else nearby sets (even in the strong Hausdorff metric topology) won't intersect. A natural condition to propose is: Ao n int Bo '# 0. We return to this topic in §7.4. The answer to the question when is the linear topology metrizable ? is hardly ever .

4.3.10 Proposition. Let be a normed linear space. metrizable if and only if the dimension of X is one.

Then is

Proof. If X is one-dimensional, then there is no loss in generality in assuming that X = R with the usual metric. Suppose 0, there exists XE H and a E A with Ix - al a. For each E < a/2, there exists Äo E A such that for all .It ~ Äo, we

=

have Id(x,A,t) - al < E and d(a,A,t) < E. By convexity of the sets A,t, it follows that ID(H,AA,) - aI < E for .It ~ Äo. By Theorem 4.3.1, we have scalar convergence, and thus TL-convergence of O.

We show that no countable

farnily of such sets can serve as a local base for TL at X, so that is not first countable and is afortiori not metrizable. Let {'B(Fj,aj): i E Z+} be a countable farnily of such sets. Set E = U Fj, and let e>O bearbitrary. If U{Sg[coF]: FeE, F finite}=X, thenbythecompactness of each set co F, there will exist a countable subset We of X such that S2g[We] = X. Thus, by nonseparability of X, for some e> 0 there exists xo E X such that for each finite subset F of E, we have d(xO,co F) ~ e. Consider . Give characterizations of the linear topology on the nonempty closed convex subsets of a normed linear space as special cases of Theorem 4.1.9 and Exercise 4.1.11. Let be a normed linear space. Prove or give a counterexample: the scalar topology on C(X) is the weak topology generated by the family of excess functionals {e(- ,H): H a closed half-space}. (a) Let be a normed linear space. Suppose hE P(X*) and epi h is a cone, i.e., h( 0*) = 0, and h is positively homogeneous and subadditive (see Exercise 1.4.13). Prove that h is the support functional of an element of C(X), namely A = {x EX: V y E X* :=:; h(y)} [Hö,Val] (Hint: to show that A is nonempty, strongly separate epi h from (0*,-1), and then translate the hyperplane upward until it supports the cone). (b) Prove this theorem ofDe Blasi and Myjak [DBM2]: let be a sequence of uniformly bounded closed convex sets in a separable reflexive space. Show that has a scalar convergent subsequence (Hint: use the ArzelaAscoli Theorem, the separability of X*, and (a)). Let X and W be normed linear spaces, and let f: X -* W be a continuous affine transformation, i.e., fex) = T(x) + Wo where T is continuous and linear and Wo E W is fixed. Prove CPF C(X) -* C(W) defined by cpfA) =clf(A) is continuous when then the convex sets in both spaces are equipped with any of the following topologies: the scalar topology, the lower Wijsman topology, the linear topology, or the Hausdorff metric topology. Show that this is not true for the Wijsman topology, the Attouch-Wets topology, or the slice topology. Let be a normed linear space with strongly separable dual with countable dense subset {Yn: n E Z+}. Define cp: [0,+00] -* [0,m'2] by cp( a) = arctan a if a is finite, and cp(+oo) = m'2. Following [SZl], define a metric p on C(X) by

(a) Express the p-metric topology as a weak topology.

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=

(b) Show that for X 12, P need not be not compatible with the scalar topology. (c) When is the scalar topology metrizable?

§4.4. Weak Topologies Determined by Intimal Value Functionals As we indicated in § 1.2, a topology 'r on a set X is expressible as a weak topology determined by a family of real-valued functions 9t if and only if it is uniformizable. For if 9t determines 'r, then a base for a compatible uniformity 'U(9t) consists of all sets of the form

where 9tO is a finite subset of 9t and e> 0. Conversely, if 'r admits a compatible uniformity, then 'r is completely regular, and we may take either C(X,R) or C(X,[O,l]) for 9t. Suppose..1 is a family of dosed subsets of X containing the singletons. In this section, we aim to resolve the following question: under what circumstances is the hitand-miss or the proximal hit-and-miss hyperspace topology determined by the family ..1 uniformizable? We will show that whenever such a topology is uniformizable, then it must arise as a weak topology determined by a family of infimal value functionals, as introduced in §4.1. Our treatment follows [BTa2]. In this section, we deviate from our usual metric space setting and present our results in the context of an arbitrary Hausdorff uniform space be a Hausdorff uniform space. We say that (disjoint) subsets A and B of X arefar provided there exists U E 'U such that U(A) n U(B) =0. Farness so defined is not really a uniform notion but a proximity notion, in that different uniformities can give rise to the same farness relation [NW, p. 71]. Given subsets A and B of X we write B++ for {A E CL(X) : A is far from Bq. A function between uniform spaces f: is called uniformly continuous if for each U2 E 'U 2, there exists Ul E 'U 1 such that (Xl,Wl) E Ul => (f(Xl).f(wl» E U2. We write UC(X,R) (resp. UC(X,l) for the dass of uniformly continuous functions defined on be a Hausdorff uniform space and let A be a nonempty subset of X.

> a, then slv (f; a) and A are far; (2) If 'U = 'U(9t) for some family 9t of real functions on X, then each member of 9t is uniformly continuous.

(1) If f: X ~ R is uniformly continuous and mf(A)

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129

We turn to the extension of our hyperspace terminology of §2.2 . Let be a Hausdorff space and let L1 be a nonempty subfamily of CL(X). The hit-and-miss hyperspace topology 't(L1) on CL(X) determined by L1 has as a subbase all sets of the form V- where V is an arbitrary open subset of X, plus all sets of the form (DC)+ where D E L1. Now suppose be a Hausdorff uniform space, and let L1 be a family of nonempty closed subsets of X containing the singletons. We call L1 a uniformly Urysohn family provided whenever D E L1 and A E CL(X) are far, there exists SE LtL1) and U E 'U such that U(D) c: Sc: AC. When L1 =CL(X), we always have L1 uniformly Urysohn, for if D and A are far, then we can U E 'U with (UoU)(D) c: AC, in which case U(D) c: cl U(D) c: AC.

When L1 =K(X), then L1 is uniformly Urysohn if and only if L1 is locally compact. When X is a metric space and L1 is a family of closed sets containing the singletons which is stable under enlargements, then L1 is uniformly Urysohn with respect to the uniformity of the metric. We chose the name "uniformly Urysohn" for such a family, because whenever D E L1 and A E CL(X) are far, one can separate the sets by a uniformly continuous Urysohn function. What is crucial for our purposes is that one can control the sublevel set structure of our Urysohn functions in a way that characterizes the uniform Urysohn property for Li. Given a uniformly Urysohn family L1, we introduce classes of uniformly continuous functions whose sublevel sets are interposed by members of LtL1):

9t 1.~ == {f E UceX,R) : whenever inf f < a< ß < sup J, 3 S E LtL1) with slv(f; a) c: S c: slv(f; ß) }, 9t~j == {f E UceX,l) : whenever inf f

< a< ß < sup J, 3 S E LtL1) with

slv(f; a) c: S c: slv(f; ß) }.

4.4.3 Lemma. Let 112 eventually. For all such A., U(D) n AA, =0, i.e., AA, is far from D. Secondly, we need to show that if A hits an open set W, then (3) and (3) => (4) are obvious. It remains to establish (4) => (1). To this end, suppose D E ..1, A E CL(X) and A n D = 0. Since (DC)+ is a 'Z(L1)-neighborhood of A, by regularity of the hyperspace, there exists a basic neighborhood Cl of A in the topology 'Z(L1) such that A E Cl c: cl Cl c: (DC)+.

We may write Cl

=(SC)+ n V1- n Vi n ... n Vn-

where SE I(L1) and {VI, V2, .

. .,Vn } are open subsets of X each meeting A. Clearly, A E Cl implies Sc: AC. It is also the case that D c: S, else choosing XE D n Sc, we would have AU {x} E Cl but A U {x} e (VC)+, an impossibility. It remains to show that D c: int S. If this inclusion fails, choose S E D n bd S. Order the family of neighborhoods D of S contained in AC by reverse inclusion, and for each V E n, pick XvE V n Sc. Then V ~ A U {xv} is a net defined on D with values in Cl that is 'Z(L1)- convergent to A U {s} II!: (DC)+, contradicting our choice of Cl. This proves that D c: int S c: S c: AC, as required. •

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135

In addition to characterizing regularity for hit-and-miss topologies, Poppe also characterized the Hausdorff separation property [popl-2].

4.4.11 Definition. Let X be a Hausdorff space and let .Q be a nonempty family of subsets of X. We call D a local family if whenever V is a neighborhood of x E X, there exists E E.Q with x Eint E c: E c: V.

4.4.12 Theorem. Let X be a Hausdorff space, and let Li be a nonempty subfamily of CL(X). Then 'Z(Li) is Hausdorff if and only if 1:(Li) is a local family. Proof. Suppose first that 1:(Li) is local, and A and B are distinct elements of CL(X).

Without loss of generality, let bEB n AC be arbitrary, and choose SE 1:(Li) with

bE int S c: Sc: AC. Then (SC)+ and (int S)- are disjoint 'Z(Li)-neighborhoods of A and B, respectively. Conversely, assume the hyperspace is Hausdorff. Then we can separate A and {x} U A by disjoint 'Z(Li)-basic open sets Cll and Cl2, defined by

where {S,T} c: I(Li)

and VI, V2, ..

"vn, Wb

W2, ... , W m are open.

Since

Cll and Cl2 are nonempty, without loss of generality, we may assume that Vi n S =0 for i ~ n and Wj

n T =0

for i ~ m. We make the following observations :

(i) XE S, else AU {x} E Cll; (ii) An S=0, else AI!! Cll; (iii) for some i ~ m, x E Wj, else A E Cl2. Let the indices for which XE Wi be {1,2, .. .,k} where 1 ~ k ~ m. We claim that for some i ~ k, we must have Wj c: S. If not then, for each i, choose Xi E Wi n Sc . Then the closed set A U {Xl, X2, x3, •.. , Xk} belongs to Cll n Cl2, a contradiction. Taking io ~ k with W io c: S, we have XE Wio c: int Sc: Sc: AC, as required. • The same condition also characterizes the Hausdorff separation property for proximal hit-and-miss topologies. The proof involves very minor modifications of the proof ofTheorem 4.4.12 and is left to the reader.

4.4.13 Theorem. Let is a Hausdorff uniform space, then a closed subset A of X fails to intersect a compact subset K if and only if A and K are far in a uniform sense. Thus, the Fell topology is at once a hit-and-miss and a proximal hit-and-miss topology (see §4.4). Initially, we develop the properties of the Fell topology in an arbitrary Hausdorff space rather than in a metrizable space, since it is no more difficult to do so. There is also considerable interest in this topology in the setting of non-Hausdorff spaces. In fact, Fell's investigations were motivated by particular applications to functional analysis in the nonHausdorff setting, and the Fell topology for non-Hausdorff spaces also arises in the study of random semicontinuous functions and randorn capacities as developed by Vervaat, Norberg, O'Brien and their associates [Norl-3,Verl-2,OBTV]. But because of some complications that arise without the Hausdorff separation axiorn- limits of nets need not be unique, compact sets need not be closed, etc.- we choose not to work at this level of generality in this introductory volume. 138

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139

As immediate consequences of Theorems 4.4.10 and 4.4.13, we obtain the following result.

5.1.2 Proposition. Let X be a Hausdorff space. The following are equivalent : (a) (b) (c) (d)

is regular; is a locally compact Hausdorff space. The proof we supply is independent of the general resu1ts of the last section. We first show that the natural extension of the Fell topology to 2x = CL(X) U {0} is without fail a compact Hausdorff space, as first observed by Fell [Fei]. This is truly an unexpected result, in that compactness occurs with no conditions whatsoever on the underlying space! The defining subbase for the extended Fell topology is no different from that specified in Definition 5.1.1, except that the miss sets may now include the empty set. A local base for the extended Fell topology at the empty set consists of all sets of the form {A E 2x : A n K = 0}, where K E K(X). We still retain our notation for hit sets and miss sets when considering the extended hyperspace. One can also make natural extensions of other hit-and-miss and proximal hit-andmiss topologies such as the Vietoris topology or the slice topology to include the possiblity of convergence to the empty set (see Exercises 4.4.6 and 4.4.7). On the other hand, there are no natural extensions of the Wijsman topology, the Attouch-Wets topology, the Hausdorff metric topology, ... to allow for the possibility of empty limits. At the heart of the problem is an appropriate definition for the distance of a point to the empty set. Although, it seems reasonable to define d(x,0) = +00 when the underlying metric space is unbounded, this convention seems highly artificial when the space is bounded. As a result, it is not clear how one should define the excess of a nonempty set over the empty set, or the gap between a nonempty set and the empty set. Following [Fla,Mat,At3], OUf compactness theorem will be established using a nontrivial result of general topology, the Alexander Subbase Theorem, which can be used to prove the Tychonoff Theorem [Wil, p. 129]. A second proof that directly uses the Tychonoff Theorem is given in §5.2.

Alexander Subbase Theorem. Let X be a Hausdorff space. Suppose each open cover of X by a family of open sets chosen from a fixed subbase for the topology has a finite subcover. Then X is compact. 5.1.3 Theorem. Let X be a Hausdorff space. Then < 2x ,'fF> is compact. Proof. Weintend to apply the Alexander Subbase Theorem with respect to the standard subbase for the Fell topology on 2x. Suppose {VA: AE A} is a family of nonempty open sub sets of X and {K 0': (J E 1:} is a family of nonempty compact subsets of X

such that the family {Vi : A E A} U {( K~) + : (J E 1:} forms an open cover in the extended Fell topology of 2x. Note that each kind of set must be represented in such an open cover, for if the index set 1: is empty, then 0 ~ U {Vi : Ä E A}, and if Ais

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140

empty, then X ~ U{(K~)+:

O'E X}.

We claim that there exists 0'0 E X with K ao c:

UÄ.e A VÄ. Otherwise, pick for each index 0' E X a point Xa E K a that fails to lie in UÄ.e A VÄ" Then the closed set cl {xa: 0' E I} lies in no hit set Vi and in no miss set (K~)+, contradicting the covering assumption .

Thus, the asserted index 0"0 exists.

By compactness of K ao' there exists indices {Al, A2, . . .,An} c: A with Ui!.l VÄj K ao

.

It is easy to check that the subfamily {(K~o)+ } U {V

subcover of 2x . 5.1.4 Corollary.

i

j

:

:::>

i::;; n} is a finite

• Let X be a locally compact Hausdorff space.

Then < 2 x,'!F>

is a compact Hausdorff space, and is already compact. To show that the hyperspace is Hausdorff, for the convenience of the reader, we repeat an argument in the proof of Theorem 4.4.13. Suppose A E 2x and BE 2X with B is {0}, and so is a compact metrizable space; (3) is metrizable.

Clearly, we can find a countable base :B for the topology of X such that for each W E :B, cl W is compact. We claim that

Proof. (1) ~ (2).

forms a countable subbase for the extended Fell topology on 2X• First, if V is open in X and A E V-, then there exists a E A and W E :B such that a E W c: V. Thus, A E W- c: V-. If A E (KC)+ where K E K(X), then for XE K there exists a compact neighborhood Cx of x with Cx n A =0. Choose Wx E :B such that XE Wx c: Cx . Note that cl Wx n A =0. By the compactness of K, we can find {Xl, x2, .. . ,xn} in K such that {WXj : i = 1,2, ... , n} is an open cover of K. It follows that

THE FELL TOPOLOGY AND KURATOWSKI-PAINLEVE CONVERGENCE

141

By Corollary 5.1.4, is an open subset of a eompletely metrizable spaee, and so by the Alexandroff Theorem (see §2.5), the hyperspaee is eompletely metrizable. Sinee a eompaet metrizable spaee is seeond eountable, , which if viewed eorreetly, paralieis Attoueh-Wets eonvergenee of nets of sets as deseribed by Theorem 3.1.7 [Be22]. A very different presentation of net eonvergenee is given in §5.2. We will use the fact that the open symmetrie elements of 'U form a base for the uniforrnity (see, e.g., [Wil, p. 241]). Reeall that if A is a subset of X and U is an entourage, we write U(A) for {x EX: 3 a E A with (x,a) EU}. 5.1.6 Theorem. Let 0 such that S oIx] is noncompact and d(x,y) > a. Let be a sequence in Sa[x] withnoclusterpoint. Then is TF-convergentto {y}, whereas limn~oo d(x,{xn,y}) *" d(x,{y}). This shows that Wijsman convergence fails for this sequence of closed sets, and so TWd *" TF. For the converse, assurne has nice closed balls. We show that each subbasic open set in the Wijsman topology beongs to {A E CL(X) : d(x,A) < a}

ball

=(Sa[x]t

TF.

Fix x E X and a> O. Clearly,

is a subbasic open set for the Fell topology. If the

S a[x] is a proper subset of X, then there exists ß>

a such that

=a, then AnS a[x] *" 0. = (S a[x]C)+ E TF. Finally, if

Sß[x] is

compact. As a result, if A E CL(X) and d(x,A) turn implies that {A X, then {A

E

E

CL(X) : d(x,A) > a}

CL(X) : d(x,A) > a} = 0

E TF.

This in

Sa[x]

=

•

5.1.11 Corollary. Let be metric space. Then only if X is compact.

THd

=TF

on CL(X) if and

Praaf. This follows easily from Theorem 3.2.3 and Theorem 5.1.10, because a totally

bounded metric space with nice closed balls can be written as a finite union of compact balls and is thus compact. •

Which metrizable spaces have a compatible metric with nice closed balls? It should come as no surprise that these are precisely the locally compact metrizable spaces. Following [BeI2], we give a paracompactness proof of this result, using the tools developed in §3.3. 5.1.12 Theorem. Let X be a metrizable space. The following are equivalent :

(a) X is locally compact; (b) X has a compatible metric d bounded by 1 such that each closed ball of radius less than 1 is compact; (c) X has a compatible metric with nice closed balls. Prao! (a) => (b). Let {Vi: i E l} be an open cover of X such that for each index i E I, cl Vi is compact. By Lemma 3.3.11, there exists a metric p for the space such that

the farnily of balls {Sf[x] : x E X} is a refinement of the cover {Vi: i E l}. Since cl Vi is compact for each index i, each closed ball of radius less than one is compact. The desired metric d is min {p, 1}. (b) => (c). Each closed ball of radius less than 1 is compact, whereas each closed ball of radius 1 or more is the entire space. (c) => (a). Let x E X. If X = {x}, then X is compact. Otherwise, there exists

a> 0 such that

Sa[x] *" X, and so Sa[x] is a compact neighborhood of

x.

•

144

CHAPTER5

Exercise Set 5.1. 1. 2. 3. 4.

Let X be a Hausdorff space. Prove that that the Fell topology is both admissible and Tl. Let X be a Hausdorff space, and let 'r be a Hausdorff hit-and-miss topology on CL(X). Using Theorem 4.4.13, show that 'r contains the Fell topology. Let X be a Hausdorff space. Prove that that the Fell topology and the Vietoris topology agree on CL(X) if and only if X is compact. Let X be a Hausdorff space. (a) Let K E K(X). Prove that {A E CL(X): An K,* 0} is compact in is

5.

the usual Alexandroff one-point compactification of .

6.

7.

Let be a locally compact Hausdorff space. Suppose 0 form a subbase for a Hausdorff topology on 2x compatible with the above convergence, called the extended Wijsman topology. (b) Describe a neighborhood base ofthe empty set. Under what circumstances is the empty set an isolated point ofthe hyperspace? (c) Prove that 2x equipped with the extended Wijsman topology is metrizable if and only if X is second countable. (d) Prove that the extended Wijsman topology is finer than the extended Fell topology. Let be a metric space. Prove that the following statements regarding the extended Wijsman topology are equivalent [FLL]: (a) closed and bounded subsets of X are compact; (b) the extended Wijsman topology reduces to the extended Fell topology; (c) the extended Wijsman topology is compact. Let X be a locally compact Hausdorff space. Let A E CL(X) and let ;'" E.A' . For each V E 13, we have fx(V) = 1 eventually if and only if fA'(V) = 1 frequently, and so is compact Hausdorff space but not sequentially compact. We use the continuum hypothesis in the following form [Wil, p. 9]: for each uncountable set C, there exists an injection qJ: {O, l}zt ~ C.

5.2.13 Proposition. Let be a metric space that is not second countable. Assuming the continuum hypothesis, there exists a sequence in CL(X) with no Kuratowski-Painleve convergent subsequence.

> 0, there exists an uncountable e-discrete subset C of X. Now let qJ: {O,l}zt ~ C be an injection, as guaranteed by the continuum hypothesis. For each n E Z+, let An C C be defined by

Proof. Arguing as in the proof of Theorem 2.1.5, for some e

An

={q)(j): JE {O,l}zt and f(n) =1}.

Each set An is an uncountable e-discrete subset of C. Now let has nice closed balls. The same proof shows that these are the same spaces for which Wijsman convergence agrees with Kuratowski-Painleve convergence of sequences or of nets of nonempty closed sets (see Exercise 5.2.9). Nevertheless, in some metric spaces - in particular, in normed linear spaces Wijsman convergence of nets can be expressed in terms of Kuratowski-Painleve convergence of closed enlargements. The next result was obtained by Dolecki [Doll].

5.2.14 Theorem. Let be a normed linear space. Let (3). Wijsman convergence of nonempty closed sets implies KuratowskiPainleve convergence without restriction (see Exercise 5.2.9). (3) => (1). Let 11-111 be any norm equivalent to the initial norm for the space. Then its unit ball Ul is a closed and bounded convex set, and for each J1 > 0 and F E CL(X), we have

SJl[F] =F E9 J1UJ,

where the enlargement is computed with respect to

IHI1. By (3) and Theorem 5.2.14, we obtain the Wijsman convergence of 0 such that for each n E Z+, IIYnll* ~ Ö . Fix xo E Y-I (a) and let E > 0 be arbitrary. By weak* a, and there exists NE Z+ such that convergence, limn~oo kYn,xo> - aI < & for each n ~ N. By the choice of Ö, each number between -

=

&

and + &

=

is a value of Yn on xo + EU, and so y~(a)

n (xo + EU) ;>!: 0

for each n ~ N. This establishes the required inclusion. The other inclusion Ls y~(a) Suppose nl < n2 < n3 < ... is an

c: y-I(a) follows easily from Lemma 5.2.8.

increasing sequence of integers and for each k

=a.

Now if IIXk - xII

~

E

Z+, Xk belongs to y ~~ (a), that is,

0, then (1) gives

from which we conclude that XE y-I(a). (2) ~ (3). This is trivial. (3) ~ (1). We first verify that {Yn: n E Z+} is norm bounded. If not, then we can find an increasing sequence of integers such that for each k, IIYnkll* > k. This means that there exists for each k E Z+ a point Xk

E

k-IU lying in Y ~~ (1).

This

yields () E Ls Y ~I(1), contradicting Ls Y ~(1) c: Y -1(1). Weak* convergence of to Y is now established exaclly as in the proof of Theorem 2.1.11.

•

5.2.17 Theorem. Let be a Banach space, and let y, YI, Y2, Y3, ... be a sequence of nonzero elements of X*. The following are equivalent : (1) Y = 0, 'Va> 0, 3 ö> J1 such that 'V x

12.

S &:x]

c:

Sa[Sll[x]].

E

X,

Let be a normed linear space. Suppose ÄeA is a net of nonempty closed subsets of X Kuratowski-Painleve convergent to A E CL(X). (a) Suppose Ä.eA is a net in X strongly convergent to x. Prove that A + x = K-lim (AA, + XA,)' (b) Show that the result remains valid if x is replaced by a compact set K and XA, is replaced by a compact set KA, with K =Hd-lim KA,.

155

THE FELL TOPOLOGY AND KURATOWSKI-PAINLEVE CONVERGENCE

13.

14. 15.

(c) Show that Kuratowski-Painleve convergence of sequences of compact convex subsets in 12 is not preserved under addition with the c10sed unit ball. Let be a normed linear space. Suppose 0 be fixed. Since Vo x (-oo,a + e) is a neighborhood of (x,a) , by definition of the upper closed limit, there exists X ~ Ao with (Vo x (-oo,a + e» n epi IA: * O. As a result,

h

Since

Ao

was arbitrary, this yields lim inf;.,mh,(Vo) :::;; a + e, and since Vo and e are

arbitrary, we have g(x):::;; a, which is to say that (x,a) E epi g. For the reverse inclusion, fix (x,a) E epi g and again fix Vo E rl(x) and e> O. It suffiees to show that Vo x (a - e,a + e) meets epih frequently. Fix Ao E A. Sinee

there exist ,1.' ;:::

A.o with

mh:(VO)

< a + e.

and as epigraphs recede in the vertical direction, we have

*0.

n epi h' * 0, (Vo x (a - e,a + e» n epi IA:

Then (Vo x (-oo,a + e»

•

Sinee the lower closed limit of a net of epigraphs is eontained in the upper closed limit, the function h that it deternlines is a larger function than the function g of Lemma 5.3.3, as g:::;; h ~ epi g => epi h. The formula that we get for this function is not entirely symmetrie with the one of the previous lemma. 5.3.4 Lemma. Let X be a Hausdorff space and let is a net in L(X) with f =K-limfA, then lim).,Hp(epifA, epij) =0; (3) whenever is a sequence in LSC(X) with f = K-limfn, then limn~oo Hp(epifn, epij) = 0 .

=

= X;

Proof (1) =::} (2). Let t: > 0 be arbitrary. Since X cl (Un':l slv (/; n»), a

eventually, which means that epi IA misses K x {a} eventually, as required.

•

In Theorem 5.1.6 we showed that when X is a uniformizable space, then 'Cpconvergence of nets in 2x could be expressed in a manner that paralIeIs Attouch-Wets convergence of sets in a metric space. This leads to adescription of Fell convergence of epigraphs in terms of sublevel sets that we present as oUf final result of this section.

THE FELL TOPOLOGY AND KURATOWSKI-PAINLEVE CONVERGENCE

167

5.3.13 Theorem. Let O. Suppose that epij = 'Z'F-limX epif;t. Fix K E K(X), U E 'U, a E R, and e> O. By Theorem 5.1.6, with C = K x {a}, there exists Ao E A such that for each A.;?: Ao, both epijn Ce: Utf...epif;t) and epijx n Ce: Utf...epif). Now let XE slv (j';a) n K be arbitrary. Clearly, (x,a) E epijn C, and so there exists (w,ß) E epif;t with (x,w) E U and la - ßI < e. Since f;t(w) ::;; ß::;; a + e, we have w E slv lfx;a + e) and so XE U(slv (j'X;a + e». We have shown that slv lf;a) n K e: U(slv (j'x;a + and the companion inclusion is established in the same way. Thus, condition (1) implies condition (2). Conversely, suppose (2) holds and CE K(X x R), U E 'U, e> 0 are given. Choose K E K(X) and p> 0 such that K x [-p,p] => c. Let -p = Jlo < JlI < ... < Jln =p be a regular partition of [-p,p] such that Jli - Jli-l < e/2 for i = 1,2, ... , n. By (2), there exists Ao E A such that for all A.;?: Ao and i = 0, 1,2, ... , n, we have

e»,

(*)

slv (j';Jli) n K e: U(slv (j'X;Jli + E/2»

and slv (j'X;Jli) n K e: U(slv (j';Jli + e/2» .

Now fix A.;?: AQ and (x,a) E epijn C. As (x,a) E K x [-p,p], there exists a smallest i such that a::;; Jli. Since .f(x)::;; a::;; Jli, we have XE slv (j; Jli). By (*), we Can find Z E slv (j'X;Jli + E/2) such that (x,z)

E

U. Then (Z,Jli + e/2) E epifA, and

This shows that (x,a)E Ue(epih), yielding epijn Ce: Ue(epih) for A.~A.Q. The inclusion epi jx n C e: UeCepi f) for A.;?: Ao is obtained in the same way. • It is also possible to speak of Kuratowski-Painleve convergence of graphs of continuous functions, as well as the Fell topology for the function space ceX,Y), where functions are identified with their graphs. Pioneering work in this direction was done by

168

CHAPTER5

H. Poppe [Pop2] (see also [Be3,HM]). But apart from Theorems 5.2.18 and 5.2.19, this theory awaits application.

Exercise Set 5.3. 1.

Let X be a Hausdorff space, and suppose fi(x) - E. Suppose that /./1,/2, ... is a sequence of real-valued l.s.c. functions. Prove that if any two of the following conditions hold, then all three hold: (1) is pointwise convergent to I; (2) 1= K-limln; (3) {fn: n E Z+} is an equi-Iower semicontinuous family. Let be a metric space, and suppose /./1,/2'/3, ... are real-valued l.s.c. functions on X. Suppose that is convergent to I uniformlyon compact subsets of X. Prove that 1= K-limln. Let X be Hausdorff space. (a) Suppose I and mf(X) , we have slv (j;a) = K-lim slv (j;.;a) [Mos2]. Show that the converse fails. Let X be a first countable Hausdorff space, and let be a sequence in L(X). Let g, h be those functions in L(X) with epi g =Ls epiln and epi h =Li epiln . (a) Prove that g is defmed by the formula g(x) =inf -+ x lim infn-+ooln(xn). (b) Prove that h is defined by the formula hex) =inf -+ x lim supn-+ooln(xn). (c) Show that in general there is no relationship between hex) and the quantity sUP -+ x lim infn-+ooln(xn), giving limiting counterexamples in the space C([O,l],R).

12.

13.

14.

Let be a complete metric space, let I E L(X), and let be a sequence in C(X,R) epi-convergent to f. As we have seen, may be nowhere pointwise convergent to f. Show, nevertheless, that fix) is a subsequentiallimit of 0 with

such that Band a + eU are disjoint. Then [int (a + eU)r and {x EX: IIx - all> e} + are disjoint 'Z"M-neighborhoods of A and B. (2) ~ (3). This is trivial. (3) ~ (1). We actually show that without reflexivity, nonempty open subsets in the Mosco topology restricted to ceX) are dense! Suppose we are given two typical basic 1:M -open subsets .QI and .Q2:

where Vll, V12, ... , Vl n, V21, V22, ... , V2m are nonempty disjoint norm open subsets of X and KI and K2 are weakly compact. By Lemma 5.4.12, choose distinct Xli E Vli and X2i E V2i such that A == co {Xll, X12, ... , Xl n, X21, X22, ... , X2m} misses KI U K2. Then A belongs to both .Q! and to .Q2, as required. •

If X is a reflexive, then each member of CB(X) is weakly compact, and so the slice topology on ceX) is contained in the Mosco topology. Since the slice topology is always Hausdorff, from our last result and Proposition 5.4.11, we get

5.4.14 Corollary. Let with j= M-limJA. Prove that whenever K cX is weakly compact, then Ls w (K n Argmin JA ) c Argmin J. Show that if some fixed weakly compact set contains some nonempty sublevel set of JA, for all Il sufficiently large, then Ls w ArgminJA is nonempty and m/(X) = limA,m/iX).

178

12.

13.

14.

CHAPTER 5

Let be a normed linear space. Prove that the Mosco topology is the weakest topology T on the weakly lower sernicontinuous functions on X such that (i) for each norm open subset V of X, f ~ mj(V) is T-U.S.C.; (ii) for each weakly compact subset K of X, f ~ mj(K) is T-l.S.C. (Hint : first identify a more tractable subbase for the topology). Let fand ?.e 11 be proper lower sernicontinuous convex functions defined on a normed linear space with f= M-limfA. Establish the following stability result for sublevel sets [Mos2,Bell]: "d a> mj(X), we have slv (j;a) = M-lim slv (fA;a). Establish appropriate analogues of Lemma 5.3.8 and Theorem 5.3.9 for Mosco convergent sequences of weakly lower sernicontinuous functions.

§5.5. Mosco Convergence versus Wijsman Convergence In this section, we pursue the interesting connections between the Mosco topology and the Wijsman topology, which features a particularly attractive marriage of hyperspace topology and Banach space geometry. There are really two separate cases : (1) the convex case; (2) the general weakly closed case. Perhaps surprisingly, the second case as settled for the most part by results of [BoFi2] is easier to handle. We write TW d for the Wijsman topology on the nonempty weakly closed subsets as deterrnined by the initial norm for the space. 5.5.1 Theorem. Let be a normed linear space. Then on the nonempty weakly closed subsets of X, (a) (b)

TWd C

TM

TWd::::> TM

if and only if X is reflexive; if and only if X is finite dimensional.

Thus, the Wijsman topology agrees with Mosco topology if and only if X is finite dimensional. (a) If X is reflexive, then by Proposition 5.4.15, A ~ Dd({X},A) = d(x,A) is continuous with respect to the Mosco topology for each XE X. Thus, TWd C TM. If X is not reflexive, we produce a sequence of compact convex sets A, Al, A2, A3, ... with A = M-lim An but A:;:. TWd -limAn. Let x be a norm one element, and choose a Proof

sequence in the ball x +

i

U with no weakly convergent subsequence, and let An

=co {xn,fJ}. Then {fJ} =M-limA n whereas d(x,{fJ}) >

i ~limsuPn~ood(x,An)'

(b) If X is finite dimensional, then strong and weak become one, and both hyperspace topologies become the Fell topology because X has nice closed balls (see Theorem 5.1.10). Now suppose X is infinite dimensional. Proposition 5.4.13 takes care of the case X infinite dimensional and nonreflexive, because the Wijsman topology is always Hausdorff. Now suppose X is infinite dimensional and reflexive. As the origin e is in the weak closure of the surface of the unit ball, by the Whitley construction [Hol,§ 18], it is in the sequential weak closure. Let be a sequence of norm one vectors converging weakly to fJ. Let Y E X* be an arbitrary norm one functional and

THE FELL TOPOLOGY AND KURATOWSKI-PAINLEVE CONVERGENCE

179

consider the weakly c10sed sets A == {x: ky ,x>1 ~ 112} and An == A U {x n }. Mosco convergence of to A fails because is weakly convergent to 8. But Wijsman convergence occurs, as we now verify. Fix x E X. Since An => A, we have

d(x,An):S d(x,A)

= max {O, 112 - ky,x>I}.

If IIxll < 112, then for each n, IIx - xnll > 112. If IIxll ~ 112, by the weak lower sernicontinuity of the norm, we have lim infn~ IIx - xnll ~ Ilxll ~ 1/2. Either way,

lim infn~oo d(x,An) = rnin {d(x,A), lim infn~oo IIx - xnll} ~

rnin {d(x,A),l/2}

=d(x,A).

This establishes Wijsman convergence, completing the proof.

•

We now turn to the convex case, in which a modification of the weak* Kadec property (see §2.4) as considered in [BoY] plays the key role.

5.5.2 Definition. Let be a normed linear space. The dual norm 11·11* is called weak*-Mackey Kadec provided whenever ;"EA is a net in X* with y = a(X* ,X)-lim;., n and lIyll* = lim;., IInll*, then converges to y uniformlyon weakly compact subsets of X. For a Banach space X, Definition 5.5.2 amounts to saying that the weak* and Mackey topologies agree on the surface ofthe unit ball of X*. For X reflexive, weak*Mackey Kadec is the same as weak* Kadec. We first show that if 'l"Wd =>'l"M on ceX), then 11·11* must be weak*-Mackey Kadec [BoY].

5.5.3 Lemma. Let be a normed linear space such that 11·11* is not weak*Mackey Kadec. Then there exists a net of closed hyperplanes in X that is Wijsman convergent but not Mosco convergent. Proof. Suppose the dual norm is not weak*-Mackey Kadec. Then we can find a net of norm one elements ;"E A in X* and a norm one element y E X*, with Y = a(X*,x)-lim;.,n, but > 2 for all /\,. For each /\" set v;., =xo + x;.,. Since IIx;.,lI:S 1, we have 1. Since 1 ::;; IIv;.,lI::;; 4, we can choose a;.,E [1/4,1] with

a;.,v;., E y-l(l).

Again by passing to a subnet, we may assurne that - s(YS-,AS-)

= - s(yo,A) =d(xo,A) > o. We conclude that eventually, xs- (20 AS-, a contradiction. Thus, Xo E A must hold, and since An K ~ 0, we get Mosco convergence. This gives TM c: TWd on ceX).

•

The proof of Theorem 5.5.1 shows that for ceX), Mosco convergence cannot guarantee Wijsman convergence unless X is reflexive. Combined with Theorem 5.5.5 we get the following corollary, which in sequential form was established using very different methods in [BoFi2].

5.5.6 Corollary. Let be a normed linear space. Then TM= TWd on ceX) if and only if X is reflexive and the dual norm is weak* Kadec. Exercise Set 5.5. 1.

Let be a Schur space. Prove that the dual norm is weak*-Mackey Kadec (Hint : weakly compact sets are norm compact).

182

2. 3. 4.

CHAPTER 5

Suppose 0

{4 }

if x :s; 0

=

x

(e) T3(X) " { E

if x

Z+: n >

:e rationals

={ -n

6.

7. 8.

9.

10.

x

if x

if x > 0

=0 c:

Z+

otherwise

Suppose X and Y are Hausdorff spaees, and T and S are multifunctions from X to Y that are o.s.c. at xo. Show that T n S and T U S are o.s.c. at xo. Suppose X and Y are Hausdorff spaces, and that T: X::::t Y. Define T': X ::::t Y by T '(x) = cl T(x) U Frae T(x). Show that T' is the smallest multifunetion containing T that has closed graph. Let {Ti: X::::t Yi : i E l} be a family of multifunetions. Suppose eaeh Ti is I Ti is o.s.e. at xo· o.s.c. at xo. Prove that

TIiE

Suppose X and Y are Hausdorff spaces, and either Y is loeally eompaet or both X and Y are first countable. Prove that T: X::::t Y is o.s.e at xo E X if and only if T(xO) is closed and whenever T(xO) misses a compact subset K of Y, then T(x) misses K for all x near xo (Hint : see Theorem 5.2.6 and Lemma 5.2.8). Let X be a Hausdorff space, and suppose ceX,x) is equipped with the topology of uniform convergence. Prove that the fixed-point multifunction of Example 6.1.7 has closed graph. Show that the assertion remains true if X is locally eompaet and ceX,x) is equipped with the weaker compaet-open topology. Consider the Banach space X of continuous real functions on [0,1] equipped with the norm of uniform convergence. Let A = {x EX: x(O) = 0 and

f~ x(t) dt = 1}. 11.

l}

=0

if x = !!:...- where {m,n} m

o 5.

z+

if x < 0

0 {n

and nE

otherwise

[x, 1 ]

(d) T4(X)

= lIn

=O?

Prove directly that A is a closed convex subset of X that is not

proximinal. Let A be a nonempty subset of a normed linear space . (a) Suppose A is a open. Prove that the visibility multifunction for A has open values.

MULTIFUNCTIONS: THE RUDIMENTS

12.

13.

14. 15.

16.

17.

18.

191

(b) Suppose A is closed. Prove that the visibility multifunction for A has closed graph. Let be a normed linear space. For each closed subset A of X, let ker (A) denote the convex kernel of A, the set of points a E A that see each point of A via A (see Exercise 4.1.5). Let 'l'Wd be the Wijsman topology for the nonempty closed subsets CL(X). Prove that A =4 ker(A) as a multifunction from to X has closed graph. Let be a normed linear space. (a) Suppose A is a closed convex subset of X and XE X. Verify that the metric projection P(x,A) is a closed and convex set. (b) Suppose A is a closed convex subset of X and x E X. Prove that ao E A belongs to P(x,A) if and only if there exists a norm one YO E X* such that = IIx - aoil and =s(yo,A), Le., YO belongs to the normal cone to A at ao (Hint: use Lemma 1.4.7). Such anormal YO is called a proximal normal [CI,BoS]. (c) A norm is called strictly convex [Cio,DS] provided the surface of the closed unit ball U contains no line segments, i.e., if IIXili =1 and IIX211 =1, then for each a E (0,1), we have lIaxl + (1 - a)x211 < 1. Prove that metric projections onto closed convex sets are without exception either empty or singletons if and only if 11-11 is strictly convex. (d) Prove that 11-11 is strictly convex if and only if each Y E X* assurnes its supremum at most one point of U (Hint : if 11·11 is not strictly convex, apply Corollary 1.4.8 at amidpoint of a segment on the surface of U). Write down the formula for the duality mapping J for the plane equipped with the box norm. Let X be a normed linear space, and let J: X =4 X* be the duality mapping. (a) Show that the values of J are nonempty weak*-compact convex sets. (b) Show that J has closed graph, when X is given the strong topology and X* is given the weak* topology. (c) Show that ifthe dual norm 11·11* is strictly convex, then J is single-valued. Let be a Banach space. Suppose X* is equipped with the weak* topology. Show that the map x =4 {y E X*: = I} has closed graph if and only if X is finite dimensional (Hint : If X is infinite dimensional, for each finite sub set F of X\{ O} and nE Z+, let Y =y(F,n) satisfy ky,x>I::;; lIn for each XE F, and lIyll* > n. Now choose x =x(F,n) satisfying IIxll < lIn and = 1). Let X be a Hausdorff space, and let L(X) be the lower semicontinuous functions on X. Suppose L(X) is equipped with the Fell topology 'l'F of §5.1, where functions are identified with their epigraphs. Prove that Argmin: L(X) ~ X has closed graph if and only if X is locally compact [BeKl]. Let Xand Y be normed linear spaces and let T: X =4 Y be a multifunction. (a) Prove that Gr T is convex if and only for each Xl and X2 in dom T and each a in [0,1], we have aT(xt} + (1 - a)T(x2) c: T(axl + (1 - a)x2). (b) Prove that Gr T is a cone if and only if the following conditions are all fulfilled: (i) 0 E T(9);

CHAPfER6

192

(ii) 'Va> 0 'V x

E

X, aT(x)

=T(ax);

(iii) 'V Xl, X2 E X, T(xt} + T(X2) c:: T(XI + X2). A multifunction whose graph is a cone is called a convex process [Rocl,AuF).

§6.2. Lower and Upper Semicontinuity for Multifunctions A single-valued function f from a Hausdorff space X to a Hausdorff space Y is continuous at xo E X if whenever V is open in Y and xo E f-I(V), then f-I(V) contains a neighborhood of X{). This formulation has a direct extension to multifunctions that is usually called lower semicontinuity (despite a different use of the same term for single-valued functions).

6.2.1 Definition. Let X and Y be Hausdorff spaces and let T: X =t Y be a multifunction. We say that T is lower semicontinuous (l.s.c.) at xo E X if whenever V is an open subset of Y and xo E T-I(V), then T-I(V) contains a neighborhood of X{). In simple English, lower semicontinuity says this: if T(xO) hits an open subset V of Y, then T(x) hits V for all X sufficiently close to xo. For singleton-valued multifunctions, lower semicontinuity is ordinary continuity, and each multifunction T is automatically lower semicontinuous at each X outside the domain of T. Recall that for a net of subsets hex) c:: T).'(x). Let T(x) nkAT(x). (a) Prove that T is an usco map. (b) Suppose that T is in addition l.s.c. Prove that O. We define subsets A(n,e) and CCe) as folIows: A(n,e) == {x EX: N(x,e)::; n}, CCe) == {x EX: 'i 0< 3e 'i W E rl(x) T(W) a::: Sß(T(x)] }.

Now if T(x) admits an e-discrete set of cardinality n, then by lower semicontinuity, so does T(w) for all w near x. This shows that A(n,e)C is open, and A(n,e) is closed. We claim that the set CCe) is also closed. To see this, suppose x E cl CCe) and let 0 < 3e be fixed. Choose r between 0 and 3e. By Lemma 6.2.6, there exists a neighborhood W of x such that for each w E W we have T(x) c: Sl'ö[T(w)], and so Sc51:T(x)] c: S~T(w)]. Choose Wo E W n C(e); since W is a neighborhood of Wo, T(W) a::: S~T(wo)], and afortiori T(W) a::: Sß(T(x)]. This shows that XE CCe). We next claim that if CC e) contains an open set V, then within V there are points x such that N(x,e) is arbitrarily large. This part is a little subtle. Let xo E V be arbitrary. It suffices to produce a point q E V such that N(q,e)':? N(xo,e) + 1. Write m =N(xo,e), and choose an e-discrete set b}, b2, b3, ..., bm in T(xo). Write 20 =min {d(bj,bj) - e} and then set 11 min {o,e}. By Lemma 6.2.6 and the definition of C(e),

=

there exists q E V such that both T(xO) c: S7J[T(q)] and

T(q) a::: S2e[T(xo)].

We may choose Yl, Y2, ... , Ym in T(q) such that d(bj,Yj) < 11 for i Then if i *" j, we have

= 1,2,3, .. .,m.

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Now choose Ym+l E T(q)\SZE[T(xO)]' For each i:5: m, d(ym+l,bi) ~ 2e and d(bi,Yi) < e, and so d(Ym+l,yi) > e. This proves that N(q,e) ~ N(xo,e) + 1, and the claim is established. We now write D(n,e) = A(n,e) n C(e), a closed subset of X for each n and e. The above paragraph shows that D(n,e) is nowhere dense, i.e., that int D(n,e) 0. For each

nE

Z+ and k E Z+, write F(n,k)

=D(n,lIk).

=

Since X is a Baire space and each

F(n,k) is nowhere dense, the set E == nn,k F(n,k)C is a dense and Go subset of X.

The proof will be complete if we can prove upper sernicontinuity at each point of E. For this purpose we again rely on Lemma 6.2.6. Fix xo E E and let e> O. We produce a neighborhood W of xo with T(W) c: SE[T(xo)]. Choose k E Z+ with 3/k< e and

let n =N(xO,lIk). Since xo e F(n,k) =A(n,lIk) n C(lIk) and xo E A(n,lIk), we conclude that xo e C(lIk). Thus, for some neighborhood W of xo and some 8< 3/k, we have T(W) c: So[T(xo)] c: SdT(xO)] , as required. •

To prove our prornised genericity result, we will use the fact that a separable metric space can be topologically embedded in a compact metric space. We outline one way to do this. Given a countable base 13 for the topology, let .1 {(VI, Vz) E 13 x 13

=

: cl VI c: Vz}. For each (Vl,vZ)

E

.1, let fVl'V2

E

C(Y,[O,I]) be a Urysohn function

2

mapping cl VI to zero and V to one. The space [0,1].1 as a countable product of metrizable spaces is metrizable, and by the Tychonoff Theorem, the product is compact. The desired embedding takes Y to y# : .1 ~ [0,1] defined by y#(Vl,vZ) = fVl'V2 (y).

6.3.10 Theorem. Let X be a Baire space and let be a separable metric space. Suppose T: X:4 Y is a lower semicontinuous multifunction with nonempty closed values. Then there exists a dense and Gö subset E of X such that at each point x of E, T is O.S.c. Proof Let be a compact metric space in which Y sits topologically, and consider the multifunction T': X:4 Y' defined by T'(x) = cl T(x), where the closure is taken in the compactification. Clearly T' is compact-valued and by Exercise 6.2.1, it remains l.s.c. By Proposition 6.3.9, there is a dense and Go subset E of X such that T' is U.S.C at each point of E. We claim that T itself is o.s.c. at each point of E. Fix xo E E and let e be positive. There exists W E rL(XO) such that T'(W) c:

~[T'(xo)]

=~[T(xo)],

where the enlargements are computed in Y'. As a result, relative to the metric subspace , we have

Frac T(xo) c: cl (T(W)\T(xo» c: cl T(W) c: Sfe[T(xo)].

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205

Since T(xO) is a closed set, it is the intersection of its open enlargements, and we have • shown that Frac T(xo) c:: T(xo). By Lemma 6.1.15, T is o.s.c. at xo. Fort [Forl] also established a dual ofProposition 6.3.9, showing that an usco map from a Baire space to a metric space is lower semicontinuous at most points. We give a result in this spirit without assuming compact or even closed values.

6.3.11 Proposition. Let X be a Baire space and let be a separable metric space. Suppose T: X ~ Y is an upper semicontinuous multifunction with nonempty values. Then there exists a dense and Gö subset E of X such that at each point x of E, T is l.s.c. Proof

Let

13

= {Vn : n E

semicontinuity, for each

Z+} be a countable base for the topology of Y. By upper

nE

Z+, the set An == T-I(cl Vn ) is a closed subset of X.

Since X is a Baire space, the set E == nn':1 (bd An)C is a dense and Gö subset of X. We claim that T is l.s.c. at each point of E. To verify the claim, let XE E and suppose T(x) n V::f:. 0 where V is open in

Y. By the regularity of metric spaces, we can find Vn E 13 and Y

E

T(x) such that

Y E Vn c:: cl Vn c:: V. Since x E An and XE E, it follows that XE intA n. Thus, W E T-I(cl Vn ) c:: T-I(V) for each w in some neighborhood of x, and so T is l.s.c. at x. • We close this section with a result regarding the upper semicontinuity of the intersection of an outer semicontinuous multifunction with an upper semicontinuous multifunction. In applications, the outer semicontinuous multifunction is frequently constant, so that the intersection amounts to a truncation of the upper semicontinuous one.

6.3.12 Proposition. Let X and Y be Hausdorff spaces and let T: X =t Y and S: X ~ Y be multifunctions. Suppose T is u.s.c at xO E X and T(xO) is compact, and S is o.s.c. at xO. Then T n S is u.s.c. at XO. Proof We modify the argument used in the proof of Proposition 6.2.8. Let V be an open neighborhood of (T n S)(xo). If T(xo) c:: V, then by the upper semicontinuity of T at xo, we get (T n S)(x) c:: V for all x near xo. Otherwise, T(xO) n VC is a nonempty compact subset of Y. Now no point of T(xo) n Vc can belong to S(xo). By compactness of T(xo) n VC and outer semicontinuity of S at xo and using condition (3) of Lemma 6.1.15, there exist neighborhoods WI of xo and VI of T(xO) n

such that S(WI) For each

XE

WI

n VI =0. Now choose n W2, we have

T(x)

n Sex) c::

(V

U VI)

vc

W2 E TL(xo) such that T(W2) c:: V U VI.

n Sex) = V n Sex)

establishing upper semicontinuity of the intersection at

XQ.

•

c:: V,

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Exercise Set 6.3. I.

A multifunction T: X::4 Y is called subcontinuous at x =xo [Sm3,DL,Lecl] if and only if whenever --7 xo and Y./t E T(x,ü, then has a convergent subnet. (a) Prove that if T is o.s.c and subcontinuous at xo, then T is u.s.c. at xo. (b) Prove that if T is u.s.c. at xo and T(xo) E K(Y), then T is subcontinuous at xo. (c) Suppose that Y is completely regular and T is subcontinuous at xo. Prove that Frac T(xo) is compact (Hint : Let .:te A be a net in Frac T(xo). Suppose'11 is a compatible uniformity for the space Y; for each (A. W,U) E A x n(xo) x '11, let x = x(A., W,U) be a point of W such that U(b).) n

2.

3.

4.

5. 6.

7.

8.

9.

T(x) n T(xo)C -:f:. 0). Let be a normed linear space and let J: X ::4 x* be the duality mapping. (a) Using Proposition 6.3.2, prove that J is norm-to-weak* usco [Cio]. (b) Suppose that the dual norm has the weak* Kadec property : the conditions Y = a(X* ,x:)-lim Y./t plus lIylI* =lim).IIY./tIl* jointly imply lim). lIy - Y./tll* =O. Prove that the duality mapping is now norm-to-norm usco. Let be a reflexive space whose norm has the Kadec property : the conditions x = a(X,x*)-lim x). plus IIxll =lim).,llx).11 jointly imply lim).llx - x).11 =

O. Prove that the metric projection P: x be a normed linear space and let T: X ~ X* be norm-to-weak* useo. Prove that the following maps are also norm-to-weak* u.s.e. : (a) Tl(X) = T(x) n a(x)U* where a: X -t [0,+00) is upper semicontinuous ; (b) T2(X) = T(x) n {y E X*: = I}, under the assumption that T is locally bounded. A Hausdorff space Y is called angelic [Pry,Flo] provided both of the following conditions are met: (i) each relatively sequentially compact subset A of Y is relatively compact; (ii) each point in the closure of a relatively eompaet subset A of Y is the limit of sequenee in A. All metrizable loeally eonvex spaees are angelie in their weak topology [Flo, p. 39]. Obtain the following result of HanselI, Jayne, Labuda, and Rogers [HJLR]: if X is first eountable and Y is angelie and T: X ~ Y is u.s.e. at xo E X, then Frae T(xO) is eompaet . Produce an example showing the need for the compaetness assumption on T(xo) for the seeond assertion in the Berge Maximum Theorem, even if f is continuous. Show that Frac for an useo map from R to R need not itselfbe u.s.e. Let be a reflexive Banach spaee whose norm has the Kadec property and whose dual norm has the weak (= weak*) Kadec property. Let be a net in C(X) and let A E C(X). Using the Exereise 6.3.3 and the results of §5.5, prove that the following eonditions are equivalent [At3,So,Ts]: (a) is Wijsman convergent to A; (b) is slice eonvergent to A; (e) foreach XE X andeach e>O, theenlargement Se[P(x,A)] contains P(x,A;.) eventually. Let B be a nonempty closed and bounded subset of ametrie spaee and let A be a nonempty sub set of X. For each X E X, let fB(X) = ed(B,{x}), where ed denotes exeess. We eall r(B;A) == infxEAfB(x) the Chebyshev radius of B in A, and we eall Cent (B;A) == ArgminfBIA the relative Chebyshev center of B in A [Gar,Amr]. (a) Explain how the metric projection is a special case of the relative Chebyshev center map, and interpret the relative Chebyshev center as ametrie projeetion itself with respeet to Hausdorff distanee. (b) Prove that Ir(Bl;A) - r(B2;A)1 ~ Hd(B},B2) where BI. B2 belong to CLB(X), the nonempty closed and bounded sub sets of X. (e) Suppose is a normed linear spaee and B E CLB(X). Prove that fB is finite-valued, lower semicontinuous, and eonvex.

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208

(d) Suppose is a reflexive Banach space and A is weakly closed. Prove that for each BE CLB(X), Cent (B;A) is nonempty, and that B::4 Cent (B;A) as a multifunction defmed on O}

U {T(Xi) : xo e T(XiH.

Then for some i we have XO E T(Xi) and, at the same time, XO e T(Xi), a contradiction. This shows that {T(x): X E X} has the finite intersection property. • With sufficient compactness, it is clear that the family {T(x): X E X} in the statement of the KKM Principle will have nonempty intersection. It is standard to call a point xo E X afixed point of a set-valued mapping S defined on X if xo E S(xo). As a first application of the KKM Principle, we give a fixed-point theorem of Fan and Browder [Fan2,Brol]. 6.4.5 Theorem. Let C be a nonempty compact convex subset of a lcs E, and let S be a set-valued function from C to C such that (i) foreach XE C, S-I({X}) ={CE C: XE S(c)} isopenin C; (ii) S(c) is nonempty and convex for each CE C.

MULTIFUNCTIONS: TIlE RUDIMENTS

211

Then S has a fixed point. Proof. Assume no fixed point exists. Then by (i) for eaeh XE C, T(x) == {c E C: x ~ S(c)} is nonempty and eompaet. Also, sinee S has nonempty values, we have Uxe C S -l({x}) = C => nxeC T(x) = 0. By the eompaetness of eaeh T(x) , the family {T(x): XE C} cannot have the finite intersection property, and so T cannot be a KKM map.

We ean thus find x}, X2, .. .,xn in C and nonnegative scalars al, a2, .. .,an such that al + a2 + ... + an = I and

Since C is eonvex, this convex eombination lies in C and so

This means that Xi E S(CO) for eaeh 1:S:; i:S:; n, and by the convexity of S(CO) , we have cO E S(CO). This contradicts our assumption that S is fixed-point free, and completes the proof. • We use Theorem 6.4.5 to establish the Schauder Theorem, a result that has many applications to differential and integral equations (see Exereise 6.4.7). First, we give a preparatory lemma that speaks to the existenee of approximate fixed points for a singlevalued mapping defined on a compact convex set whose values almost stay in the set [Las]. 6.4.6 Lemma. Let C be a nonempty eompaet convex sub set of a lcs E and let W be an open convex neighorhood of origin. Suppose J: C ~ E is continuous, and f( C) c: C + W. Then there exists CO E C satisfying f( CO) E co + W.

Proof Let S be the set-valued function from C to C defined by S(c) = (j(c) - W) n c. Since S(c) is the intersection oftwo convex sets, it is eonvex, and by the eontinuity of J, S -l( {x}) is open in C for each XE C. Also, since f(c) E C + W, we see that S(c) is nonempty for eaeh CE C. By the Theorem 6.4.5, S has a fixed point CO, and

by eonstruction f( co) E cO + W.

J:

•

6.4.7 Schauder Theorem. Let C be a nonempty convex subset of a lcs, and let C ~ C be eontinuous with elf(C) a compact subset of C. Then J has a fixed point.

Proof Let TL be the family of symmetrie open convex neighborhoods of the origin,

ordered by reverse inelusion.

Fix W E TL; we elaim that there exists cw

f(cw)

E

C with

E cw+ W. To see this, choose by the compactness of elf(C) a finite subset F of C such that el f( C) c: F + W. Then eo F is a eompact convex subset of C, and

f(co F) c: elf(C) c: F + W c: co F + W.

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Applying Lemma 6.4.6, we see that for some ewE co F, we have j{CW) E CW+ W. Now consider the net W ~ j{cw) from TI to j{ C). By the relative compactness of j{ C), the net has a cluster point co E C. We claim j{co) = co. If not choose V E TI such that (co + V) n (((co) + V) =0. By continuity of f at co choose W E TI such that j{co + W) c: j{co) + V, and then W' E TI such that W' + W' c: W n V. Since co is a cluster point of our net of function values, there exists W" c: W' such that j{cw") E co + W'. But then cw" E j{cw")

+ W" c:

co + W'

+ W" c:

co + W,

and so j{cw") c: j{co) + V. We have shown that j{cw") with this contradiction, the proof is complete. •

E

(co + V)

n (((co) + V),

and

Notice that we now have retrieved (much more than) Brouwer's Theorem from the KKM principle, and so the two results are in essence equivalent (for a more direct derivation, see Exercise 6.4.3 ). As a second application ofthe Theorem 6.4.5, we obtain a coincidence theorem of Fan [Fan3]. This result gives conditions under which two multifunctions have values that intersect.

6.4.8 Theorem. Let EI and E2 be locally convex spaces, and let C and K be two nonempty compact convex subsets of EI and E2, respectively. Suppose T and S are two set-valued functions from C to K satisfying the following conditions : (i) for each

XE

C, S(x) is open and T(x) is a nonempty convex set;

(ii) foreach yE K, T-I({y}) isopen and S-I({y}) is anonemptyconvex set.

Then for some xo E C, we have S(xo)

n T(xo) =F- 0.

Proof We define a product set-valued function L from C x K to C x K by L(x,y)

=

Note that L-I({(x,y)})

S-I({y}) X T(x).

= T-l({y}) x S(x).

By Theorem 6.4.5, L has a fixed point

(XO,YO), and for this pair, xo E S-l({yO}) and YO E T(xO), and so YO E S(XO)

as required.

n T(xO) ,

•

Let A and B be sets without structure, and suppose f: A x B function. We compare the following expressions : (i)

SUPaeA

~

R is any

infbeB j{a,b);

(ii) infbeB SUPaeA f(a,b).

Let ao E A be fixed. For each bEB, we have j{ao,b)::;; SUPaeA f(a,b), and so

MULTIFUNCTIONS: TUE RUDIMENTS

infbeB.f(ao,b)

~ infbeB

213

SUPaeA .f(a,b).

Since ao is arbitrary, this gives SUPaeA infbeB .f(a,b) ~ infbeB SUPaeA .f(a,b). We will presently state sufficient conditions for equality. We call an extended real function g defined on a convex set C quasi-convex provided it has convex sublevel sets. In analytic terms, g is quasi-convex if and only if for each Xl and X2 in C and a E [0,1], we have g(axI + (1 - a)x2) ~ max {g(X}),g(X2)}. Clearly, convex functions are quasi-convex. On the other hand, each monotone function of a real variable is quasi-convex. We call ! quasi-concave if -! is quasi-convex. The following result is due to Sion [Sio]. 6.4.9 Minimax Theorem. Let C and K be two nonempty compact convex sets in the locally convex spaces EI and E2. Suppose!: C x K ~ R satisfies the following conditions: (i) ' to a metric space . (1) If T is measurable and closed-valued, then the inverse image of each compact set is measurable; (2) If the inverse image of each closed set is measurable, then T is measurable. Proof The assertion of (1) follows from the formula T-l(K)

= nn:l T-l(SlIn[K]),

valid

for each K E K(Y). Since T-l preserves unions, condition (2) follows from the fact that each proper nonempty open sub set V of Y can be written as a countable union of closed sets, namely, V= Un: 1 {y: d(y,VC) ~ lIn}. • Examples exist requiring some background in descriptive set theory which show that neither implication of Proposition 6.5.4 is reversible (see, e.g., [HPV]). But for multifunctions with closed values in a sigma compact metrizable space, both are reversible (recall that a Hausdorff space is called sigma compact provided it can be expressed as a countable union of compact subsets).

6.5.5 Proposition. Let T be a multifunction with closed values from a measurable space to a sigma compact metric space . The following are equivalent : (1) T is measurable; (2) For each K E K(y), T-l(K) is measurable;

(3) For each BE eL(y), T-l(B) is measurable. Proof By Proposition 6.5.4, we need only prove (2) ::::} (3). By sigma compactness, we

may write X = Un: 1 K n with K n E K(Y). Then if BE eL(Y) and (2) holds, we get T-l(B)

= Un: 1 T-l(B n K n ) E

Cl.

•

We also record the following fact.

6.5.6 Proposition. Let T be a compact-valued multifunction from a measurable space to a metric space . Then T is measurable if and only if for each BE eL(y), T-l(B) is measurable. Proof

If T is measurable and Bey is closed, then T-l(B)

The converse follows from Proposition 6.5.4.

= n;l T-l(Slln[B]).

•

Measurability is preserved under countable unions (see Exercise 6.5.1), but not under finite intersections, even if the intersections are nonempty and the target space is Polish (see again [HPV]). Intersections are preserved with some compactness. The proof of the next result is taken from Himmelberg [Hirn].

6.5.7 Proposition. Let be a measurable space and let be a separable metric space.

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220

(1) If is a sequence of compact-valued measurable multifunctions from X to Y, then n;l Tn is measurable; (2) If T : X::::f Y is compact-valued and measurable, and S: X::::f Y ·is closedvalued and measurable, then T n S is measurable.

Proof. For (1), we consider the compact-valued product multifunction L ==

U':1 T n,

whose measurability is easily checked (see Exercise 6.5.2). By Proposition 6.5.6, for each product closed set E, we have L-l(E) E Cl. Let D denote the diagonal of the product, and let B be a closed subset of Y. We have (n;l Tn )-l(B)

=L-l(D n BZ+)

E Cl, and by either Proposition 6.5.4 or Proposition 6.5.6, the interseetion multifunction is measurable. To establish (2), ftrst topologically embed Y into a compact metric space E (see clES is measurable as a p. 204), and replace S by clE S. By (1), x::::f T multifunction from X to E. But for each x E X, T(x) n clES(X) =T(x) n S(x), and so T n S is measurable. •

n

Measurability of a multifunetion with nonempty closed values in a separable metric space can be comfortably linked with distance functionals and the Wijsman topology 'Z'Wd on CL(Y). Recall that

'Z'Wd

is the weak topology on CL(Y) generated by the family of

distance functionals {d(y,') : YE Y}, where each functional is a function of a closed set variable. Dur ftrst result in this direction is

6.5.8 Proposition. Let T be a multifunction with nonempty values from a measurable space to a separable metric space . Then T is measurable if and only if for each YE Y, X -+ d(y,T(x)) is a measurable function.of x. Proof. First, suppose that T is measurable. For each Y E Y, let gy: X -+ [0,+00) be deftned by gy(x) =d(y,T(x)). Fix YE Y and let a> 0; we have {x: gy(x) < a}

= {x: d(y,T(x)) < a} =T-l(Sa[Y]) E

Cl,

and so gy is measurable. Conversely, suppose each gy is measurable. Let V be a nonempty open subset of Y and write V as a countable union of open balls, say

We then have T-l(V)

=U';l g ;!([O,En)) E

Cl, and this yields measurability of T.

•

We declare a sequence of multifunctions with values in CL(Y) to be Wijsman convergent to a multifunction T with values in CL(Y) if we have T(x) = 'Z'Wdlim Tn(x) at each XE X. convergence [Hesi].

We next show that measurability is preserved by Wijsman

MULTIFUNCTIONS: THE RUDIMENTS

221

6.5.9 Proposition. Let T, Tl, T2, . .. be a sequence of multifunctions from a measurable space to a separable metric space each with values in eL(Y). Suppose each Tn is measurable and is Wijsman convergent to T. Then T is measurable. Proof By the definition of Wijsman convergence, for each YE Y, we have d(y,T(x» = li.mn-+>od(y,Tn(x». By Proposition 6.5.8 and Lemma 6.5.1, the single-valued function x ~ d(y,T(x» is measurable. Again by Proposition 6.5.8, T is measurable. • Hausdorff metric convergence, Kuratowski-Painleve convergence, etc., for sequences of closed-valued multifunctions are also defined in the same pointwise way. By Proposition 6.5.9, measurability is preserved by any convergence stronger than Wijsman convergence, e.g., Hausdorff metric convergence. With respect to the weaker Painleve-Kuratowski convergence of §5.2, we give the following result [SW3,Tom].

6.5.10 Proposition. Let T, Tl, T2, T3, ... be multifunctions each with closed values from a measurable space to a sigma compact metric space . Suppose each Tn is measurable and is Kuratowski-Painleve convergent to T. Then T is measurable. Proof By assumption, at each x E X, we have T(x) =Li Tn(x) =Ls Tn(x); so, we need only show that Ls Tn is measurable. By Proposition 6.5.5, it suffices to show that (Ls Tn)-I(K) is measurable for each K E K(Y). This follows from the formula

For positive results regarding the measurability of the weak upper closed limit of a sequence of measurable multifunctions, the reader rnay consult [Hes2]. We now show that measurability of a multifunction into a separable metric space implies measurability of its graph with respect to a natural sigma algebra on X x Y. This is the smallest sigma algebra containing all measurable rectangles of the form A x B where A E Cl and BE :B(Y). We denote this product sigma algebra by Cl ® :B(Y). First, we give a lemma of interest in its own right.

6.5.11 Lemma. Let be a separable metric space and be a metric space. Suppose is a measurable space and f: X x Y ~ E has the following properties: (i) for each x E X, fix, .) E C(Y,R); (ii) for each Y E Y, f(- ,y) is Clmeasurable. Then f is measurable with respect to the sigma algebra Cl ® :B(Y). Proof Let {Yn: n E Z+} be a countable dense subset of Y. By condition (ii), for each nE Z+, gn: X x Y ~ E defined by gn(x,y) = f(x,Yn) is Cl ® :B(Y) - measurable, for if V is open in E, then

(Ui.::\ SlIk[yj])c).

Fix k E Z+, and for each nE Z+ restriet 8n to X x (SlIk[Yn] n we amalgamate these restrietions, we get a globally defined function

!k

If

which by Lemma

222

CHAPTER6

6.5.1 is Cl ® B (Y) - measurable. For fixed x and y there exists Yn such that d(Y,Yn) < l/k for which Ik(X,y) =l(x,Yn), and by continuity of fix, .), we get f(x,y) = limk~oo!k(x,y). Again invoking Lemma 6.5.1,1 is Cl ® B(Y)-measurable. • A function f(x,y) that is separately continuous in one variable and measurable in the other is called a Caratheodory function .

6.5.12 Theorem. Let T be a closed-valued measurable multifunction from a measurable space to a separable metric space . Then Gr T belongs to Cl ® B(Y). Proof By Proposition 6.5.3, we may assume that T has nonempty values. values of T are closed, we have

Gr T= ((x,y)

E

Xx Y: d(y,T(x»

Since the

=O}

But f(x,y) =d(y,T(x» is continuous in y for each fixed x, and by Proposition 6.5.8, it is also measurable in x for each fixed y. By Proposition 6.5.11, 1 is Cl ® B(X)measurable, and in particular Gr T =1-1(0) lies in Cl ® B(X), as required. • We now study the connection between the measurability of a multifunction T:

:::::t with values in CL(Y) and the measurability of the associated singlevalued function

f

into CL(Y) equipped with an appropriate topology.

Are there

convenient, easily manipulated topologies r on CL(X) such that r-measurability of T from to be a normed linear space. Then T: X:::::t Y admits a continuous e-approximate selection for each e > o if and only if for each e> 0 and each xo E X, there exists W E :rl(xo) such that

n w Se[T(x)] '# 0. XE

For necessity, fix xo E X and e> O. Let g be a continuous E/3-approximate selection for T, and choose W E :rl(xo) such that XE W => IIg(x) - g(xo)1I < E/3.

Proof.

229

MULTIFUNCTIONS: THE RUDIMENTS

Choosing YO E T(xO) such that IIg(xO) - YOli < &3, we see that YO E Se[T(x)] for each x E W, as required.

e> :F. 0.

For sufficiency, we use the convexity and paracompactness assumptions. Fix

o and for each

x E X, let Wx E TL(x) be chosen such that

Choose by Theorem 3.3.9

a partition of unity {Px: x E X} subordinated to the open

cover {Wx : x E X}, and for each x E X let Yx now form the weighted average function g ==

2.xE

nWE Wx Se[T(w)]

E

nWE Wx Se[T(w)]

be arbitrary.

We

X PX'Yx

which by the definition of partition of unity is continuous (see Exercise 3.3.6). Fix xo E there exists a finite subset {Xl, x2, ... , xn } of X such that

X;

and such that for each i::; n, p4xo) > O. Since {Px: x E X} is subordinated to the open cover {Wx : x

E

X}, we have xo E WXj for each i::; n. By the choice of the YXj' we

have YXj E SdT(xo)] for each i::; n, and since

:L:l

p4xo)

= 1,

we see that g(XO)

as a convex combination of points in Se[T(XO)] is in SdT(xO)] itself. We conc1ude that d(g(xO),T(xO» < E, and this shows that g is an e-approximate selection for T. •

6.6.3 Michael's Selection Theorem. Let X be a paracompact space and let be a Banach space. Suppose T: X =4 Y is a lower sernicontinuous multifunction with nonempty c10sed convex values. Then T has a continuous selection.

Proof. We shall construct recursively a sequence {

c:

qx,y)

such that for each n

(i) fn is a continuous 2- n -approximate selection for T;

(ii) V x

E

X V n ;::: 1 Ilfn+ I (x) - fn(x)11 < 2- n+ I

It is evident that a lower sernicontinuous multifunction with nonempty values satisfies the continuity condition of Deutsch and Kenderov; so by Proposition 6.6.2, we can find an initial 112-approximate selection 11 for T. Now suppose 11,12,13, .. .In have been constructed satisfying (i) and (ii). By (i), for each x E X, Tn(x) == S2-n[fn(x)] n T(x) is nonempty, and it is easy to check that Tn is lower sernicontinuous and convex-valued. Applying Proposition 6.6.2 with respect to Tn , there exists a continuous 2-(n+l) approximate selection for Tn, which we denote by fn+l. Since d(fn+I(X),S2-n[fn(X)] )::; d(fn+I(X),Tn(x» < 2-(n+l) , we have l!fn+l(X) -fn(x) 11 < 2- n + 2-(n+l) < 2-n+1 ,

and since Tn(x) c: T(x), fn+l is a 2-(n+l) approximate selection for T as weIl. By (ii), the sequence is uniformly Cauchy, and since the metric deterrnined by the norm of Y is complete, is uniformly convergent to some continuous function

CHAPTER6

230

f

Finally, fex) = limn-toofn(x) and (i) imply for each x and since T(x) is closed, f is aselection for T. •

E

X that dlf(x),T(x» = 0,

We now give some applications ofMichael's Selection Theorem.

6.6.4 Proposition. Let X be a paracompact Hausdorff space, let be a Banach space and let A be a nonempty closed sub set of X. Suppose g: A -7 Y is a continuous function. Then g has a continuous extension to all of X. Proof. Define T: X =4 Y by

1

{g(x)}

T(x) =

Y

if x

E

A

if x

~

A

By Michael's Theorem, T has a continuous selection f, and

f is an extension of

g.

•

We next prove a theorem of Bartle and Graves [BrG], that speaks to the existence of a continuous one-sided inverse to a surjective continuous linear transformation.

6.6.5 Proposition. Suppose and are Banach spaces, and let f: Xl -7 X2 be a continuous linear transformation such that j(XI) = X2. Then there exists a continuous function g : X2 -7 Xl such that gof= idx1 .

Proof. By the Open Mapping Theorem (see, e.g., [HoI2, p. 141]), f maps open sets to open sets. By Exercise 6.2.4, y -7 j-I( {y}) is lower sernicontinuous. Since the inverse image of each point is a translate of the kernel of f and is therefore a closed convex set, a continuous selection g exists, and we have gof= idx1 . • We finish OUT discussion of continuous selections with an example due essentially to Michael [Mic3] which shows that a lower semicontinuous multifunction with closed convex values in a noncomplete normed linear space need not have a continuous selection.

°

6.6.6 Example. Let be an enumeration of the set of rationals in [0,1]. Let Y be the noncomplete subspace of .Q 2 consisting of the sequences that are eventually. Define T: [0,1] =4 Y by T(x)

{y E Y : y(n)

={

~

l/n}

Y

otherwise

Clearly, T has closed convex values, and it is left to the reader to verify that T is l.s.c. Now suppose that g were a continuous selection for T. For each finite sub set F of Z+, let EF = {x

E

[0,1]: g(x)(n) =

°for all

n ~ F}

MULTIFUNCTIONS: THE RUDIMENTS

231

By continuity of g, EF is closed. Let W be a nonempty open sub set of [0,1]; since W contains qn for some nE Fe and qn e EF, we see that int EF = 0. By Baire's Theorem, there exists XE [0,1] that belongs to no EF and for this x, we have g(x)(n) 7:- frequently. With this contradiction, no continuous selection can exist.

°

We now turn to the existence of measurable selections for measurable multifunctions with values in a Polish space, a separable space that is metrizable with a complete metric.

6.6.7 Theorem. Let

g(xo) is real-valued.

Prove Dieudonm!'s

Interposition Theorem [Ddn] : there exists a continuous function j: X ~ R such 3.

4.

5.

that for each x, hex) ~f(x) ~ g(x) (Hint : let T(x) = {a ER: hex) ~ a ~ g(x)}). Let X be paracompact, let be a Banach space, and T: X::::t Y have nonempty closed convex values. Prove that T is l.s.c if and only if for each xo E X and YO E T(xo), there exist a continuous selection j for T with f(xo) =YO. Let X be a Hausdorff space and let be a metric space. Suppose T: X::::t Y is a multifunction with nonempty values. (a) Suppose T admits a continuous selection. Show that for each x E X, there

exists Y E T(x) such that for each e> 0, T-1(Se[Y]) contains a neighborhood of x. (b) Show that the continuity condition of (a), introduced by Christensen [Chr2], implies the continuity condition ofDeutsch and Kenderov, and that the two notions coincide for compact-valued multifunctions. (c) Construct a compact convex-valued multifunction on [0,1] with values in R2 that satisfies this continuity condition but which has no continuous selection. Let be a Banach space and let Y be a closed subspace. Suppose T: X ::::t Y has values in ceY), and is additive modulo Y: \:;f x E X \:;f Y E Y T(x + y) = T(x) + y. Let H(T) =T-1( {On. (a) Show that H(T):t= 0. (b) Suppose H(T) is closed. Prove the following result of Deutsch, Indumathi, and Schnatz [DIS]: T is l.s.c. if and only if T has a continuous selection f with fix) = 0 for each XE H(T) (Hint : Use Exercise 6.2.1(d)). (c) Use (b) to obtain Krüger's Theorem [Kr]: the metric projection ,y) onto a proximinal subspace Y is l.s.c. if and only if P admits a continuous selection f such that fix) =0 whenever d(x,y) =IIxll. Let C be a convex subset of a Banach space X. Suppose T: C::::t C is l.s.c., has values in ceX), and cl T(C) is a compact subset of C. Use Michael's Selection Theorem and Theorem 6.4.7 to prove that T has a fixed point. It is a result of A. H. Stone (see, e.g., [Dug, p. 168]) that a Hausdorff space X is paracompact if and only if each open cover '\i of X has an open refinement 'Uj

pe·

6. 7.

such that for each x E X, U{W E 'Uj : x E W} is contained in some element of v. Such a refinement 'Uj is called a barycentric refinement of '\i.

234

CHAPTER6

(a) Prove the following theorem of Cellina [Cell,Rei,DuG] : Let be a metric space, and let be a normed linear space. Suppose T: X 4 .y is upper semicontinuous with nonempty c10sed convex values. Then there exists I E C(X,y) such that Gr I c: Se[Gr 11 and fiX) c: co T(X). (Hint for each x E X, choose Vx E rL(x) such that Vx c: Se[x] and T(Vx) c:

8.

9.

10.

11.

12.

13.

14.

Ss[T(x)]. Take an open barcycentric refinement of {Vx : x E X} and then make use of a partition of unity subordinated to this refinement). (b) Use (a) and the Schauder Theorem to give an alternative proof of the Kakutani Fixed-Point Theorem for convex-valued usco maps from a compact convex set into itself. Cellina [CeI2] has shown that a convex-valued usco map can be graphically approximated in Hausdorff distance by continuous functions. Each metric space can be isometrically embedded in the Banach space of bounded continuous functions on Y equipped with the usual uniform norm [Dug, p. 286]. Using this fact, prove that Theorems 6.6.7 and 6.6.8 remain valid for functions with nonempty complete values in a separable metric target space. Let a, then (x,ß)

E

epi f.

Hyperspace topologies defined on C(X x R) naturally induce

topologies on nX). In this chapter, we look in some depth at an important topology on nX) that arises in this way: the Attouch-Wets topology. For one thing, the convergence of a sequence of c10sed convex sets corresponds to the convergence of the associated sequences of indicator functions, support functions, and distance functions. Furthermore, convergence of a sequence of proper lower semicontinuous convex functions in the Attouch-Wets topology implies and is implied by the Attouch-Wets convergence of functions dual to the originals in J'*(X*). Continuity of polarity for convex sets easily falls out of this result for convex functions. Thus, the Attouch-Wets topology is stable with respect to duality. Moreover, the strength of Attouch-Wets topology, as weIl as its overall tractability in terms of estimation, make it a highly potent convergence concept applicable to convex optimization and approximation problems.

§7.1. Attouch-Wets Convergence of Epigraphs Let be a metric space. Recall that the Attouch-Wets topology on CL(X) is the topology that CL(X) inherits from C(X,R) equipped with the metrizable topology of uniform convergence on bounded subsets of X, under the identification A ~ d(· ,A). Since sequences determine this topology, when we consider Attouch-Wets convergence of sets, we confine our attention to sequences rather than considering general nets of sets. In §5.3, we studied Kuratowski-Painleve convergence of lower semicontinuous functions, as associated with their epigraphs, under the name epi-convergence. By the Attouch-Wets convergence of a sequence in LSC(X) to I E LSC(X), we mean the convergence of to epil with respect to Attouch-Wets topology as determined by the box metric p on X x R, defined by

We will also write 1= 'Z"AW p -limln when what we really mean is Attouch-Wets convergence of epigraphs. Obviously, Attouch-Wets convergence of functions in LSC(X) implies their epi-convergence, and the two notions coincide when has compact c10sed balls (see Exercise 5.1.10 and Theorem 5.2.10). A primary focus of the first papers written on Attouch-Wets convergence of sets and functions was estimation [AW3-4,APl]. Here, Theorem 3.1.7 provides the key too1. For functions, fixing xo in X, the Attouch-Wets convergence of a sequence in LSC(X) to I E LSC(X) means that for each c10sed ball B in X x R with center (xo,O), there exists NE Z+ such that for each n > N, we have both 235

CHAPTER 7

236

and

ep(epifn n B, epij) < €.

e,

When X is a nonned linear space, it is convenient to take xo = and Attouch-Wets convergence of to f may be expressed simply by the following condition: for each /1 > 0, we have limn~oo hausJl (epiJ, epifn) = 0, where, again, hausJl (epiJ, epifn) = max {ep(epifn (/1UX [-/1,/1]), epifn), ep(epifn n (/1UX [-/1,/1]), epij)}. We illustrate hausJl (epiJ, epi g) with respect to the box metric in Figure 7.1.1 below .

........ = hausdepiJ, epi g)

/1U x [-/1,/1] I ... _________ I

FIGURE 7.1.1 A thorough treatment of the Attouch-Wets convergence of arbitrary proper lower semicontinuous functions is outside the scope of this book. However, many of the results we wish to give for lower semicontinuous convex functions hold without convexity. We now present a few general results that we will apply to lower semicontinuous convex functions in the sequel.

7.1.1 Proposition. Let be a metric space. Then the map A -+ J(- ,A), assigning to each c10sed set its indicator fuction, is an embedding of '

Proof. Fix xo E X. If B is a c10sed ball in X x R with center (xo,O) and with radius /1, then by the definition of the product metric, we have B = {x: d(x,xo) ~ /1} x [-/1,/1]. If Al and A2 are nonempty c10sed subsets of X, then

The result now follows from Theorem 3.1.7.

•

7.1.2 Lemma. Let be a metric space. Suppose g, gl, g2, g3, ... is a sequence in LSceX) such that g is real-valued. If converges uniformly to g on bounded subsets of X, then g = 'rA Wp-lim gn .

THE ATIOUCH-WETS TOPOLOGY FüR CONVEX FUNCTIONS

237

e> 0, j.l > 0, and set B = {x: d(x,xo) ~ j.l} x [-j.l,j.l]. For all n sufficiently large, we have sup {Ig(x) - gn(x)l: d(x,xo) ~ j.l} < e, from which it follows that max {ep(epi g n B, epi gn), ep(epi gn n B, epi g)} < e. This gives Attouch-Wets • convergence of to g. Proof. Fix xo

E X,

7.1.3 Proposition. Let be a metric space and let fo,JI,h, ... be a sequence of lower semicontinuous real-valued functions on X such that fo is finitevalued and Lipschitz continuous on bounded subsets of X and is eventually equi1'AWp-limfn. Then Lipschitzian on bounded subsets of X. Suppose that fo converges to fo uniformly on bounded subsets of X.

=

Proof. Fix xo E X, j.l> 1 and e E (0,1). We produce a positive integer N such that whenever n ~ N, we have sup {!fo(x) - fn(x) I : XE S.u[xo]} ~ e. There exists NI E Z+ and Ä, > 1 such that Ä, is a uniform Lipschitz constant for the farnily lfn IS2.u[XO] : n = or n ~ Nd. By the epi-convergence of to fo, implied by Attouch-Wets convergence, there exists a sequence convergent to xo such that N', we have Ig(w) - gn(w)1 < El3. Thus, (w, ß+ g(x) + 2E13) E epi lfn + gn), and p[(x,a), (w,

ß+ g(x) + 2E13)) =max {d(x,w), la - g(x) - ß-

2E131}

::; max {El3, la - g(x) - ßI + 2E13} < 8 + 2E13 < e.

This establishes the claim. Similarly, one can show that for this choice of N' and for each n > N', we have ep(epi lfn + gn) n B, epi lf+ g)) ::; e. • 7.1.6 Example. Theorem 7.1.5 fails if is only Attouch-Wets convergent to g, rather than uniformly convergent on bounded subsets, even if all functions are continuous and uniformly bounded and X is compact. Let X =[0,1) as a subspace of the line. For each n, let gn be the piecewise linear spike function whose graph connects the following points in succession :

1 1 3 2 2n-I (0,0), (2n,I), (n'O) , (2n,I), (n,o), ... , (211,1), (1,0). For each n, let In = 1 - gn' Then both and are Attouch-Wets convergent to the zero function. On the other hand, for each n, In + gn == 1, and so 'l'AWp-limln +

'l'AWp-lim gn

::f=

'l'AWp-lim lfn + gn).

Again, our particular interest here is in Attouch-Wets convergence in I1x) where X is a normed linear space, or in T*(X*), the weak* lower semicontinuous proper convex functions on X*. For notational simplicity, in the rest of this chapter, we will write just 'l'A W far the Attouch-Wets topology, whether C(X), C*(X*), T(X) or T*(X*) is the hyperspace under consideration. We close this section by showing that Attouch-Wets convergence of convex functions implies convergence of sublevel sets at fixed heights above the infima! value of the limit. 7.1.7 Proposition. Let be a normed linear space. Suppose IE J'(X) and is a sequence in I1x) with 1= 'l'A w-limln. Then for each a> m/(X), we have slv lf; a) ='l'A W -lim slv lfn; a) .

Prool. Choose ß strictly between a and mJ lai such that for some xo Eint J.LoU we have f(xo) < ß. By the epi-convergence of to J, there exists Nt E Z+ such that n > Nt implies epi/n n [int J.LoU X (-oo,ß)) ::f= 0. Fix J.L > J.Lo; we produce N E Z+ such that for each n > N, both of the following conditions hold: (1) slv lf; a)

n J.LU c::

(2) slv lfn; a)

slv lfn; a) + EU;

n JIU c::

slv lf; a) + EU.

CHAPI'ER 7

240

Choose Ö> 0 such that

0+

2oJ.L N2, we have hausJl (epif, epifn) < Ö. We claim that the choice N = NI + N2 works. We verify that condition (1) holds; verification of (2) is exactly the same and is left to the reader. Fix n > N and x E slv (f, a) n J.LU. Since J.L > laI, we obtain (x,a) E J.LU x

[-J.L,J.L]. Since n > N2, there exists (wn,an) E epifn with IIwn - xII< Ö and lan - aI < Ö. Since n > NI and J.L > J.LQ, epifn hits int J.LU x (-oo,ß); so, there exists zn E J.LU with fn(Zn) < ß. Set A = (a - ß)/( a - ß + 8); we will show (i) AWn + (1 - A)Zn (ii) IIAwn

E

slv (fn;a);

+ (1 - A)Zn - xII ~ E.

Condition (i) follows easily from fn(wn) ~ an< a + Ö and fn(Zn) < ß:

=

(a - ß)(a + Ö) + Öß = a. a-ß+ö

Condition (ü) follows from the choice of 0 and the fact that both Zn and x lie in J.LU: IIAwn + (1 - A)Zn - xII ~ ;Wwn - xII + (1 - A)lIzn - xII

~

IIwn - xII + (

0 ) Ilzn - xII a-ß+ö

2ÖJ.L < Ö+ ------''-- and E> 0; we show for all sufficiently large n that haus.u (H C , H Cn) < E. Fix Ö E (0,1) such that

n

2öJi

Ö

lfll* + - - < y

1- Ö

E

n

and

N implies hausl(C, Cn ) < öIJillyll*, whence haus.u(C, Cn) < &lIyll*. Now fix n ~N; we show that

There exists NE Z+ such that n

C nH

n JiU c.

Fix w E C n H

n JiU;

(Cn

~

n H) + EU .

since C n JiU c. (Cn n JiU) + IIY~*U' there exists

XE

JiU

n

Ö

Cn with IIx - wll::;; lIyll*. Note that Ilxll ::;; IIx - wll + IIwll < 2Ji. Since = 1, we have 1 - Ö< < 1 + ö. Write a Cn, and we compute

n

IIx - er 1xII

=~ Ilxll a

=;

then a-1x E H

< _Ö_ 11x1l < 2ÖJi 1-ö

1-ö

Finally, the triangle inequality and the choice of Ö yield IIw - a-1xll::;; IIw - xII + IIx - a-1xll
-00, then f(jl,') is finite-valued and Lipschitz continuous with constant jl. Moreover, f(jl,') is the largest Lipschitz continuous function with constant jl that f majorizes. Suppose that f(jl,xI) =-00 for some Xl each nE Z+, and since eil is a cone, we have

Proof.

Xx R

=U;1 [(xI,-n) + eil]

E

X. Then (xr,-n)

E

epif + eil for

c:: (epif + eil) + eil c:: epif(jl, .),

contradicting (xo.f(jl,xO) - 1) e epif(jl, .). Thus f(jl,') is finite-valued. Now to say that a real-valued function g on X is Lipschitz continuous with constant jl is to say

CHAPTER 7

252

simply that epi g + CJl c: epi g. Thus, if 1 majorizes such a Lipschitz function g, we have epi g::::> epi g +CJl::::> epi/+ CJl, and so epi g ::::> epiftJ.L, .). It remains to show that f{J.L,.) is itselfLipschitz with constant J.L. To tbis end, fix x and w in X; we show ftJ.L,x) 5,/(J.L,w) + J.Lllx - wll. Let a> .!tJ.L,w) be arbitrary and choose z E X with ftz) + J.Lllw - zll

< a. We compute

ftJ.L,x) 5,ftz) + J.Lllx - zll 5,ftz) + J.L(l1x - wll + IIw - zll) < a + J.L11x - wll.

Since a >.!tJ.L,w) was arbitrary, .!tJ.L,x) 5,.!tJ.L,w) + J.Lllw - xII immediately follows.

•

Returning to our general discussion, there are at least four properties shared in common by the three families of smoothing kemeIs that we presented, wbich we signal out in the following definition.

7.3.5 Definition. Let X be a normed linear space, and let Q = {gA. : A > O} be a family of nonnegative lower semicontinuous convex functions on X. We call Q a regularizing lamily 01 smoothing kemels if (1) for each A., gA. is nonnegative;

(2) for each A., g.t is continuous at the origin ;

=

(3) for each A, gA.( 6) 0; (4) O} be a is uniformly convergent on regularizing family of smoothing kerneis. Then

bounded subsets of X* to the zero function as A, ----7 O.

Proof Let g be the indicator function of the origin. Fix k E Z+. Choose e > 0 such that for all A, < e, we have haus2(epi g, epi g).) < 1/k2, which implies that inf {g).(x) : Ilxll = 1/k2 } ~ 1. Since g).((}) = 0 for each parameter A" by convexity, we conclude that for each J1 ~ 1/k2 , we have

We claim that whenever lIylI*:5 k and A, E (O,e), we have 0 ~ g l(y) ~ 1/k. kU*.

First, g l(y) ~ 0

because g).(9)

=O.

Fix Y E

We now show that g). majorizes the

affine functional x ----7 - l/k. First, if Ilxll ~ 1/k2 , we have 1

*

1

- k :5l1yll 11xl1- k

1

~ k· k2 -

k1

=0 ~

g).(x).

On the other hand, if IIxll ~ 1/k2 , then 1 g).(x) ~ k211xll ~ kllxll ~ lIyll*lIxll > - k We have shown that for all x E X, 1/k ~ - g).(x). By the definition of the Fenchel conjugate, it follows that g 1(y) ~ 1/k, and the proof is complete. • We finally come to our promised result regarding regularizations. 7.3.8 Theorem. Let be a normed linear space, and let {g).: A, > O} be a regularizing farnily of smoothing kerneis. Let JE I'(X) be fixed, and for each A" let

254

CHAPTER 7

h)., be the lower envelope of J ~ g)." i.e., epi h)., =cl epi lf ~ gA,). Then h)., is majorized by J, and for all A. sufficiently small, h)., belongs to I'(X) and is continuous at each point of dom! Moreover, converges pointwise and in the Attouch-Wets topology to! ProoJ. Fix YO E dom.f". By Lemma 7.3.7, there exists e > 0 such that for all

we have YO E dom g

1.

As a result, both

J and

A. < e,

g)., have a common continuous affine

minorant, and it follows that J ~ g)., has a continuous affine minorant (see Exercise 7.3.6). But J ~ g)., and h)., have the same continuous affine minorants, and so h)., E I'(X). The condition g;.( 0) =0 yields J ~ g Jl 5. J, and since h Jl 5. J ~ g)." we conclude that J majorizes each hJl. Fix A. < e; by continuity of gJl at 0, there exists (3 > 0 such that g)., is bounded above by 1 on (3U. Then for each XE domf, sup {h).,(w) : w E x + (3U} 5. sup {(f

~ g).,)(w) :

w E x + (3U} 5.f(x) + 1.

But a proper convex function bounded above in a neighborhood of a point x in its effective domain must be continuous at such a point (see Exercise 7.3.3), and we have continuity of hJl at each point of dom! To show Attouch-Wets convergence of to f, we use the continuity of the inverse of the Fenchel transform. By Lemma 7.3.6, we have for each A. > 0,

1

Since < g > is uniformly convergent to the zero function on bounded sets, we obtain

1>

1

from Theorem 7.1.5 the Attouch-Wets convergence of x . Viewing the set A as a constraint set to which the function I is subject, it is easy to see that some form of "constraint qualification" is required to obtain positive results, even in finite dimensions. On the line, let g be the indicator function of the ray [0, +00) and let C == (-00, 0]. Then g is the 'l"A w-lirnit of the sequence of continuous convex functions defined by

gn(x) ==

{

I - nx

0

if x ::;; l/n if x > l/n

But g + [(. ,C) == [(. ,( O}), whereas 'l"A w-lim gn + [(. ,C) == [L{ O}) + 1, as illustrated in Figure 7.4.1. At the heart of our theorem, as established in [BeL2], is the stability of the interior of a convex a set with respect to the Attouch-Wets topology, which fails for the weaker slice topology (see Exercise 7.4.2). This in turn follows immediately from a celebrated cancellation principle for convex sets [Ra].

THE ATTOUCH-WETS TOPOLOGY FOR CONVEX FUNCTIONS

257

epi gn + 1(- ,(-00,0]) 'l"AW

(1,0)

( 1,0)

FIGURE

7.4.1

7.4.1 Rädström Cancellation Principle. Let A, Band C be closed convex subsets of a normed linear space X with B bounded. Suppose A + B c:: C + B. Then A c:: C.

Proof Suppose A is not a subset of C. Then for some a E A and y E X*, we have > s(y,C). Since B is bounded, s(y,B) is finite, and there exists bEB with > s(y,B) - «y,a> - s(y,C». It follows that

= + >

which contradicts a + b E C + B.

s(y,C) + s(y,B)

=s(y, C + B),

We conclude that Ac:: C.

•

7.4.2 Stability of Interior Theorem. Let A be a nonempty closed convex subset of a normed linear space X, and suppose a + 8U c:: /lU n A. Then for each 8 E (0,8) and each CE C(X) with hausjl (A,C) < 8 - 8' , we have a + 8V c:: C. Proof.

We have the following inclusions : (a

+ o'U) + (0 - o')U = a + oU C /lU nA

C

C + (8 - 8')U .

By the Radström cancellation principle with B =(8- 8' )U, we get a + 8'u c:: C.

•

Theorem 7.4.2 is often used in the following form : Let A, A), A2, . .. be nonempty closed convex subsets of a normed linear space X, with A = 'l"A w-lim An. Suppose a + 8U cA. Then for each 8'E (0,8) there exists NE Z+ such that for each n > N, we have a + 8'U c:: An. We will need the fact that an Attouch-Wets convergent sequence of convex functions is uniforrnly bounded below on bounded sets. This is true much more generally.

7.4.3 Lemma. Let X be a normed linear space and let fo,f),h, ... be a sequence in JtX) such that is epi-convergent to fo . Then for each /l > 0, the family {JO, ft, f2, . .. } is uniforrnly lower bounded on /lU. Proof Fix xo E domfo, and choose strongly convergent to xo with 0 and NE Z+ such that whenever n

> N and IIxll:::; 0, we have gn(x) > -1.

Since (0,0) E epi gn for each n > N, it now follows that whenever IIxll > 0, we have gn(x) > -8- 1Ilxll, else epi gn would intersect oU

{-I} (see Figure 7.4.2 above). It now follows that {gn: n > N} is uniformly bounded below on bounded sets. But the finite family {gn: n:::; N} also has this property, because each such function majorizes a continuous affine function. The result now follows. • X

For our continuity result, we also will use the next technical statement.

7.4.4 Lemma. Let X be a normed linear space, and let CE ceX). Suppose a E C n pD and b + oU c C n pD. Suppose that e> 0 is chosen so that A == c/8/1 is less than 1. Then the set C contains the ball B with center (l - A)a + Ab with radius AO, and for each wEB, we have lIa - wil :::; c/4.

Prao!. We have the following inclusion string : B = [(1 - A)a + Ab] + Mu = (l - A)a + ACb + oU)

c co ({a} U (b If wEB, then for some Z E b + oU, we have

W

+ oU)) ce.

= (l - A)a + Al. But then

THE ATTOUCH-WETS TOPOLOGY FOR CONVEX FUNCTIONS

lIa - wll = 11(1 - A)a + k

as required.

259

- all ::; Allall + Alizil ::; 2AJ..l = e/4,

•

The main result of this section was initially established in finite dimensions by McLinden and Bergstrom [MB]. The argument that we give comes from [BeL2]. For more precise quantitative estimates, the reader may consult [AP1].

7.4.5 Theorem. Let X be a normed linear space. Suppose JE reX), gE r(X), and g is continuous and real-valued at some point Zo E dom! Suppose that

where IfI, gI,f2, g2, ... } c: r(X). Then

ProoJ. Take (zo,ßo) E (int epi g) n epi! Choose by Theorem 7.4.2 and Theorem 5.3.5 a scalar 8 E (0, 1) and NI E Z+ such that for each n > NI (zo,ßo) + (28U x [-28,28]) c: epi gn and

epiJn n [(zo,ßo) + (8U x [-8,8])]

'* 0.

Choosing for each n > NI a point (Zn,ßn) in epiJn n [(zo,ßo) + (8U x [-8,8])], we have

(Zn,ßn) + (8U x [-8,8]) c: epi gn, and

lI(zn,ßn) - (zo,ßo)lI::; 8. Pick 1]0> 1 so large that 1]oU x [-1]0,1]0] contains all balls with radius 8 whose centers come from {(Zn,ßn): n > NIl. To verify Attouch-Wets convergence of and e> both (i) ep(epi (f + g)

°

be fixed. We must show that for all n sufficiently large,

n (1]U x [-1],1]]), epi (fn + gn))::; e,

(ii) ep(epi (fn + gn)

n (1]U x [-1],1]]), epi (f + g)) ::; e,

260

CHAPTER7

where, as usual, p is the box metric on the product. Without loss of generality, we may assume that 11 > 110. By Lemma 7.4.3 there exists (1 > 0 such that -(1 is a common lower bound for the restriction to 11U of each function in {f, g,J}, gl,J2, g2, . .. }. Let J.L = (1+ 11, and pick N2 E Z+ so large that for each n > N2, we have both

eo , hausJl (epl. gn, epl.g) < -eo . hausJl (epifn, epij)
N. We may write a = g(x) + a' where a' ~f(x). Since g(x) ~ -(1, a' ~ -(1, and a ~ 11, we have both -J.L ~ g(x) ~ J.L

and

-J.L ~ a' ~ J.L .

We now apply our technical Lemma 7.4.4 with the following values for the variables C, a, and b that appear in its statement: C=epig,

a

=(x,g(x»,

b =(Zo,ßo).

The ball B with center PO == (1 - Ä)(x,g(x» + Ä(zo,ßo) and radius Äo = e0/8J.L is contained in epi g, and each point in the ball has distance at most El4 from (x,g(x». We now invoke our stability of interior theorem: for n > N> N2, the ball with the center PO and with radius eOl16J.L is contained in epi gn because hausJl (epi gn, epi g) < eOlI6J.L. Since both (x,a') and (zo,ßo) are in epif n (J.LU x [-J.L,J.L]) , a convex set, so is PI == (1 - Ä)(x,a') + Ä(zo,ßo). As hausJl (epifn, epij) < e0I16J.L, for each n > N, there is a point (xn,a~) E epifn whose distance from PI is at most e0I16J.L< e/16. Since lI(x,a') - PIII ~ Ä· max {11xI1 + IIzoll, la'l + IßoI}, the triangle inequality yields

" ,e e lI(x, a') - (xn,a n )1I ~ H(x,a ) - PIII + IIpl - (xn,an)1I < 4" + 16

= Se 16

.

Since the points PO and (xn, Äßo + (1 - Ä)g(x» have the same second coordinates, the distance between them is

eo

IIxn - (1- Ä)x- koll < - . 16J.L

261

THE ATTOUCH-WETS TOPOLOGY FOR CONVEX FUNCTIONS

We see that (xn, A,ßo + (1 - A,)g(x)) lies in epi gn' Since a ~ ?In(xn), we also may say that (xn, A,ßo + (1 - A,)g(x) + a ~ )

E

epi lfn + gn). This point is e-close to our initial

point (x,a) of epilf+g)n (1]Ux[-1],1]]), aswenowverify: lI(xn, A,ßo + (1 - A,)g(x) + a~) - (x,a)1I = max {ILxn - xII, IA,ßo + (1 - A,)g(x) + a ~ - al} ~ max {lIxn - xII, IA,ßo + (1 - A,)g(x) - g(x)1 + la ~ -

al}

~ max {lIxn - xII, IA,ßo + (1 - A,)g(x) - g(x)l} + max {lIxn - xII, la~ - a'l}

= lI(xn, A,ßO + (1 - A,)g(x)) - (x,g(x))11 + lI(xn, a ~ ) - (x,a')11 ~

e Se 4" + 16 < e.

This proves that ep(epi lf + g) n (1]U x [-1],1]]), epi lfn + gn)] ~ e, for each n > N.

•

As we observed at the beginning of this section and in § 1.3, the restriction of a function I to a set C is achieved by adding the indicator function of C to f Thus, Theorem 7.4.5 immediately yields a result regarding continuity of restrictions. In the special case that the functions themselves are indicators, we get a continuity result for set intersection (see Exercise 7.4.1).

7.4.6 Theorem. Let {Cl, C2, C3, ... ,Ck} be nonempty closed convex subsets of a normed linear space X, and let g E nX). Suppose g = 'l'A w-lim gn and for each i ~ k, wehave Ci='l'AW-limCin. Thenifeither domgn int(nf=l Ci);t0 or g isreal-

valued and continuous at some point of nf=1 Ci, we have

Prool. By Proposition 7.1.1, A

~ I(·,A) is 'l'Aw-continuous. The result now follows from the previous theorem using mathematical induction, noting that (i) points of continuity of an indicator function for a set A are precisely the interior points of A, and (ii) the indicator function of an intersection of sets is the sum of their indicators. •

We apply our sum theorem to link Attouch-Wets convergence of a sequence in nX) to I E nX) to the uniform convergence on bounded sub sets of the Lipschitz regularizations of the functions. We find it convenient to work with the reciprocal of the parameter A, used in §7.3 to index such regularizations. As in Proposition 7.3.4, we shall write f(/l,') for

I

~

/lll-li in the sequel.

7.4.7 Theorem. Let X be a normed linear space and let sequence in

nX). The following are equivalent :

IO.!I, 12, ...

be a

262

CHAPfER 7

(2) V P > d( 8*, domf~),

0, because hausJl is monotone in the parameter P for fixed set arguments. 'l"Aw-limf~ , and so fo = 'l"Aw-limfn'

Thus,

fo =

•

When the functions in the hypotheses of Theorem 7.4.7 are indicator functions of closed convex sets, we recover the definition of Attouch-Wets convergence for sequences of convex sets: uniform convergence of the associated sequence of distance functionals on bounded subsets of the underlying space. Finally, we note in closing that Attouch-Wets convergence can also be explained in terms of the uniform convergence on bounded sub sets of Moreau-Yosida regularizations with respect to any fixed positive parameter (see Exercise 7.4.6). Exercise Set 7.4. 1.

Let X be a normed linear space. Let and be two sequences in C(X) with A = 'l"A w-lim An and B = 'l"Aw-1im Bn . Suppose further that A int B *" 0. Prove that An B = 'l"Aw-lim An n B n .

n

THE ATTOUCH-WETS TOPOLOGY FOR CONVEX FUNCTIONS

2.

263

(a) Show that stability ofthe interior fails for the slice (= Mosco) topology on the closed convex subsets of 12. (Hint : for nE Z+, define An E c(12) by

An = {x: I : 1 2 ~ 1 and =0 for i > n}). (b) Show that the conclusion of Exercise 7.4.1 is no longer valid if, in the hypotheses, "A = 'l'A W-lim An" is replaced by "A = 'l's-lim An"

3.

4.

Let X be a normed linear space. Suppose 10./1,/2, ... is a sequence in reX) such that 10 ='l'Aw-limln. Suppose for each ).L > 0, there exists N J.1 E Z+ such that the family lfn: n -00, A = Argminj, and An = slv (j; a + lIn) for n = 1,2,3, . .. . If an E An for eaeh n, then is a minimizing sequenee that by (1) has a cluster point. Condition (2) now follows from the equivalenee of eonditions (a) and (b) above. (2) ~ (3). This is obvious from the nested nature of sublevel sets. (3) ~ (1). Suppose is a minimizing sequenee for j, and write an =flan). By passing to a subsequenee, we may assurne that is a noninereasing sequenee. With A = Argmini, and An = slv (j; an), the equivalenee of eonditions (a) and (e) above yields a cluster point for , and I is therefore g.w.p. • Retuming to oUf initial question, we intend to show here that if we approximate OUf initial eonvex objeetive funetion in the Attoueh-Wets sense, then well-posedness of the objeetive funetion in the generalized sense is not only neeessary but also sufficient to ensure that sequenees of approximate minimizers of approximating funetions have eonvergent subsequenees. Again, it is then automatie by Proposition 1.3.6 that the limit of this subsequenee is a minimizer of the initial objeetive funetion, and that infimal values eonverge. As a first step, we show that if I E T(X) has a bounded sublevel set at some height greater than mfiX), then the Attoueh-Wets eonvergenee of in T(X) to I implies convergence of infimal values. To this end, we use a technical fact about AttouchWets eonvergenee, established in §3.2 : Attoueh-Wets eonvergenee of a sequenee of closed eonneeted sets to a closed and bounded set ensures Hausdorff metrie convergence. 7.5.3 Lemma. Let e gives a point (w)",ß) E epij)., lying above H with w}., E xo + EU. By our choice of j1, the line segment joining these points, which lies entirely in epi JA., hits H at a point within the cylinder j1U x R. This contradicts j;., E (s(j1,Y,T/)C)++. Thus, for all A ~ AQ, we • have slv (j).,; a) E (BC)++ . limn~oo d(wn,B) = O.

8.1.5 Lemma. Let X be a normed linear space. Suppose is a net of positively homogeneous functions in T(X), and jE T(X) is positively homogeneous. Then j = 'l's-lim j;., if and only if flU = 'l's-lim j;.,IU. Similarly, if is a net of positively homogeneous functions in T*(X*), and h E T*(X*) is positively homogeneous, then h = rs-lim h)., if and only if hlU* = 'l's-lim h).,IU*. Proof We againjust prove the first statement. Suppose that j= 'l's-limj;.,. If V is open

in X x R, and flU E V-, i.e., (epiflU)

n V * 0,

then by positive homogeneity,

CHAPTER8

274

epijn (V n (int U X R)::t 0. Since j= 'Z"s-limJA., epiJA. meets V eventually.

vn

(int Ux R) eventually, whence epifAJU meets

epi jlU E (BC)++ where B is a closed, bounded and

Now suppose

convex subset of X X R. Let us write D(B, epijlU) = 0> o. If for some p > 1, we have B c:: (pU X R)C, then for all A. in the underlying directed set, we have f;.,IU E (BC)++. Otherwise, choose ß> 1 such that for each (x,a) E B, we have Ia! < ß, and pick J.l > 0 satisfying both J.l < min {1/2,0I2} and 2J.l 0 ----''-- (ß + 1) < 2" . 1 + 2J.l We claim that with BI = B n «1 + J.l)U X R)::t 0, we have D(Bl, epij) ~ J.l. Suppose not. Then we can choose (x,a) E epij with d«x,a), BI) < J.l. Clearly, IIxll < (1 + J.l) +

J.l = 1 + 2J.l < ß + 1, and lai< ß + J.l < ß + 1. Since epij is a cone, the point (xQ,ao) given by (xQ,ao)

=

1 1 + 2J.l

(x,a)

lies in epi J, and thus in epijlU because IIxll < 1 + 2J.l. We compute d«xQ,ao),B) S; d«xQ,ao),BI) S; lI(xQ,ao) - (x,a)1I + d«x,a), BI) =


O.

But this means that glU E (BC)++

eventually, we have f;.,IU E (BC)++

for g E Cl.

Since JA. must be in Cl

eventually.

The other direction is much easier. Assume jlU = 'Z"s-limfAJU.

First suppose

jE V- where V is norm open. Pick p> 0 such that epijn V n «int pU) X R) ::t

0. Set W =p-1.(V n «int pU) x R); then W- is a 'Z"s-neighborhood of jlU, so

that f;JU is in W- eventually. For all such Ä, we have by positive homogeneity epiJA. n V ::t 0, Le., JA. E V . On the other hand, if fE (BC)++ where B is closed bounded and convex, choosing p> 0 with and for all A. sufficiently large,

hJU E

~PU x R;:)

B, we have jlU E «p-lB)C)++,

«fTIB)C)++ which yields JA. E (BC)++ .

•

THE SLICE TOPOLOGY FOR CONVEX FUNCTIONS

275

Exercise Set 8.1. 1. 2.

3.

4.

5.

6.

Give an example of a sequence of proper lower semicontinuous convex functions defined on .0.2 that is slice convergent but is not Attouch-Wets convergent. Let be a normed linear space and let f,!1,f2, ... be in I'(X). Prove that is slice convergent to f if and only if both conditions below are satisfied: (i) for each x E X there exists convergent strongly to x for which f(x) =limn~oofn(xn); (ii) for each (y,1'/) E epif' with 1'/ > f'(y) and each bounded sequence , there exists NE Z+ such that for each n > N, we have fn(xn) > - 1'/. Using Exercise 8.1.2 and Theorem 5.4.18, show directly that slice convergence for a sequence in I'(X) implies Mosco convergence, and that the converse holds when the space is reflexive. Let be a normed linear space. Show that finite suprema of continuous affine functions are dense in . Show that this fails in an infinite dimensional space if the slice topology is replaced by the stronger Attouch-Wets topology (Hint : consider f(x) =IIxll). Let be a separable normed linear space. Let {xn : n E Z+} be a countable dense subset of X, and for each nE Z+, let An be the subspace spanned by {Xk : k ~ n }. Suppose fE T(X). Show that f is the slice limit of its finite dimensional approximations , where for each n, fn = f1An =f + I(· .An). Is the same true for the Attouch-Wets topology? In .0.2, let An bethesubspacespannedby {ek:k~n}, andlet BnE C(X) be defined by Bn

7.

8. 9.

10.

= {x: =~

and =0 for all i > n+I}.

(a) Prove that An and Bn are c10sed subspaces both convergent to .0.2 in the slice topology. What is the limit in the slice topology of ? (b) Show that Theorem 7.4.5 and Theorem 7.4.6 fail in .0.2 if Attouch-Wets convergence is replaced by slice convergence. Obtain the following result of Lahrache [LahI]: Let be a normed linear space and let f,fI,f2,f3, . .. and g, gl, g2, ... be sequences in I'(X) such that f= 7:s-limfn and g = 7:s-lim gn. Suppose is uniformly bounded above in some neighborhood of some point zo of dom! Then f + g = 7:s-limfn + gn. Using the previous exercise, state and prove a result regarding continuity of intersection for the slice topology (see Exercise 7.4.1) Let Y E X* be a norm one element of a nonreflexive Banach space which is not norm achieving on U. Pick xo E X with =2. Let C be the smallest cone containing the epigraph of g == d(· ,y-I(I)) + Ie ,xO+ U), i.e., C = U {A,·epi g : A,;:: O}.

(a) Verifythat cU {(O,a):a;::O} istheepigraphofsomefEI'(X). (b) Showthat Argminf={O}, and Argminf* 7:w-lime~o+slv(f;inff+e). Let be a Banach space. Using Exercise 8.1.9, prove that the following are equivalent : (1) X is reflexive;

CHAPTER8

276

(2) for each JE T(X) and each a for which slv (/; a)"# 0, we have slv (/;a)

= 'l"s-lime~o+ slv (/; a + e);

(3) for each JE T(X) and each a for which slv (/; a)"# 0, we have slv (/;a)

= 'l"w-lime~o+ slv (/; a + e).

§8.2. Convex Duality and the Slice Topology Let be a general normed linear space. In §7.2 we showed that the Attouch-Wets topology for convex functions is stable with respect to duality, in that the Fenchel trans form from to X x R* is lower semicontinuous. Evidently, J =4 epiJ is also a lower semicontinuous multifunction on and into o. Proof Suppose that the infimum is a positive number 8. Let us set g == 1- y + 1]. The

nonemptiness of domln int.uU allows us to find a point xo Eint.uU such that g(XO) < 28. If Ir/:. (s(.u,y,1])C)++, then since epigraphs recede in the vertical direction, we can

find a sequence in X such that IIx nll -7.u and lim sUPn~oo g(xn) all n sufficiently large, we have

2

1

3' Xn + 3' xo E .uU

8 and gCxn) < 2"

~

O. Now for

By convexity of

g, we get

which contradicts the definition of 8. We have shown that the gap between s(.u,Y,1]) and epil is indeed positive. The converse is obvious. •

8.2.11 Theorem. Let be a normed linear space, let be a net in I E T(X). The following are equivalent :

T(X), and let (1)

1= Ts-limfll;

(2)

V.u > max (d(O*, domf*), deO, dom.!)), V x E X, V Y E X*, we have

I(.u,x) = lim)"fIlC.u,x) and f*(.u,y) = lim)., 11c.u,y); (3)

there exists sequences of positive scalars -7 += and -7 += such that V x E X, V Y E X*, V n E Z+, we have IC.un,x)

= lim)"I)"C.un,x)

and f*(.u~,y) = lim).,11C.u~,y). Proof (1) => (2). Fix.u > max (d(O*, domf*), dCO, dom.!)}. Then

epiln int (.uU x R)

:;t:

0

and

epif* n int (.uU* x R)

:;t:

0.

284

CHAPTER8

By Theorem 8.2.2, there exists an index AQ such that for each 1.;::: AQ, both domfA, n int IlU =f. 0 and domfl n int IlU* =f. 0. By Lemma 8.2.8, all functions of the form

f(ll, .), /*(11, .), hJIl, .), and fl(Il,') are continuous convex functions when 1.;::: AQ. Now fix y E X* and c> O. We produce an index 1.2;::: AQ such that for all 1.;::: 1.2, we have

This would establish pointwise convergence of + a - e. By the definition of a, we have inf (fex) - a(x) : X E IlU} > O. Applying Lemma 8.2.10, we see that epifE (s(ll,y,-a+ e)C)++, and so there exists 1.2;::: Al such that for all 1.;::: 1.2, we have epijA, E (S(Il,y,-a + e)C)++. For all 1.;::: 1.2, we have inf f),,(x) - - a+ 10> 0 IIxll::;;1l which is to say that a),,> a - e.

This establishes pointwise convergence of be a normed linear space and let f,JI,J2,h, ... be a sequence in reX). Using Theorem 8.2.2 and Proposition 8.1.2, show that is slice convergent to f if and only if both of the following conditions are satisfied [At3,AB1]: (i) for each x E X, there exists a sequence strongly convergent to x such that fex) = limn~oofn(xn); (ii) for each y E X*, there exists a sequence strongly convergent to y such that .f*(y) =limn~oof~(Yn). Let be a normed linear space. Using Proposition 8.1.2 and Lemma 8.2.9, prove that the slice topology is the weakest topology '0 on ItX) with these three properties: (a) for each norm open subset W of X, f -7 mf(W) is 'O-upper sernicontinuous; (b) for each A

3. 4. 5.

6.

7.

8.

9.

E

f -7 mf(A) is 'O-Iower sernicontinuous; X*, f -7 f + y is 'O-continuous. CB(X),

(c) for each y E Establish the first duality formula of Lemma 8.2.8 (Remark : this depends on the involutional character of the Fenchel transform rather than on Moreau's formula). Let be a normed linear space. Prove that X is reflexive if and only if the Fenchel transform is a continuous function from - a on X x R x X* and Fenchel's equality .f(x) + f"(y) = , valid for (x,y) E a.r. An immediate consequence of Theorem 8.3.7 is of course

an

8.3.8 Corollary.

Let be a Banach space.

Then f

~

Jf is a lower

semicontinuous multifunction on

n---p>

(Hint: Use Exercise 8.2.1 and Proposition 8.3.6). § 8 .4. Stability of the Geometrie Ekeland Principle

A key tool in the results of last seetion was the Geometrie Ekeland Principle for a proper lower semicontinuous function f defined on a Banach space, which asserts that if the set epifn [(xoJ(xo» - Cp ] is lower bounded for some xo E dom/, then there is a point (XlJ(X!» that is f.l-cone maximal for epi f: epifn [(xI!(x]» - C p ] = {(XlJ(Xj)} (see Figure 8.3.3). It is clear (see Exercise 8.3.5) that f.l-cone maximality of (Xl J(X!» is equivalent to saying that the point X =Xl is the unique minimizer of the perturbed function JI1(x)

= fex) +

f.llix - x!11.

Let us now write Mlf,f.l) for the set of all such f.l-cone maximal points. In view of our lower semicontinuity results for the operators ()f and,1f on n:X) equipped with the slice topology, it is natural to make the following conjecture: if is slice convergent to f in n:X), and (wJ(w» is f.l-cone maximal for epif, then we can find a sequence of points «xnJn(xn»> convergent to (wJ(w» such that for each n, (xnJn(x n » E M(fn,f.l). Perhaps surprisingly, this is not the case, even if slice convergence is replaced by the stronger Attouch-Wets convergence. But if JI1 has a strong minimum at X = w, in the sense that the function JI1 is well-posed as defined in §7.5, then the asserted approximation property is valid. In this seetion, following [AB2], we provide supporting details. Failure of lower semicontinuity of Mlf,f.l) for the Attouch-Wets topology follows easily from the celebrated Bishop-Phelps Theorem [BiP,Die, Ph], which is itself an easy consequence of Theorem 8.3.3.

8.4.1 Bishop-Phelps Theorem. Let C be a nonempty closed bounded convex subset of a Banach space . Then {y E X* : 3 X E C with =s(y,C)} is dense in X*.

CHAPTER8

300

Prool. Let

E > 0 and let

YO

*' 0*

be an otherwise arbitrary element of X*.

By the

boundedness of C, there exists xo E C such that > s(yo,C) - E2. Since xo E C, we may rewrite this inequality as folIows:

J(xo,C) + s(YO,C) - E2 < · Since the conjugate of the indicator function for C is the support function of C, this says that YO is an E2- subgradient for the indicator at xo. By Theorem 8.3.3 there exists (x,y) E

JI(- ,C) with lIy - YOIl*

< E. By Fenchel's equality, we have

= .

J(x,C) + s(y,C)

In particular J(x,C) is finite, and as a result, J(x,C) = 0 and s(y,C) = .

•

8.4.2 Proposition. Suppose is a nonreflexive Banach space. Then there exists a sequence I,JI,fz,/3, ... of uniformly lower bounded functions in reX) and J,l > 0 such that 1= TA w-limln but Mif,J,l) is not contained in Li Mifn,J,l) .

Proof If X is not reflexive, then by a theorem of James (see, e.g., [DS] or [HoI2,§19]), there exists a norm one element y of X* such that y is not norm achieving on the unit ball. We claim that (0,0) E M(y,I), that is, epi y n -Cl = {(O,O)}. To see this, fix x

*' O.

By the choice of y, we have < li-xII, which is to say that + IIxll > O.

Thus, the origin is the unique minimizer of x -? + 1· IIx - eil, as required. By the Bishop-Phelps Theorem and a normalization, there is a sequence of norm achieving elements in x* strongly convergent to y such that IIYnll* 1 for each n. As we have seen in §3.4, y = TAW-lim Yn (this also follows from Lemma 7.1.2). Now let I be the following restriction of our linear functional y:

=

1= yl{x : ~ -2} =y + J(. ,{x: ~ -2}). Similarly, set In = Ynl{x : ~ -2}, n = 1,2,3, . . . . Clearly, all restricted functions are bounded below by -2. Proposition 7.1.7 applied to yields

{x:

~

-2}

= TAW-lim

{x: ~ -2}.

Applying Theorem 7.4.6 that speaks to continuity of restriction, we have Now epi/c: epi y implies that (0,0)

E

1= TAW-limln.

Mif,I). On the other hand, we claim that for

each fixed index n, (x.tn(x» E Mifn,l) implies = -2.

To see this,

suppose

that = -2 + E for some positive E. Since Yn is norm attaining, we may choose

Zn

E

X with Ilznll = 1 such that = -1. Then

(x + ez n, -2)

E

[(x,{n(x» - CIl

n epiln .

This shows that (x.tn(x» e Mifn,I), unless =-2. By the norm convergence of to y, it is clear that 0 e Li {x : = -2}, and in particular, we have (0,0) e Li M(fn,I). •

THE SLICE TOPOLOGY FOR CONVEX FUNCTIONS

301

The key to the subsequent analysis is a geometrie expression of well-posedness of the perturbed function j11(x) = fix) + ,ullx - xoll, as defined in §7.5, where (xo.f{xO)) is ,u-cone maximal. To this end, for each t > 0, we define the horizontal disc C~ in the product X x R by the formula c~

= -Cp. n

{(x,-t): x EX}.

The disc C~ is obtained by taking a horizontal section ofthe inverted vertical cone -Cp. at height -t. For convenience, we inc1ude a simple geometric fact as a lemma, whose proof is left to the reader in Exercise 8.4.3.

8.4.3 Lemma. Let be a normed linear space and suppose ,u> 0, t> O. Let Hp. denote the boundary of the cone -Cp., and let «wn,an» be a sequence in X x R. Then {

d((wn,an), Hp.) ---7 0

a n ---7 -t

imply

8.4.4 Lemma. Let j be a proper lower semicontinuous convex function defined on a normed linear space with (xo.f{xO)) ,u-cone maximal for epif For each t ~ 0, define ({X..t) by the formula ((X..t)

=D(epif, (xo.f{XO)) + C~ ).

Then qJ is a nonnegative convex increasing function on [0,+00) with qJ(O)= O. Furthermore, the perturbed function j11(x) =j(x) + ,ullx - xoll has a strong minimum at xo if and only if ({X..t) > 0 for each positive t. Proof Nonnegativity of qJ is obvious, and ((X..O) =0 follows from (xo,f(xo)) E epij. Without loss of generality, we now assume that xo = () and f((}) = O. For convexity, fix tl ~ 0, t2 ~ 0, e> 0, and ß E [0,1]. Choose (wI,al) and (w2,a2) in epij and

points (Xl, -tI) E d~ and (X2, -t2) E d~ with

Now we form convex combinations: W3 = ßWI + (1 - ß)W2, X3 = ßXI + (1 - ß)X2, a3 =ßal + (1 - ß)a2, and t3 =ßtl + (1 - ß)t2 . We c1early have (w3,a3) E epij and (X3, -t3) E

dJ. We now compute

302

CHAPTER8

:::; ß lI(wl,ar) - (Xl, -tj)1I + (l - ß) II(W2,a2) - (X2, -t2)11 < ß((i..tr) + (l - ß)((i..t2) + E. This establishes convexity, and monotonicity follows from nonnegativity along with ((i..0) =0. We now turn to the well-posedness of the perturbed function, again under the normalization assumption xo = (J and f( (J) = O. Suppose first that q>(t) > 0 for each positive t, but that j11 has a minimizing sequence that stays away from the origin (J. By convexity of fand f( (J) =0, we may assume that for each n, we have Ilxnll = 8

> O. With t =Jl8, we have

As a result, we have ((i..t) = 0 for t = Jl8, an obvious contradiction. Thus (J is indeed a strong minimum of the perturbed function. For the converse, suppose that q>(t) 0 for some positive t. Then we find for

=

each positive integer n a point (xn , -t)

E

lim 11 (x n, -t) - (w n, an) 11

C~ and a point (w n , an)

E

epif with

=O.

n~oo

Since the sequence «w n, an» must approach the boundary of allows us to conclude that Jlllwnll ~ t. We compute lim sup (f(wn) + Jlllwnll) :::; lim sup an + lim sup Jlllwnll n~

n~

-C}1'

Lemma 8.4.3

=- t + t =O.

n~

This shows that is a minimizing sequence for the perturbed function j11, while at the same time, IIwnll

~.i > O. Thus, j11 fails to be well-posed. Jl

•

8.4.5 Corollary. Let be a reflexive Banach space, let fE T(X), and let Jl > O. Suppose that j11(x) = f(x) + Jlllx - xoll has a unique minimizer at X = xo. Then the minimum is actually a strong minimum. Proof. Fix t> O. Since (xo/(xo)) + C~ is a weakly compact sub set of X x R disjoint from epi f, the gap between the two sets must be positive by the weak lower semicontinuity of the norm. Now apply Lemma 8.4.4. •

303

THE SLICE TOPOLOGY FOR CONVEX FUNCTlONS

We now come to our announced positive stability result.

8.4.6 Theorem. Let be a Banach space, let fE l(X), and let /1 > O. Suppose that (xoJ(XO)) E M(J,/1), i.e., that (XOJ(XO)) is /1-cone maximal for epi f Furthermore, suppose that jP(x) =fex) + /1l1x - xoll has a strong minimum at x =xo. Then whenever is a sequence in nX) convergent to f in the slice topology, we have (XOJ(xo)) E Li M(Jn,/1).

Praof It suffices to prove the following: given f > 0, there exists Ne E Z+ and for each n > Ne, a point (wnJn(w n)) E M(Jn,/1), with lI(w nJn(w n)) - (XOJ(XO)) 11 < f . Since slice convergence of epigraphs implies Wijsman convergence which in turn implies Kuratowski-Painleve convergence, there exists a sequence convergent strongly to xo for which f(xo) =limn~oofn(xn). Fix t> 0; we claim that there exists N =N(t) E Z+ such that for each n > N we have (#)

We assurne that no such N exists and arrive at a contradiction. This means that we can find an increasing sequence of positive integers and a sequence of points such that for each index k, both IIz nk - xnkll s;; 1//1 and fnk(znk) s;;fnk(xnk ) - I . By the boundedness of there exists A> 0 such that for each k, znk

E AU.

By Lemma

8.4.4 and the well-posedness of jP, we have D(epij, (XOJ(XO)) + C~) > O. Thus, by Lemma 8.1.1 we may strongly separate epi f from the horizontal disc (xoJl..xO)) + C~ by the graph of a continuous affine functional as shown in Figure 8.4.1 : there exists y E X* and 1] > f*(y) such that the graph G of the affine functional x ~ - 1] satisfies D(G, (xoj(XO)) + C~) > 0 .

-.I I I I

:t I I I I I

FIaURE 8.4.1

304

CHAPTER8

The disc lies below G because epif lies above, and in particular, fE (s(Ä,Y,1])C)++ . By the strong convergence of «xnJn(xn»> to (XOJ(XO», we actually have Hausdorff metric convergence of the sequence of discs «xnJn(xn» + e~> to (XOJ(XO» + e~, so that for all n sufficiently large, we have (xnJn(xn» + e~ below G. In analytic terms, this means that whenever IIz - xnll::;; all k sufficiently large, we have

i, Jl

then

and since znk E ÄU, we conc1ude that fnk

(i!:

fn(x n) - t < - 1]. In particular, for

(s(Ä,y,1J)C)++, in violation of Proposition

8.1.2. This establishes (#). Choose t> 0 so small that max {t, tlJl} < e/2. Now take NE> N(t) so large that for each n > NE both IIxn - xOIi < e/2 and !fn(xn) - f(xo) I < e/2. Fix n > NE' Since n > N(t), by (#) and the convexity of fn, the set [(xnJn(x n eil] n epifn is lower bounded by fn(x n) - t. By the Geometrie Ekeland Principle, there exists (wnJn(w n E (xnJn(x n eil such that (wnJn(w n E M(jn,Jl). By the lower boundedness requirement, we have both

»-

»-

and

»

»

t IIw n - xnll < - . Jl

By the choice of t and NE, this yields

< max {t, tlJl} + e/2 < e. This completes the proof of the theorem.

•

Exercise Set 8.4. 1.

2. 3. 4.

Let be a Banach space and let fE JtX). Establish this inc1usion string [AR]: {x : 3 Y E dJ(x) with lIylI* < Jl} C {x : (xJ(x» E M(j,Jl)} C {x: 3 y E (}f(x) with lIyll*::;; Jl}. Show that none of the inc1usions is in general reversible. Let be a Banach space and let fE reX). Prove or give a limiting counterexample: M(f,Jl) -:t. 0 if and only if d( ()*, dom.f') < Jl. Prove Lemma 8.4.3. Give a direct proof of Corollary 8.4.5 without using Lemma 8.4.4.

THE SLICE TOPOLOGY FOR CONVEX FUNCTIONS

5.

Let f be a proper lower semicontinuous convex function defined on a normed linear space and let Jl> O. For (x.f(x» E Mlf,Jl) define ({Jx :[0,+00) ~ [0,+00) by ((Jx(t)

6. 7.

8.

9.

305

=D(epif, (xo.f(xo»

+ C~). Obtain the following result of

Attouch and Beer [AB2] : {x EX: 3 Y E (}J(x) with lIylI* < Jl} = {x EX: (x.f(x» E Mif,Jl) and ({J~(O) > O}. On the line, produce a sequence of lower bounded continuous functions convergent uniformlyon bounded subsets (and hence Attouch-Wets convergent) to f= 0 such that Mif,l) a:: Li Mifn,l). Establish the following result of Attouch and Riahi [AR]: Let be a uniformly lower bounded sequence of proper weakly lower semicontinuous functions defined on a reflexive Banach space X that is Mosco epi-convergent to a proper weakly lower semicontinuous function f. Then for each Jl > 0, Mlf,Jl) c:: Li Mifn,Jl). Let be a Banach space, let CE C(X), and let xo E C. Recall from Exercise 1.4.7 that the normal cone to C at xo is given by Nc(xo) = {y E X* : s(y,C) =}. (a) Show that Y E Nc(xo) if and only if Y E ()/(xo,C). (b) Suppose xo E bd C and e> O. Use Theorem 8.3.3 to show that there exists Xl E C with IIXI - xOIi < e and such that Nc(XI) contains a nonzero vector (Hint: take w (i!: C with IIw - xoll < e2, and separate w from C by a norm one functional y). (c) Present a general Bishop-Phelps Theorem to include the possibility that C is unbounded (Hint: work with the effective domain of the support functional). Let be a Banach space, let C, Cl, C2, C3, . ... be a sequence in C(X). Using Theorem 8.3.7 and Theorem 8.3.9, prove that the following are equivalent : (a) is convergent to C in the slice topology; (b) {(x,y) :XE C andYE Nc(x)} c:: Li {(x,y) :XE Cn andYE Ncn(x)};

(c) {(X,Y):XE C andYENc(x)}=K-lim{(x,y):xE Cn andYENcn(X)}.

Notes and References Chapter 1 The material in § 1.1 is very well-known. There are many accessible accounts of general topology; for our purposes, we recommend the monographs of Dugundji [Dug] and Willard [Wil]. Somewhat more imposing is the encyclopedic monograph of [En], which gives an unmatched account of the literature. The monograph of Berge [Bg], although omitting many aspects of general topology, presents many of the constructs of set-valued analysis that the reader will encounter in subsequent chapters, and at the same time presents significant material on convex sets and the elements of nonlinear analysis. In some sense, our text seeks to maintain the flavor of [Bg], often walking the border between point-set topology and geometric functional analysis. For the latter subject, we recommend the monographs of Holmes [Holl-2] and Giles [Gil]. The ball measure of noncompactness as introduced in § 1.1 and its relatives are considered in the monograph of Banas and Goebel [BaG]. Kuratowski's Theorem [Ku1] presented in Exercise 1.1.4 has numerous applications; it is a curious fact that for a decreasing sequence of closed convex subsets of a Banach space, limn --+ oo X(An\An+l) = 0 is enough to guarantee nonempty intersection, as shown by Cellina [CeI4]. One of the most striking aspects of the theory of topologies on closed sets is that the most important of them have attractive presentations as weak: topologies determined by geometric set functionals. Although weak: topologies are standard tools in modem linear analysis, it is difficult to find facts about them collected in one place. For this reason § 1.2 exists. We mention that the idea of weak: topology as it is used in functional analysis originates with von Neumann [Neu]. In this book we only consider uniformities as described by entourages; altematively, they may be approached through uniform covers v [ls] or through pseudo-metrics [Dug]. For accounts of Baire spaces and Cech complete spaces, the reader may consult [En] and Haworth and McCoy [HaM]. Lower The notion of semicontinuous function originates with Baire [Bai]. semicontinous functions rather than upper semicontinuous functions are usually considered primitive in one-sided analysis, as the theory has its origins in the study of convex rather than concave functions. The systematic construction of proofs for results involving semicontinuous functions based on the manipulation of epigraphs, rather than treating such functions as transformations, owes a great deal to the monograph of Rockafellar [Roc4]. Theorem 1.3.7 for real-valued functions defined on a metric space is due to Hahn [Hah!]; a Tl space admits such continuous approximations if and only if it is perfectly normal [Ton]. There exists a literature on semicontinuous functions with values in a partially ordered topological space, where there is some compatibility between the order and the topology (see, e.g., [PT,Be6,How,McC2]). The Separation Theorem is the corners tone of geometric functional analysis, and it is of course possible to formulate it much more generally, even in a linear space without topological structure. It may be used to prove the general Hahn-Banach Theorem, or it may be derived from this extension result [HoI2,Gil]. Modem convex function theory in a sense originated with the lecture notes of Fenchel [Fen], and reached its maturity much through the efforts ofRockafellar and Moreau in the 1960's [Roc4,Mor2]. The classical approach to the subject, with a particularly attractive treatment of convex functions and inequalities, is documented in [RV]. There is an enormous literature on convex cones and their application; the reader may consult Aubin and Frankowska [AuF]. Purely geometrical aspects of the theory of convex sets in locally convex spaces are considered in the historically important volume of Valentine [Val]. For an introduction to the theory of 306

NOTES AND REFERENCES

307

convex sets in finite dimensions, we recommend [Lay,MS]. The differentiation of convex functions has been a focal point of research in recent years (see, e.g., [Gil,Ph]). The idea of excess (ecart) originates with Pompieu in 1905 [Porn]. The name "gap functional" was forcefully suggested to the author by Penot. Lemma 1.5.1 can be found in [Cnt], although it was known much earlier for bounded sets.

Chapter 2 In view of the centrality of weak topologies in modern functional analysis, it is a little odd that the the creation of a theory of hyperspaces based on weak topologies was not given adecent try until weIl after 1980. Efforts in this direction have their roots in the study of topologies on lower semicontinuous functions, both convex and nonconvex [Verl-2,Jol], but it was the 1985 paper of Francaviglia, Levi and Lechicki [FLL], who considered distance functionals with a fixed point argument as a function of a set variable, that focused attention on the general method. The notion of strict d-inclusion originates with Costantini, Levi and Zierninska [CLZ]. Paralleling Theorem 2.1.10, necessary and sufficient conditions for Wijsman topologies to agree for closed convex sets have been identified by Corradini [Cor2]. The Vietoris topology was introduced in the early 1920's by Vietoris in [Viel-2J, and its basic topological properties were identified by Michael [Mic 1] in a 1951 paper that set the agenda for the study of hyperspaces for the next twenty years. In terms of later developments, we cite Keesling's deep study of compactness in the hyperspace [Keel-2J, as weIl as Smithson's interesting results on first countability [Sm4]. The term "proximal hit-and-miss topology" originates with Naimpally and his collaborators [BLLN, DCNS], although the proximal topology of §2.2 is implicit in the work of Nachmann [Nac]. Sequential convergence with respect to the proximal topology was studied by Fisher [Fis] and is sometimes called Fisher convergence in the literature (see, e.g., [BaP]). A number of authors had tried to pin down the precise relationship between the Wijsman topology, the ball proximal topology, and the topology of Exercise 2.2.12 (see, e.g., [FLL,Be8, DMN,BTa1] before Hola and Lucchetti [HL] clarified the issue, using the constructs of [CLZ]. Hit-and-miss topologies in the abstract were studied by Poppe [Popl-2] in papers that should have attracted more attention than they received. Given a farnily of weak topologies on a set, the supremum of these topologies is again a weak topology, as determined by the amalgamation of the defining functionals. In particular, suprema of hyperspace topologies can often be attractively expressed as weak topologies. Clean presentations of infima of hyperspace topologies are generally difficult to produce. One problem is a lack of distributivity within the lattice of hyperspace topologies. With distributivity, the infimum of the Vietoris topology and the Hausdorff metric topology ought to be the proximal topology. But as Levi, Lucchetti, and Pelant have shown, this occurs if and only if the underlying metric space is either UC or totally bounded [LLP]. UC spaces perhaps originated with Nagata [Nag] and are periodically rediscovered every few years. It is of course possible to consider UC spaces in the context of uniform spaces [Ats2,NS]. Theorem 2.3.4 is a variant of a result of the author [Be4], who used the Hausdorff metric topology in lieu of the proximal topology. For a general discussion of hyperspaces and function spaces, consult [Nai3]. The slice topology was anticipated by a hit-and-miss topology compatible with Mosco convergence of sequences of convex sets [Be 10-11]. The topology itself was first introduced by Sonntag and Zalinescu [SZ2] as a weak topology determined by a farnily of gap functionals (see Corollary 4.l.7) as part of an overall program to classify hyperspace topologies. Around the same time, the author [Be19-20] introduced the slice topology in

308

NOTES AND REFERENCES

much the same way in an attempt to come up with a topology a good deal stronger than the Wijsman topology that remained compatible with the measurability of convex-valued multifunctions (see §6.5). For lower semicontinuous convex functions identified with their epigraphs, the topology agrees with a topology of Joly [Jol] defined in terms of the lower semicontinuity of epigraphieal multifunctions, as we show in §8.2 The author does not know whether the Wijsman topology corresponding to an arbitrary compatible metrie for a Polish space is necessarily Polish. This is true in the special case that the metrie d is totally bounded, as established by Effros [Eff]. The author is hopeful that more transparent proofs of the results of §2.5 can be found, as weH as a useful complete metrie for the Wijsman topology for the closed subsets of a Polish space.

Chapter 3 Attouch-Wets convergence appears in Mosco's initial 1969 paper on Mosco convergence [Mos1], but was not studied in depth until many years later by Attouch and Wets [AW1-2]. Theorem 3.1.3 was first proved by Attouch, Lucchetti, and Wets [ALW], modifying the standard proof of completeness for the Hausdorff metric topology, as established for bounded sets by Hahn [Hah2]. The Hausdorff metric topology originates with Pompieu [Pom] , but not the Hausdorff metrie itself, as Pompieu worked with the equivalent metric p(A, B) =ecJCA,B) + ed(B,A). Without question, the Hausdorff metrie topology is the most weH-studied hyperspace topology, and §3.2 merely scratches the surface. A great deal of attention has been focused on the Hausdorff metric applied to metric continua [Nad]. A central result in this theory is the Curtis-Schiori Theorem [CS]: the space of closed subsets of a locally connected metrie continuum is homeomorphic to the Hilbert cube. An equal amount of attention has been given to the space of closed and bounded convex sets in finite dimensions, equipped with the Hausdorff distance. In this direction, curious Baire category results have been obtained, especially by Zarnfirescu (see, e.g., [Gru]). As another result within this framework, Vitali [Vit2] has obtained a Hörmandertype theorem replacing uniform distance between support functions by LP-distances, computed through integration on the surface of the unit ball equipped with normalized Lebesgue measure. Moreover, such a distance is bounded below by an expression involving the uniform distance, Le., ordinary Hausdorff distance. The Bulgarian school of approximation under the leadership of Sendov [Sen] has developed a theory of constructive approximation of functions based on Hausdorff distance. This theory not only produces analogues of the classieal theory in the continuous case, but also comfortably handles discontinuous bounded real functions by identifying such functions with the set of points lying between the lower and upper envelopes of the function (see §6.1). Hausdorff distance naturally arises in the theory of fractals [Hut,Wie]. There is also literature on descriptive set theoretic questions involving Hausdorff distance. The Hausdorff uniform topology was studied along with the Vietoris topology by Michael [Micl], and many facts cited in this monograph about both topologies come from this source. It has been shown by Ward [War] that distinct uniformities may determine the same Hausdorff uniform topology. Theorem 3.3.12 which appeared in [BHPV] was perhaps the first result giving a nontrivial presentation of a supremum of related hyperspace topologies, and influenced the direction of subsequent research in the field. The locally finite topology itself was

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309

eonsidered earlier and independently by Marjanovic [Mar] and by Feiehtinger [Fei]. For a reeent variant ofthe locally finite topology, the reader may eonsult [DC]. The equivalenee between strong eonvergenee of linear funetionals and AttouehWets eonvergenee of graphs is anticipated by the notion of "Kato gap" between subspaees [Kat, pp. 197-204].

Chapter 4 Continuity of various set funetionals (diameter, measures of noneompaetness, gap, exeess, . . .) with respeet to Hausdorff distanee has been known for many years, and attempts to prove eontinuity of set funetionals with respeet to weaker topologies are seattered throughout the literature (see, e.g., [BaP,SoD. A general program aimed at the eharaeterization of topologies in terms of the eontinuity of set funetionals was earried out by the author and his assoeiates over the last six years [BelO,Be13,BeP,BLLN,BeL3]. In a similar direetion, Sonntag and Zalineseu [SZ2-3] foeused on classes of uniformities eompatible with hyperspaee topologies, subsuming not only weak topologies but also Cornet's program [Cnt] of embedding CL(X) into C(X,R) , under the identifieation A H d(- ,A). Particular uniforrnities provide the maehinery for quantitative estimation and approximation, as aetively pursued by Attoueh, Wets, Az6, and Penot. One eould easily take as primitive objeets in a study of hyperspaee topology the upper and lower halves of topologies as presented in §4.2 (see, e.g., [FLLD. We mention that the upper Wijsman topology when speeialized to points has been studied in the eontext ofBanaeh spaee geometry by Godefroy and Kalton [GK]. The precise history of the eonneetion between infimal value funetionals and eonvergenee is obseure, and early results in this direetion often appear in multifunetion form [Cho2,DRl]. Attention was foeused on this eonneetion only reeently, again through the work by Sonntag and Zalineseu [SZ2], who euriously did not explicitly bring this within their uniforrnity framework. Their results were eonfined to the metrie setting. We anticipate in the near future topological results involving proximity and eompaetifieation in whieh infimal value funetionals playa pivotal role. Interest in sealar eonvergenee sterns not only from the fundamental 1954 paper of Hörmander [Hö] but also from the expository paper of Salinetti and Wets [SW2]. As a variant of sealar eonvergenee, one ean study pointwise eonvergenee of support funetionals for truneated sets, following Theorem 2.4.8 [BZ].

Chapter 5 The Fell topology was introdueed by Fell in 1962 [Fel] in eonneetion with eertain nonHausdorff but loeally eompaet spaees arising in the theory of C*-algebras. His topology was antieipated by the lbe topology ofMrowka [Mrl]; for both papers, Miehael's article [Miei] is again influential. The notions of upper and lower closed limits of sequenees of sets in §5.2 are unifQrrnly attributed to Painleve, and perhaps first appear in print in articles by his student Zoretti [Zorl-2]. Set eonvergenee of sequenees defined as equality of these limits was subsequently popularized by Hausdorff [Hau] and then by Kuratowski [Ku3]. Convergenee of filters of sets was first eonsidered by Choquet [Chol] in 1947. Convergenee of nets of sets in an equivalent formulation was studied about ten years later by Mrowka [Mrl], who was unaware ofChoquet's work. Shortly thereafter, KuratowskiPainleve convergence of nets was rediscovered by Frolik [Fro]. Without question,

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Choquet's article and the general dissemination of much of its contents through the monograph of Berge [Bg] had a huge influence on the development of the subject, and when convergence is considered in applications, filters rather than nets are traditionally used. In this matter, and despite the equivalence between filter and net convergence (see, e.g., [Wil, § 12]), the present monograph is heretical. Mrowka' s Theorem was earlier established for regular spaces by Frolik [Fro]. The compatibility of Kuratowski-Painleve convergence with the Fell topology in locally compact spaces is a result of Mrowka [Md]; for sequences, this holds in k-spaces [DM2], i.e., spaces in which a set is open if and only if its intersection with each compact subset is relatively open. In relation to Wijsman convergence, Kuratowski-Painleve convergence of nets of nonempty closed sets can be shown to be dual to Vietoris convergence, as expressed by Theorem 2.2.5: in a metrizable space X, A = K-Lim A;., if and only if there exists a compatible metric d such that