Geometric functional analysis and its applications, richard b holmes

Graduate Texts in Mathematics 24 Editorial Board: F W Gehring P R Halmos (Managing Editor) C C Moore Richard B Holmes Geometric Functional Analysis and its Applications Springer-Verlag New York Heidelberg Berlin Richard B Holmes Purdue University Division of Mathematical Sciences West Lafayette, Indiana 47907 Editorial Board P R Halmos Indiana University Department of Mathematics Swain Hall East Bloomington, Indiana 47401 F W Gehring c University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 University of California at Berkeley Department of Mathematics Berkeley, California 94720 C Moore AMS Subject Classifications Primary: 46.01, 46N05 Secondary: 46A05, 46BI0, 52A05, 41A65 Library of Congress Cataloging in Publication Data Holmes, Richard B Geometric functional analysis and its applications (Graduate texts in mathematics; v 24) Bibliography: p 237 Includes index I Functional analysis I Title II Series QA320.H63 515'.7 75-6803 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1975 by Springer-Verlag New York Inc Softcover reprint of the hardcover 1st edition 1975 ISBN 978-1-4684-9371-9 ISBN 978-1-4684-9369-6 (eBook) DOl 10.1 007/978-1-4684-9369-6 To my mother and the memory of my father Preface This book has evolved from my experience over the past decade in teaching and doing research in functional analysis and certain of its applications These applications are to optimization theory in general and to best approximation theory in particular The geometric nature of the subjects has greatly influenced the approach to functional analysis presented herein, especially its basis on the unifying concept of convexity Most of the major theorems either concern or depend on properties of convex sets; the others generally pertain to conjugate spaces or compactness properties, both of which topics are important for the proper setting and resolution of optimization problems In consequence, and in contrast to most other treatments of functional analysis, there is no discussion of spectral theory, and only the most basic and general properties of linear operators are established Some of the theoretical highlights of the book are the Banach space theorems associated with the names of Dixmier, Krein, James, Smulian, Bishop-Phelps, Brondsted-Rockafellar, and Bessaga-Pelczynski Prior to these (and others) we establish to two most important principles of geometric functional analysis: the extended Krein-Milman theorem and the HahnBanach principle, the latter appearing in ten different but equivalent formulations (some of which are optimality criteria for convex programs) In addition, a good deal of attention is paid to properties and characterizations of conjugate spaces, especially reflexive spaces On the other hand, the following (incomplete) list provides a sample of the type of applications discussed: Systems of linear equations and inequalities; Existence and uniqueness of best approximations; Simultaneous approximation and interpolation; Lyapunov convexity theorem; Bang-bang principle of control theory; Solutions of convex programs; Moment problems; Error estimation in numerical analysis; Splines; Michael selection theorem; Complementarity problems; Variational inequalities; Uniqueness of Hahn-Banach extensions Also, "geometric" proofs of the Borsuk-Dugundji extension theorem, the Stone-Weierstrass density theorem, the Dieudonne separation theorem, and the fixed point theorems of Schauder and Fan-Kakutani are given as further applicati6ns of the theory viii Preface Over 200 problems appear at the ends of the various chapters Some are intended to be of a rather routine nature, such as supplying the details to a deliberately sketchy or omitted argument in the text Many others, however, constitute significant further results, converses, or counterexamples The problems of this type are usually non-trivial and I have taken some pains to include substantial hints (The design of such hints is an interesting exercise for an author: he hopes to keep the student on course without completely giving everything away in the process.) In any event, readers are strongly urged to at least peruse all the problems Otherwise, I fear, a good deal of the total value of the book may be lost The presentation is intended to be accessible to students whose mathematical background includes basic courses in linear algebra, measure theory, and general topology The requisite linear algebra is reviewed in §1, while the measure theory is needed mainly for examples Thus the most essential background is the topological one, and it is freely assumed Hence, with the exception of a few results concerning dispersed topological spaces (such as the Cantor-Bendixson lemma) needed in §25, no purely topological theorems are proved in this book Such exclusions are warranted, I feel, because of the availability of many excellent texts on general topology In particular, the union of the well-known books by J Dugundji and J Kelley contains all the necessary topological prerequisites (along with much additional material) Actually the present book can probably be read concurrently with courses in topology and measure theory, since Chapter I, which might be considered a brief second course on linear algebra with convexity, employs no topological concepts beyond standard properties of Euclidean spaces (the single exception to this assertion being the use of Ascoli's theorem in 7C) This book owes a great deal to numerous mathematicians who have produced over the last few years substantial simplifications of the proofs of virtually all the major results presented herein Indeed, most of the proofs we give have now reached a stage of such conciseness and elegance that I consider their collective availability to be an important justification for a new book on functional analysis But as has already been indicated, my primary intent has been to produce a source of functional analytic information for workers in the broad areas of modern optimization and approximation theory However, it is also my hope that the book may serve the needs of students who intend to specialize in the very active and exciting ongoing research in Banach space theory I am grateful to Professor Paul Halmos for his invitation to contribute the book to this series, and for his interest and encouragement along the way to its completion Also my thanks go to Professors Philip Smith and Joseph Ward for reading the manuscript and providing numerous corrections As usual, Nancy Eberle and Judy Snider provided expert clerical assistance in the preparation of the manuscript Table of Contents Chapter I § § § § § § § § Convexity in Linear Spaces Linear Spaces Convex Sets Convex Functions Basic Separation Theorems Cones and Orderings Alternate Formulations of the Separation Principle Some Applications Extremal Sets Exercises Chapter II § §10 §11 §12 §13 §14 §15 Convexity in Linear Topological Spaces Linear Topological Spaces Locally Convex Spaces Convexity and Topology Weak Topologies Extreme Points Convex Functions and Optimization Some More Applications Exercises Chapter III §16 §17 §18 §19 §20 §2l Completion, Congruence, and Reflexivity The Category Theorems The Smulian Theorems The Theorem of James Support Points and Smooth Points Some Further Applications Exercises Chapter IV §22 §23 §24 §25 Principles of Banach Spaces Conjugate Spaces and Universal Spaces The Conjugate of C(Q, R) Properties and Characterizations of Conjugate Spaces Isomorphism of Certain Conjugate Spaces Universal Spaces Exercises 1 10 14 16 19 24 32 39 46 46 53 59 65 73 82 97 109 119 119 131 145 157 164 176 191 202 202 211 221 225 231 x Table of Contents References 235 Bibliogmphy 237 Symbol Index 241 Subject Index 243 Chapter I Convexity in Linear Spaces Our purpose in this first chapter is to establish the basic terminology and properties of convex sets and functions, and of the associated geometry All concepts are "primitive", in the sense that no topological notions are involved beyond the natural (Euclidean) topology of the scalar field The latter will always be either the real number field R, or the complex number field C The most important result is the "basic separation theorem", which asserts that under certain conditions two disjoint convex sets lie on opposite sides of a hyperplane Such a result, providing both an analytic and a geometric description of a common underlying phenomenon, is absolutely indispensible for the further development of the subject It depends implicitly on the axiom of choice which is invoked in the form of Zorn's lemma to prove the key lemma of Stone Several other equally fundamental results (the "support theorem", the "subdifferentiability theorem", and two extension theorems) are established as equivalent formulations of the basic separation theorem After indicating a few applications of these ideas we conclude the chapter with an introduction to the important notion of extremal sets (in particular extreme points) of convex sets §1 Linear Spaces In this section we review briefly and without proofs some elementary results from linear algebra, with which the reader is assumed to be familiar The main purpose is to establish some terminology and notation A Let X be a linear space over the real or complex number field The zero-vector in X is always denoted bye If {xJ is a subset of X, a linear combination of {Xi} is a vector X E X expressible as x = LAiXi, for certain scalars Ai' only finitely many of which are non-zero A subset of X is a (linear) subspace if it contains every possible linear combination of its members The linear hull (span) of a subset S of X, consists of all linear combinations of its members, and thus span(S) is the smallest subspace of X that contains S The subset S is linearly independent if no vector in S lies in the linear hull of the remaining vectors in S Finally, the subset S is a (Hamel) basis for X if S is linearly independent and span(S) = X Lemma subset of S S is a basis for X if and only ifS is a maximal linearly independent Theorem Any non-trivial linear space has a basis; infact, each non-empty linearly independent subset is contained in a basis 231 Exercises has also been shown that any separable universal Banach space contains a constrained subspace which is congruent to C(K, C) Finally, we remark that while no (separable) reflexive space can be universal for all separable Banach spaces (as a consequence of exercise 4.33), it is possible for such a space to be universal for the class of all finite dimensional spaces (but not for the class of all separable reflexive spaces) Indeed, there is an example due to Szankowski of a separable reflexive space X such that every finite dimensional Banach space is congruent with a constrained subspace of X Exercises 4.1 4.2 4.3 4.4 4.5 4.6 Prove formula (22.3) (To prove the inclusion from left to right consider first the case where ¢ E U(X*) has norm one.) Let X be an order unit normed linear space with order unit e If ¢ E X* satisfies II¢II = ¢(e) then ¢ is a positive linear functional Show that the correspondence 11 I > t:P I' of 22B is bipositive in the sense that 11 is a positive Borel measure if (and only if) Ja x dl1 ~ for all non-negative x E C(Q, R) Let Q be a compact Hausdorff space Suppose that {xn} is a bounded sequence in C(Q, F) that is pointwise convergent to e: limn xn(t) = 0, t E Q Show that Xn converges weakly to e Show by example that this conclusion may fail if {xn} is replaced by a bounded pointwise convergent net in C(Q, F) Let Q be an extremally disconnected topological space a) Any two disjoint open subsets of Q have disjoint closures b) If Q is metrizable (more generally, first countable) then Q is discrete c) No sequence in Q can converge unless it is eventually constant a) Use the Riesz-Kakutani theorem to give a new proof of the fact that a Banach space X is reflexive if U(X) is weakly compact (16F) (Given t:P E U(X**), define a Borel measure 11 on U(X) by t:P(¢) = SU(X) ¢IU(X)dl1, ¢EX* Then 1111(U(X)) = 111111v:!( By 13B,E, 11 is the weak* -limit of a net of atomic measures of the form C\a)Dxl'), where {x\a)} is, for each rt., a finite subset of U(X), and c\a) ~ 0, c\a) = Now consider any weak cluster point in U(X) of the net {Ic\a)x\a)}.) b) Use the fact that reflexivity of a Banach space is equivalent to the weak compactness of its unit ball to give a new proof ofthe reflexivity of all closed subs paces and quotient spaces of a reflexive space (For the quotient space argument use 15B and 161.) Let v be a positive regular Borel measure on a compact Hausdorff space Q Prove the two assertions made about the support of v in 22E For any real Banach space X the set £!P(X) was defined in 20E For any compact Hausdorff space Q show that £!P( C(Q, R) ) can be identified with the set Clfmeasures 11 E Ar(Q, B, R) such that a(I1+) n a(I1-) = 0· (11+ and 11- were defined in 22B.) ° Li 4.7 4.8 Li 232 Conjugate Spaces and Universal Spaces Prove that the space of all polynomial functions on [0, 1J normed by the uniform norm is not subreflexive 4.10 Let xbe a normalized peak function in m == m(Q, R), so that Ix(t)1 = Ilxlloo = for a single t E Q Show that xE sm(U(I1'i») (This can be done in two ways: either directly by use of the representation of m* (16H), or by use of the congruence m ~ C(P(Q), R), and the result of 4.9 20F, Ex d.) 4.11 Generalize the example of 22F to compact spaces other than [0, 1] What must be assumed about such spaces for that proof to still apply? 4.12 Use the Dixmier-Ng theorem to show that the spaces m(Q, F) and Cl(Q, F) are conjugate spaces 4.13 Show that any reflexive space has a unique pre-dual 4.14 Determine a pre-dual of the Lipschitz space Lip(Q, d, F) (Consider the linear span of the evaluation functionals {b t : t E Q} in Lip(Q, d, F) This space can in turn be identified with the free vector space generated by Q.) 4.15 Let X be a Banach space and Va subspace of X* a) Suppose that J v == J x, v has a bounded inverse Prove 4.16 4.17 4.18 4.19 where 0' == O'(X, V) b) Suppose that X** = Jx(X) EB VO and let P:X** ~ Jx(X) be the associated projection Prove that IIJviii = IIPII· Let X be a Banach space a) Show that X is reflexive if and ont'y if X* contains no proper total closed linear subspace b) Assume that X is separable Show that X is reflexive if and only if every total sequence in X* is fundamental (9F) Let X be a separable Banach space a) Show that X* contains a separable duxial subspace b) Let X be the Lebesgue space Ll([O, 1J, ]F) Show that the space C([O, 1J, F) as a subspace of VIJ([O, 1J, ]F) is a (separable) duxial subspace of X* Let M be a (closed) complemented linear space of a Banach space X, and suppose that M is a conjugate space Then there exists a minimal projection on M, that is, a projection: X ~ M whose norm is ~ that of any other projection of X on M (Use the method of the example in 23B.) Show that the sequence space Co is not complemented in the space m, thereby proving anew that Co is not isomorphic to any conjugate space (A simple proof can be constructed along the following lines Suppose that Z is a complementary subspace for Co in m: Co EB Z = m Then Z is isomorphic to tn/co (exercise 2.2) Now there exists a countable total set 'in m*, hence there is such a set in Z*, and therefore also in (m/c o)* This last assertion leads to a contradiction To obtain it, we Exercises 233 make use of a fact about any countable set N: there exists an uncountable family {U oj of infinite subsets of N such that UIX n Up is finite whenever rt =1= /3 Applying this fact to the case where N = {I, 2, } we let fa be the coset in 1n/co that contains the characteristic function of the set UIX • Show that for any