Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences

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VOLUME 12

A Graduate-Level Book Series

The titles published in this series are listed at the end of this volume. Fundamentals of Functional Analysis

by S. S. Kutateladze Sobolev Institute ofMathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress

ISBN 978-90-481-4661-1 ISBN 978-94-015-8755-6 (eBook) DOI 10.1007/978-94-015-8755-6

Printed on acid-free paper

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents

Preface to the English Translation ix Preface to the First Russian Edition x Preface to the Second Russian Edition xii Chapter 1. An Excursion into Set Theory 1.1. Correspondences ...... 1 1.2. Ordered Sets ...... 3 1.3. Filters ...... 6 Exercises ...... 8 Chapter 2. Vector Spaces 2.1. Spaces and Subspaces ...... 10 2.2. Linear Operators ...... 12 2.3. Equations in Operators ...... 15 Exercises ...... 18 Chapter 3. Convex Analysis 3.1. Sets in Vector Spaces ...... 20 3.2. Ordered Vector Spaces ...... 22 3.3. Extension of Positive Functionals and Operators 25 3.4. Convex Functions and Sublinear Functionals 26 3.5. The Hahn-Banach Theorem ...... 29 3.6. The KreIn-Milman Theorem ...... 31 3.7. The Balanced Hahn-Banach Theorem ...... 33 3.8. The Minkowski Functional and Separation ...... 35 Exercises ...... 39 Chapter 4. An Excursion into Metric Spaces 4.1. The Uniformity and Topology of a Metric Space 40 4.2. Continuity and Uniform Continuity ...... 42 4.3. Semicontinuity ...... 44 4.4. Compactness ...... 46 4.5. Completeness ...... 46 4.6. Compactness and Completeness ...... 49 4.7. Baire Spaces ...... 52 4.8. The Jordan Curve Theorem and Rough Drafts 54 Exercises ...... 55 VI

Chapter 5. Multinormed and Banach Spaces 5.1. Seminorms and Multinorms ...... 56 5.2. The Uniformity and Topology of a Multinormed Space ...... 60 5.3. Comparison Between Topologies ...... 62 5.4. Metrizable and Normable Spaces ...... 64 5.5. Banach Spaces ...... 66 5.6. The Algebra of Bounded Operators ...... 73 Exercises ...... 79 Chapter 6. Hilbert Spaces 6.1. Hermitian Forms and Inner Products ...... 80 6.2. Orthoprojections ...... 84 6.3. A Hilbert Basis ...... 87 6.4. The Adjoint of an Operator ...... 90 6.5. Hermitian Operators ...... 93 6.6. Compact Hermitian Operators ...... 95 Exercises ...... 99 Chapter 7. Principles of Banach Spaces 7.1. Banach's Fundamental Principle ...... 100 7.2. Boundedness Principles ...... 102 7.3. The Ideal Correspondence Principle ...... 105 7.4. Open Mapping and Closed Graph Theorems 107 7.5. The Automatic Continuity Principle ...... 112 7.6. Prime Principles ...... 114 Exercises ...... 118 Chapter 8. Operators in Banach Spaces 8.1. Holomorphic Functions and Contour Integrals 120 8.2. The Holomorphic Functional Calculus ...... 126 8.3. The Approximation Property ...... 132 8.4. The Riesz-Schauder Theory ...... 134 8.5. Fredholm Operators ...... 137 Exercises ...... 144 Chapter 9. An Excursion into General Topology 9.1. Pretopologies and Topologies ...... 146 9.2. Continuity ...... 148 9.3. Types of Topological Spaces ...... 151 9.4. Compactness ...... 155 9.5. Uniform and Multimetric Spaces ...... 159 9.6. Covers, and Partitions of Unity ...... 164 Exercises ...... 168 Vll

Chapter 10. Duality and Its Applications 10.1. Vector Topologies ...... 169 10.2. Locally Convex Topologies ...... 171 10.3. Duality Between Vector Spaces ...... 173 10.4. Topologies Compatible with Duality ...... 175 10.5. Polars ...... 177 10.6. Weakly Compact Convex Sets ...... 179 10.7. Reflexive Spaces ...... 180 10.8. The Space C(Q, R) ...... 181 10.9. Radon Measures ...... 187 10.10. The Spaces ~(n) and ~/(n) ...... 194 10.11. The Fourier Transform of a Distribution ...... 201 Exercises ...... 211 Chapter 11. Banach Algebras 11.1. The Canonical Operator Representation ...... 213 11.2. The Spectrum of an Element of an Algebra 215 11.3. The Holomorphic Functional Calculus in Algebras 216 11.4. Ideals of Commutative Algebras ...... 218 11.5. Ideals of the Algebra C(Q, C) ...... 219 11.6. The Gelfand Transform ...... 220 11.7. The Spectrum of an Element of a C*-Algebra 224 11.8. The Commutative Gelfand-Nalmark Theorem 226 11.9. Operator *-Representations of a C*-Algebra 229 Exercises ...... 234 References 237 Notation Index 255 Subject Index 259 Preface to the English Translation

This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz-Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. With standard provisos the presentation is self-contained, exposing about a hun- dred famous "named" theorems furnished with complete proofs and culminating in the Gelfand-Nalmark-Segal construction for C*-algebras. The first Russian edition was printed by the Siberian Division of "Nauka" Pub- lishers in 1983. Since then the monograph has served as the standard textbook on functional analysis at the University of Novosibirsk. This volume is translated from the second Russian edition printed by the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences· in 1995. It incorporates new sections on Radon measures, the Schwartz spaces of distributions, and a supplementary list of theoretical exercises and problems. This edition was typeset using AMS-'lEX, the American Mathematical Society's 'lEX system. To clear my conscience completely, I also confess that := stands for the definor, the assignment operator, signifies the end of the proof.

s. K utateladze Preface to the First Russian Edition

As the title implies, this book treats functional analysis. At the turn of the century the term "functional analysis" was coined by J. Hadamard who is famous among mathematicians for the formula of the radius of convergence of a power series. The term "functional analysis" was universally accepted then as related to the calculus of variations, standing for a new direction of analysis which was intensively developed by V. Volterra, C. Arzela, S. Pincherle, P. Levy, and other representatives of the French and Italian mathematical schools. J. Hadamard's contribution to the present discipline should not be reduced to the invention of the word "functional" (or more precisely to the transformation of the adjective into a proper noun). J. Hadamard was fully aware of the relevance of the rising subject. Working hard, he constantly advertised problems, ideas, and methods just evolved. In particular, to one of his students, M. Frechet, he suggested the problem of inventing something that is now generally acclaimed as the theory of metric spaces. In this connection it is worth indicating that neighborhoods pertinent to functional analysis in the sense of Hadamard and Volterra served as precursors to Hausdorff's famous research, heralding the birth of general topology. Further, it is essential to emphasize that one of the most attractive, difficult, and important sections of classical analysis, the calculus of variations, became the first source of functional analysis. The second source of functional analysis was provided by the study directed to creating some algebraic theory for functional equations or, stated strictly, to simplify- ing and formalizing the manipulations of "equations in functions" and, in particular, linear integral equations. Ascending to H. Abel and J. Liouville, the theory of the latter was considerably expanded by works of I. Fredholm, K. Neumann, F. Noether, xii Preface A. Poincare, et al. The efforts of these mathematicians fertilized soil for D. Hilbert's celebrated research into quadratic forms in infinitely many variables. His ideas, developed further by F. Riesz, E. Schmidt, et al., were the immediate predecessors of the axiomatic presentation of Hilbert space theory which was undertaken and implemented by J. von Neumann and M. Stone. The resulting section of mathematics h.as vigor- ously influenced theoretical physics, first of all, quantum mechanics. In this regard it is instructive as well as entertaining to mention that both terms, "quantum" and "functional," originated in the same year, 1900. The third major source of functional analysis encompassed Minkowski's geometric ideas. His invention, the apparatus for the finite-dimensional geometry of convex bod- ies, prepared the bulk of spatial notions ensuring the modern development of analysis. Elaborated by E. Helly, H. Hahn, C. Caratheodory, I. Radon, et al., the idea of convexity has eventually shaped the fundamentals of the theory of locally convex spaces. In turn, the latter has facilitated the spread of distributions and weak derivatives which were recognized by S. L. Sobolev as drastically changing all tools of mathematical physics. In the postwar years the geometric notion of convexity has conquered a new sphere of application for mathematics, viz., social sciences and especially economics. An exceptional role in this process was performed by linear programming discovered by L. V. Kantorovich. The above synopsis of the history of functional analysis is schematic, incomplete, and arbitrary (for instance, it casts aside the line of D. Bernoulli's superposition principle, the line of set functions and integration theory, the line of operational calculus, the line of finite differences and fractional derivation, the line of general analysis, and many others). These demerits notwithstanding, the three sources listed above reflect the main, and most principal, regularity: functional analysis has synthesized and pro- moted ideas, concepts, and methods from classical sections of mathematics: algebra, geometry, and analysis. Therefore, although functional analysis verbatim means analysis of functions and functionals, even a superficial glance at its history gives grounds to claim that functional analysis is algebra, geometry, and analysis of functions and functionals. A more viable and penetrating explanation for the notion of functional analysis is given by the Soviet Encyclopedic Dictionary: "Functional analysis is one of the principal branches of modern mathematics. It resulted from mutual interaction, unifi- cation, and generalization of the ideas and methods stemming from all parts of classical mathematical analysis. It is characterized by the use of concepts pertaining to various Preface xiii abstract spaces such as vector spaces, Hilbert spaces, etc. It finds diverse applications in modern physics, especially in quantum mechanics." The S. Banach treatise Theorie des Operationes Lineares, printed half a century ago, inaugurated functional analysis as an essential activity in mathematics. Its influ- ence on the development of mathematics is seminal: Omnipresent, Banach's ideas, propounded in the book, captivate the realm of modern mathematics. An outstanding contribution toward progress in functional analysis was made by the renowned Soviet scientists: I. M. Gelfand, L. V. Kantorovich, M. V. Keldysh, A. N. Kolmogorov, M. G. KreIn, L. A. Lyusternik, and S. L. Sobolev. The characteristic feature of the Soviet school is that its research on functional analysis is always con- ducted in connection with profound applied problems. The research has expanded the scope of functional analysis which becomes the prevailing language of the applications of mathematics. The next fact is demonstrative: In 1948 even the title of Kantorovich's insightful article Functional Analysis and Applied Mathematics was considered paradoxical, but it provided a basis for the numerical mathematics of today. And in 1974 S. L. Sobolev stated that "to conceive the theory of calculations without Banach spaces is just as impossible as trying to conceive of it without the use of computers". The exponential accumulation of knowledge within functional analysis is now observed alongside a sharp rise in demand for the tools and concepts of the discipline. The resulting conspicuous gap widens permanently between the current level of analysis and the level fixed in the literature accessible to the reading community. To alter this ominous trend is the purpose of the present book.

Preface to the Second Russian Edition

For more than a decade the monograph has served as a reference book for compul- sory and optional courses in functional analysis at Novosibirsk State University. This time span proves that the principles of compiling the book are legitimate. The present edition is enlarged with sections addressing the fundamentals of distribution theory. Theoretical exercises are supplemented and the list of references is updated. Also, inaccuracies, mostly indicated by my colleagues, have been corrected. I seize the opportunity to express my gratitude to all those who helped me in the preparation of the book. My pleasant debt is to acknowledge the financial support of xiv Preface the Sobolev Institute of Mathematics of the Siberian Division of the Russian Acade- my of Sciences, the Russian Foundation for Fundamental Research, the International Science Foundation and the American Mathematical Society during the compilation of the second edition. March, 1995 S. Kutateladze