Graduate Texts in Mathematics 96 Editorial Board F W Gehring P R Halmos (Managing Editor) C C Moore Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKE Un/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMM BACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEcn/ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Al)!ebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLI.ER Rings and Categories of Modules GOLUBITSKy/GuILLFMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol l ZARISKUSAMUEL Commutative Algebra Vol 11 JACOBSON Lectures in Abstract Algebra I: Basic Concepts JACOBSON Lectures in Abstract Algebra 11: Linear Algebra JACOBSON Lectures in Abstract Algebra Ill: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARYESON An Invitation to C*-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEYE Probability Theory I 4th ed LOEYE Probability Theory 11 4th ed MOISE Geometric Topology in Dimensions and continued after Index John B Conway A Course in Functional Analysis Springer Science+Business Media, LLC John B Conway Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A Editorial Board P R Halmos F W Gehring c C Moore Managing Editor Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A AMS Classifications: 46-01, 45B05 Library of Congress Cataloging in Publication Data Conway, John B A course in functional analysis (Graduate texts in mathematics: 96) Bibliography: p Includes index Functional analysis I Title II Series 84-10568 QA320.C658 1985 515.7 With illustration ©1985 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc in 1985 Softcover reprint of the hardcover I st edition 1985 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC Typeset by Science Typographers, Medford, New York 987 543 ISBN 978-1-4757-3830-8 ISBN 978-1-4757-3828-5 (eBook) DOI 10.1007/978-1-4757-3828-5 For Ann (of course) Preface Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other The common thread is the existence of a linear space with a topology or two (or more) Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both In this book I have tried to follow the common thread rather than any special topic I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts The word "course" in the title of this book has two meanings The first is obvious This book was meant as a text for a graduate course in functional analysis The second meaning is that the book attempts to take an excursion through many of the territories that comprise functional analysis For this purpose, a choice of several tours is offered the reader-whether he is a tourist or a student looking for a place of residence The sections marked with an asterisk are not (strictly speaking) necessary for the rest of the book, but will offer the reader an opportunity to get more deeply involved in the subject at hand, or to see some applications to other parts of mathematics, or, perhaps, just to see some local color Unlike many tours, it is possible to retrace your steps and cover a starred section after the chapter has been left There are some parts of functional analysis that are not on the tour Most authors have to make choices due to time and space limitations, to say nothing of the financial resources of our graduate students Two areas that Vlll Preface are only briefly touched here, but which constitute entire areas by themselves, are topological vector spaces and ordered linear spaces Both are beautiful theories and both have books which them justice The prerequisites for this book are a thoroughly good course in measure and integration-together with some knowledge of point set topology The appendices contain some of this material, including a discussion of nets in Appendix A In addition, the reader should at least be taking a course in analytic function theory at the same time that he is reading this book From the beginning, analytic functions are used to furnish some examples, but it is only in the last half of this text that analytic functions are used in the proofs of the results It has been traditional that a mathematics book begin with the most general set of axioms and develop the theory, with additional axioms added as the exposition progresses To a large extent I have abandoned tradition Thus the first two chapters are on Hilbert space, the third is on Banach spaces, and the fourth is on locally convex spaces To be sure, this causes some repetition (though not as much as I first thought it would) and the phrase" the proof is just like the proof of " appears several times But I firmly believe that this order of things develops a better intuition in the student Historically, mathematics has gone from the particular to the general-not the reverse There are many reasons for this, but certainly one reason is that the human mind resists abstraction unless it first sees the need to abstract I have tried to include as many examples as possible, even if this means introducing without explanation some other branches of mathematics (like analytic functions, Fourier series, or topological groups) There are, at the end of every section, several exercises of varying degrees of difficulty with different purposes in mind Some exercises just remind the reader that he is to supply a proof of a result in the text; others are routine, and seek to fix some of the ideas in the reader's mind; yet others develop more examples; and some extend the theory Examples emphasize my idea about the nature of mathematics and exercises stress my belief that doing mathematics is the way to learn mathematics Chapter I discusses the geometry of Hilbert spaces and Chapter II begins the theory of operators on a Hilbert space In Sections 5-8 of Chapter II, the complete spectral theory of normal compact operators, together with a discussion of multiplicity, is worked out This material is presented again in Chapter IX, when the Spectral Theorem for bounded normal operators is proved The reason for this repetition is twofold First, I wanted to design the book to be usable as a text for a one-semester course Second, if the reader understands the Spectral Theorem for compact operators, there will be less difficulty in understanding the general case and, perhaps, this will lead to a greater appreciation of the complete theorem Chapter III is on Banach spaces It has become standard to some of this material in courses on Real Variables In particular, the three basic Preface ix principles, the Hahn-Banach Theorem, the Open Mapping Theorem, and the Principle of Uniform Boundedness, are proved For this reason I contemplated not proving these results here, but in the end decided that they should be proved I did bring myself to relegate to the appendices the proofs of the representation of the dual of LP (Appendix B) and the dual of Co( X) (Appendix C) Chapter IV hits the bare essentials of the theory of locally convex spaces -enough to rationally discuss weak topologies It is shown in Section that the distributions are the dual of a locally convex space Chapter V treats the weak and weak-star topologies This is one of my favorite topics because of the numerous uses these ideas have Chapter VI looks at bounded linear operators on a Banach space Chapter VII introduces the reader to Banach algebras and spectral theory and applies this to the study of operators on a Banach space It is in Chapter VII that the reader needs to know the elements of analytic function theory, including Liouville's Theorem and Runge's Theorem (The latter is proved using the Hahn-Banach Theorem in Section IlLS.) When in Chapter VIII the notion of a C*-algebra is explored, the emphasis of the book becomes the theory of operators on a Hilbert space Chapter IX presents the Spectral Theorem and its ramifications This is done in the framework of a C*-algebra Classically, the Spectral Theorem has been thought of as a theorem about a single normal operator This it is, but it is more This theorem really tells us about the functional calculus for a normal operator and, hence, about the weakly closed C*-algebra generated by the normal operator In Section IX.S this approach culminates in the complete description of the functional calculus for a normal operator In Section IX.lO the multiplicity theory (a complete set of unitary invariants) for normal operators is worked out This topic is too often ignored in books on operator theory The ultimate goal of any branch of mathematics is to classify and characterize, and multiplicity theory achieves this goal for normal operators In Chapter X unbounded operators on Hilbert space are examined The distinction between symmetric and self-adjoint operators is carefully delineated and the Spectral Theorem for unbounded normal operators is obtained as a consequence of the bounded case Stone's Theorem on one parameter unitary groups is proved and the role of the Fourier transform in relating differentiation and multiplication is exhibited Chapter XI, which does not depend on Chapter X, proves the basic properties of the Fredholm index Though it is possible to this in the context of unbounded operators between two Banach spaces, this material is presented for bounded operators on a Hilbert space There are a few notational oddities The empty set is denoted by D A reference number such as (S.lO) means item number 10 in Section S of the present chapter The reference (IX.S.lO) is to (S.lO) in Chapter IX The reference (A.1.l) is to the first item in the first section of Appendix A x Preface There are many people who deserve my gratitude in connection with writing this book In three separate years I gave a course based on an evolving set of notes that eventually became transfigured into this book The students in those courses were a big help My colleague Grahame Bennett gave me several pointers in Banach spaces My ex-student Marc Raphael read final versions of the manuscript, pointing out mistakes and making suggestions for improvement Two current students, Alp Eden and Paul McGuire, read the galley proofs and were extremely helpful Elena Fraboschi typed the final manuscript John B Conway Bibliography 391 J B Conway [1981] Subnormal Operators Boston: Pitman R Courant and D Hilbert [1953] Methods of Mathematical Physics New York: In terscience A M Davie [1973] The approximation problem for Banach spaces Bull London Math Soc., 5, 261-266 A M Davie [1975] The Banach approximation problem Approx Theory, 13, 392-394 W J Davis, T Figel, W B Johnson, and A Pelczynski [1974] Factoring weakly compact operators J Functional Anal., 17, 311-327 L de Branges [1959] The Stone-Weierstrass Theorem Proc A mer Math Soc., 10, 822-824 W F Donoghue [1957] The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation Pacific J Math., 7, 1031-1035 R G Douglas [1969] On the operator equation S * XT = X and related topics Acta Sci Math (Szeged), 30,19-32 J Dugundji [1966] Topology Boston: Allyn and Bacon N Dunford and J Schwartz [1958] Linear Operators I New York: Interscience N Dunford and J Schwartz [1963] Linear Operators II New York: Interscience J Dyer, E Pedersen, and P Porcelli [1972] An equivalent formulation of the invariant subspace conjecture Bull Amer Math Soc., 78, 1020-1023 P Enflo [1973] A counterexample to the approximation problem in Banach spaces Acta Math., 130, 309-317 J Ernest [1976] Charting the operator terrain Memoirs Amer Math Soc., Vo! 71 P A Fillmore, J G Stampfli, and J P Williams [1972] On the essential numerical range, the essential spectrum, and a problem of Halmos Acta Sci Math (Szeged), 33, 179-192 B Fuglede [1950] A commutativity theorem for normal operators Proc Nat A cad Sci., 36, 35-40 T W Gamelin [1969] Uniform Algebras Englewood Cliffs: Prentice-Hall L Gillman and M Jerison [1960] Rings of Continuous Functions Princeton: Van Nostrand Reprinted by Springer-Verlag, New York F Greenleaf [1969] Invariant Means on Topological Groups New York: Van Nostrand A Grothendieck [1953] Sur les applications lineaires faiblement compactes d'espaces du type C(K) CanadianJ Math., 5,129-173 P R Halmos [1951] Introduction to Hilbert Space and the Theory of Spectral Multiplicity New York: Chelsea Pub! 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invariance of the essential spectrum under a change of boundary conditions of partial differential operators Indag Math., 21, 142-147 W Zelazko [1968] A characterization of multiplicative linear functionals in complex Banach algebras Studia Math., 30,83-85 List of Symbols o ix IF, IR, C (x,y) /2(1), /2 B(a; r) L~(G) A L A::; £, PAt 10 VA 11 L{h;: i E I} 16 [) 21 len) 22 f , EEl {£;: i E I} 24 !!4(£, f), !!4(£) 27 M 28,70 £~ EEl; A; 38 Ae % 38 AlA 40 op(A) 45 /OO(C) 58 A ~ 59 A ~ B 61 Cb(X), Co(X), C(X) 67 /00 (1), /00 68 fP(1), fP 68 Co 68 c(n)[O,l] 68 396 List of Symbols ~n[o, 1] 68 c 69 Cc(X) 69 spt I 69 PA(!!f, c?9'), PA(!!f) 70 ffip!!f" ffio!!fn 74 !!f* 77 LOO(M(X)) 79 A(K) 84 P 87 vf{.L 91 graA 94 C(X), H(G) 104 a(!!f, !!f*), a(!!f*,!!f) 104 [a, b] 104 co(A), co(A) 104 s 107 cl" 114 C«OO)(Q), '@(Jf"), '@(Q) 120 'T1!!f 121 (x, x*), (x*, x) 127 wk, a(!!f, !!f*) 127 wk*, a(!!f*,!!f) 128 cl A, wk-cl A 128 AO, °B, DAD 129 A.L, L B 130 ball !!f 134 ~x 140 f3X 141 ext K 145 j 149 hf-L 149 lx' xl, 1# c?9" 159, 229 170 A* 171 PAo(!!f, c?9'), PAo(!!f), !JBoo(!!f, OJ/) 178 Lat T 182 L l( G) 194, 228 a(a), (J,(a), (Jr(a) 199 p(a), p,(a), Pr(a) 199 r(a) 201 n(y;a) 203 ins y, out y 204 ins r, out r 204 Hol(a) 206 List of Symbols a ,.(a) 210 lIillA 211 K'211 aap(A) 213 E(Ll) = E(Ll; A), ?It 215 E(A),?IA 215 2:, 2: ( a) 224 rad d 225 7# 225 Il 228 G 232 C*(a) 241 f(a) 243 Red 245 d+ 246 a+, a_ 246 a :$ b 246 IAI 248 Yl'(n), A(n), 7T(n) 254 S , 258 Eg,h 263 f dE 264 B(X, Q) 265 N", 271 ess-ran(