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Linear Functional Analysis Joan Cerdà Graduate Studies in Mathematics Volume 116 American Mathematical Society Real Sociedad Matemática Española http://dx.doi.org/10.1090/gsm/116 Linear Functional Analysis Linear Functional Analysis Joan Cerdà Graduate Studies in Mathematics Volume 116 American Mathematical Society Providence, Rhode Island Real Sociedad Matemática Española Madrid, Spain Editorial Board of Graduate Studies in Mathematics David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani Editorial Committee of the Real Sociedad Matem´atica Espa˜nola Guillermo P. Curbera, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Rosa Mar´ıa Mir´o Bernardo Cascales Pablo Pedregal Javier Duoandikoetxea Juan Soler 2010 Mathematics Subject Classification. Primary 46–01; Secondary 46Axx, 46Bxx, 46Exx, 46Fxx, 46Jxx, 47B15. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-116 Library of Congress Cataloging-in-Publication Data Cerd`a, Joan, 1942– Linear functional analysis / Joan Cerd`a. p. cm. — (Graduate studies in mathematics ; v. 116) Includes bibliographical references and index. ISBN 978-0-8218-5115-9 (alk. paper) 1. Functional analysis. I. Title. QA321.C47 2010 515.7—dc22 2010006449 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To Carla and Marc Contents Preface xi Chapter 1. Introduction 1 §1.1. Topological spaces 1 §1.2. Measure and integration 8 §1.3. Exercises 21 Chapter 2. Normed spaces and operators 25 §2.1. Banach spaces 26 §2.2. Linear operators 39 §2.3. Hilbert spaces 45 §2.4. Convolutions and summability kernels 52 §2.5. The Riesz-Thorin interpolation theorem 59 §2.6. Applications to linear differential equations 63 §2.7. Exercises 69 Chapter 3. Fr´echet spaces and Banach theorems 75 §3.1. Fr´echet spaces 76 §3.2. Banach theorems 82 §3.3. Exercises 88 Chapter 4. Duality 93 §4.1. The dual of a Hilbert space 93 §4.2. Applications of the Riesz representation theorem 98 §4.3. The Hahn-Banach theorem 106 vii viii Contents §4.4. Spectral theory of compact operators 114 §4.5. Exercises 122 Chapter 5. Weak topologies 127 §5.1. Weak convergence 127 §5.2. Weak and weak* topologies 128 §5.3. An application to the Dirichlet problem in the disc 132 §5.4. Exercises 138 Chapter 6. Distributions 143 §6.1. Test functions 144 §6.2. The distributions 146 §6.3. Differentiation of distributions 150 §6.4. Convolution of distributions 154 §6.5. Distributional differential equations 161 §6.6. Exercises 175 Chapter 7. Fourier transform and Sobolev spaces 181 §7.1. The Fourier integral 182 §7.2. The Schwartz class S 186 §7.3. Tempered distributions 189 §7.4. Fourier transform and signal theory 195 §7.5. The Dirichlet problem in the half-space 200 §7.6. Sobolev spaces 206 §7.7. Applications 213 §7.8. Exercises 222 Chapter 8. Banach algebras 227 §8.1. Definition and examples 228 §8.2. Spectrum 229 §8.3. Commutative Banach algebras 234 §8.4. C∗-algebras 238 §8.5. Spectral theory of bounded normal operators 241 §8.6. Exercises 250 Chapter 9. Unbounded operators in a Hilbert space 257 §9.1. Definitions and basic properties 258 §9.2. Unbounded self-adjoint operators 262 Contents ix §9.3. Spectral representation of unbounded self-adjoint operators 273 §9.4. Unbounded operators in quantum mechanics 277 §9.5. Appendix: Proof of the spectral theorem 287 §9.6. Exercises 295 Hints to exercises 299 Bibliography 321 Index 325 Preface The aim of this book is to present the basic facts of linear functional anal- ysis related to applications to some fundamental aspects of mathematical analysis. If mathematics is supposed to show common general facts and struc- tures of particular results, functional analysis does this while dealing with classical problems, many of them related to ordinary and partial differential equations, integral equations, harmonic analysis, function theory, and the calculus of variations. In functional analysis, individual functions satisfying specific equations are replaced by classes of functions and transforms which are determined by each particular problem. The objects of functional analysis are spaces and operators acting between them which, after systematic studies intertwining linear and topological or metric structures, appear to be behind classical problems in a kind of cleaning process. In order to make the scope of functional analysis clearer, I have chosen to sacrifice generality for the sake of an easier understanding of its methods, and to show how they clarify what is essential in analytical problems. I have tried to avoid the introduction of cold abstractions and unnecessary terminology in further developments and, when choosing the different topics, I have included some applications that connect functional analysis with other areas. The text is based on a graduate course taught at the Universitat de Barcelona, with some additions, mainly to make it more self-contained. The material in the first chapters could be adapted as an introductory course on functional analysis, aiming to present the role of duality in analysis, and xi xii Preface also the spectral theory of compact linear operators in the context of Hilbert and Banach spaces. In this first part of the book, the mutual influence between functional analysis and other areas of analysis is shown when studying duality, with von Neumann’s proof of the Radon-Nikodym theorem based on the Riesz representation theorem for the dual of a Hilbert space, followed by the rep- resentations of the duals of the Lp spaces and of C(K), in this case by means of complex Borel measures. The reader will also see how to deal with initial and boundary value problems in ordinary linear differential equations via the use of integral operators. Moreover examples are included that illustrate how functional analytic methods are useful in the study of Fourier series. In the second part, distributions provide a natural framework extend- ing some fundamental operations in analysis. Convolution and the Fourier transform are included as useful tools for dealing with partial differential operators, with basic notions such as fundamental solutions and Green’s functions. Distributions are also appropriate for the introduction of Sobolev spaces, which are very useful for the study of the solutions of partial differential equations. A clear example is provided by the resolution of the Dirichlet problem and the description of the eigenvalues of the Laplacian, in combi- nation with Hilbert space techniques. The last two chapters are essentially devoted to the spectral theory of bounded and unbounded self-adjoint operators, which is presented by us- ing the Gelfand transform for Banach algebras. This spectral theory is illustrated with an introduction to the basic axioms of quantum mechanics, which motivated many studies in the Hilbert space theory. Some very short historical comments have been included, mainly by means of footnotes. For a good overview of the evolution of functional analysis, J. Dieudonn´e’s and A. F. Monna’s books, [10] and [31], are two good references. The limitation of space has forced us to leave out many other important topics that could, and probably should, have been included. Among them are the geometry of Banach spaces, a general theory of locally convex spaces and structure theory of Fr´echet spaces, functional calculus of nonnormal operators, groups and semigroups of operators, invariant subspaces, index theory, von Neumann algebras, and scattering theory. Fortunately, many excellent texts dealing with these subjects are available and a few references have been selected for further study. Preface xiii A small number of references have been gathered at the end of each chapter to focus the reader’s attention on some appropriate items from a general bibliographical list of 44 items. Almost 240 exercises are gathered at the end of the chapters and form an important part of the book. They are intended to help the reader to develop techniques and working knowledge of functional analysis. These exercises are highly nonuniform in difficulty. Some are very simple, to aid in better understanding of the concepts employed, whereas others are fairly challenging for the beginners. Hints and solutions are provided at the end of the book. The prerequisites are very standard. Although it is assumed that the reader has some a priori knowledge of general topology, integral calculus with Lebesgue measure, and elementary aspects of normed or Hilbert spaces, a review of the basic aspects of these topics has been included in the first chapters. I turn finally to the pleasant task of thanking those who helped me during the writing. Particular thanks are due to Javier Soria, who revised most of the manuscript and proposed important corrections and suggestions. I have also received valuable advice and criticism from Mar´ıa J. Carro and Joaquim Ortega-Cerd`a. I have been very fortunate to have received their assistance.
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