Linear functional analysis PDF
Preview Linear functional analysis
Linear Functional Analysis Joan Cerdà Graduate Studies in Mathematics Volume 116 American Mathematical Society Real Sociedad Matemática Española 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 Espan˜ola Guillermo P. Curbera, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Rosa Mar´ıa Miro´ Bernardo Cascales Pablo Pedregal Javier Duoandikoetxea Juan Soler 2010 Mathematics Subject Classification. Primary 46–01;Secondary 46Axx, 46Bxx, 46Exx, 46Fxx, 46Jxx, 47B15. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-116 Library of Congress Cataloging-in-Publication Data Cerd`a,Joan,1942– Linearfunctionalanalysis/JoanCerda`. p.cm. —(Graduatestudiesinmathematics;v.116) Includesbibliographicalreferencesandindex. ISBN978-0-8218-5115-9(alk.paper) 1.Functionalanalysis. I.Title. QA321.C47 2010 515(cid:2).7—dc22 2010006449 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Requests for such permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by [email protected]. (cid:2)c 2010bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://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
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