REAL AND COMPLEX ANALYSIS

1 real AND COMPLEX ANALYSIS real AND COMPLEX ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison ) McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogota Hamburg London Madrid Mexico Milan Montreal New Delhi Panama Paris Sao Paulo Singapore Sydney Tokyo Toronto real AND COMPLEX ANALYSIS INTERNATIONAL EDITION 1987 Exclusive rights by McGraw-Hill Book Co., Singapore for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. 0123 45678920 BJE9876 Copyright 1987, 1974, 1966 by McGraw-Hill, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or a retrieval system, without the prior written permission of the publisher.

2 This book was set in Times Roman. The editor was Peter R. Devine. The production supervisor was Diane Renda. Library of Congress Cataloging-in-Publication Data Rudin, Walter, 1921 - real and COMPLEX ANALYSIS . Bibliography: p. Includes index. 1. Mathematical ANALYSIS . I. Title. 1987 515 86-7 ISBN 0-07-054234-1 When ordering this title use ISBN 0-07-100276-6 Printed in Singapore ABOUT THE AUTHOR Walter Rudin is the author of three textbooks, Principles of Mathematical ANALYSIS , real and COMPLEX ANALYSIS , and Functional ANALYSIS , whose widespread use is illustrated by the fact that they have been translated into a total of 13 languages. He wrote the first of these while he was a Moore Instructor at , just two years after receiving his at Duke University in 1949.

3 Later he taught at the University of Rochester, and is now a Vilas Research Professor at the University of Wisconsin-Madison, where he has been since 1959. He has spent leaves at Yale University, at the University of California in La Jolla, and at the University of Hawaii. His research has dealt mainly with harmonic ANALYSIS and with COMPLEX vari ables. He has written three research monographs on these topics, Fourier ANALYSIS on Groups, Function Theory in Polydiscs, and Function Theory in the Unit Ball ofC0 CONTENTS Preface Xlll Prologue: The Exponential Function 1 Chapter 1 Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, oo]

4 Integration of positive functions Integration of COMPLEX functions The role played by sets of measure zero Exercises Chapter 2 Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3 LP-Spaces Convex functions and inequalities The LP -spaces Approximation by continuous functions Exercises 5 6 8 15 16 18 19 24 27 31 33 33 35 40 47 49 55 57 61 61 65 69 71 vii viii CONTENTS Chapter 4 Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5 Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L 1-functions The Hahn-Banaeh theorem An abstract ap6roach to the Poisson integral Exercises Chapter 6 COMPLEX Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on LP The Riesz representation theorem Exercises Chapter 7 Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8 Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9 Fourier Transforms Formal properties

5 The inversion theorem The Plancherel theorem The Banach algebra L 1 Exercises 76 76 82 88 92 95 95 97 100 103 104 108 112 116 116 120 124 126 129 132 135 135 144 150 156 160 160 163 164 167 170 172 174 178 178 180 185 190 193 Chapter 10 Elementary Properties of Holomorphic Functions COMPLEX differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11 Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12 The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13 Approximation by Rational Functions Preparation Runge's theotem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14 Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class f/ Continuity at the boundary Conformal mapping of an annulus Exercises CONTENTS ix 196 196 200 204 208 214 217 224 227 231 231 233 237 239 245 249 253 253 254 256 260 262 264 266 266 270 273 274 276 278 278 279 281 282 285 289 291 293 X CONTENTS Chapter 15 Zeros of Holomorphic Functions Infinite products The Weierstrass factorization theorem An

6 Interpolation problem Jensen's formula Blaschke products The Mtintz-Szasz theorem Exercises Chapter 16 Analytic Continuation Regular points and singular points Continuation along curves The monodro y theorem Construction of a modular function The Picard theorem Exercises Chapter 17 HP-Spaces Subharmonic functions The spaces HP and N The theorem of F. and M. Riesz Factorization "theorems The shift operator Conjugate functions Exercises Chapter 18 Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19 Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20 Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises 298 298 301 304 307 310 312 315 319 319 323 326 328 331 332 335 335 337 341 342 346 350 352 356 356 357 362 365 369 371 371 372 377 380 383 386 386 387 390 394 CONTENTS xi Appendix.

7 Hausdorff's Maximality Theorem 395 Notes and Comments 397 Bibliography 405 List of Special Symbols 407 Index 409 PREFACE This book contains a first-year graduate course in which the basic techniques and theorems of ANALYSIS are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of " real ANALYSIS " and " COMPLEX ANALYSIS " are thus united; some of the basic ideas from functional a alysis are also included. Here are some examples of the way in which these connections are demon strated and exploited. The Riesz representation theorem and the Hahn-Banach theorem allow one to "guess" the Poisson integral formula. They team up in the proof of Runge's theorem.

8 They combine with Blaschke's theorem on the zeros of bounded holomorphic functions to give a proof of the Miintz-Szasz theorem, which concerns approximation on an interval. The fact that 13 is a Hilbert space is used in the proof of the Radon-Nikodym theorem, which leads to the theorem about differentiation of indefinite integrals, which in turn yields the existence of radial limits of bounded harmonic functions. The theorems of Plancherel and Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in the Denjoy-Carleman theorem about infinitely differentiable functions on the real line. The maximum modulus theorem gives information about linear transform ations on I! -spaces. Since most of the results presented here are quite classical (the novelty lies in the arrangement, and some of the proofs are new), I have not attempted to docu ment the sour ce of every item.

9 References are gathered at the end, in Notes and Comments. They are not always to the original sources, but more of ten to more recent works where further references can be fo und. In no case does the absence of a ref erence imply any claim to originality on my part. The prerequisite fo r this book is a good course in advanced calculus (settheoretic manipulations, metric spaces, uniform continuity, and uniform convergence). The first seven chapters of my earlier book "Principles of Mathe matical ANALYSIS " furnish sufficient preparation. xiii xiv PREFACE Experience with the first edition shows that fir st-year graduate students can study the first 15 chapters in two semesters, plus some topics from 1 or 2 of the remaining 5.

10 These latter are quite independent of each other. The first 15 should be taken up in the order in which they are presented, except for Chapter 9, which can be postponed. The most important difference between this third edition and the previous ones is the entirely new chapter on differentiation. The basic facts about differen tiation are now derived fro the existence of Lebesgue points, which in turn is an easy consequence of the so-called "weak type" inequality that is satisfied by the maximal functions of measures on euclidean spaces. This approach yields strong theorems with minimal effort. Even more important is that it familiarizes stu dents with maximal functions, since these have become increasingly useful in several areas of ANALYSIS .