Rudin w real and complex analysis

McGRAW-HILL SERIES IN HIGHER MATHEMATICS E IP Spanier Conrrulting Editor I Ausbnder and MaeKeruie Introduction to Differentiable Manifolds Foundations of Mathematical Logic Curry Goldberg Unbounded Linear Operators Guggenheirner Differential Geometry Rogers ( Theory of Recursive Functions and Effective Computability Rudin Real and Complex AnaIysia Spanier Algebraic Topology Valentine I Con&ta I I I Real and Complex Analysis W a l t e r Hudla Professor of Mathematics University of Wisconsin International Student Edition McGRAW-HILL London N e w York Sydney Toronto Dusseldorf Mexico Johannesburg Panama Singapore MLADINSKA KNJIGA Ljubljana REAL AND COMPLEX ANALYSIS International Student Edition Exclusive rights by McGraw-Hill Publishing Company Limited anc hhadinska Knjiga for manufacture and export from Yugoslavia This book cannot be re-exported from the country t o which it is consigned by McGraw-Hill Publishing Company Limited or by Mlaalns~r Knjiga or by McGraw-Hill Book Company or any of its subsidiaries Copyright @ 1970 by McGraw-Hill Inc All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, recording or otherwise, without the prior permission of the publisher Library of Congress Catalog Card Number 65-27982 Printed and bound by MLADiNSKA KNJIGA, LJUBWANA, YUGOSLAVIA I n this book I present an analysis course which I have t a w to first+ yem graduate students at the Univereity of Wisconsin since 1962 The course was developed for two reasons The first was a belief that one could present the basic techniques and theorems of analysis in one year, with enough applications to make the subject interesting, in such a way that students could then specialize in any direction they choose The second and perhaps even more important one was the desire to away with the outmoded and misleading idea that analysis consists of two distinct halves, "real variables" and "complex variables.'' Traditionally (with some oversimplification) the first of these deals with Lebesgue integration, with various types of convergence, and with the pathologies exhibited by very discontinuous functions; whereas the second one concerns itself only with those functions that are a s smooth rts can be, namely, the holomorphic ones That these two areas interact most intimately has of course been well known for at least 60 years and is evident to anyone who is acquainted with current research Nevertheless, the standard curriculum in most American universities still contains a year course in complex variables, followed by a year course in real variables, and usually neither of these courses acknowledges the existence of the subject matter of the other I have made an effort to demonstrate the interplay among the various parts of analysis, including some of the basic ideas from functional analysis Here are a few examples 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, from which the homol6gy version of Cauchy's theorem follows easily 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 fsct that LZis a Hilbert space is used in the proof of the W o n - N i i y m theorem, which leads to the theorem ,about differentiation of indefinite integrals (incidentally, daerentiation seems to be unduly slighted in most modern texts), which in turn yields the v vi Preface 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 lime The maximum modulus theorem gives information about linear transformations on Lp-spsces 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 document the source of every item References are gathered at the end, in Notes and Comments They are not always to the original sources, but more often to more recent works where further references can be found I n no case does the absence of a reference imply any claim to originality on my part The prerequisite for this book is a good course in advanced calcuIus (set-theoretic manipulations, metric spaces, uniform continuity, and uniform convergence) The first seven chapters of my earlier book "Principles of Mathematical A d y s i s " furnish s m c i e n t preparation Chapters to and 10 to 15 should be taken up in the order in which they are presented Chapter is not referred to again until Chapter 19 The last five chapters are quite independent of each other, and probably not all of them should be taken up in any one year There are over 350 problems, some quite easy, some more challenging About half of these have been -signed to my classes a t various times The students' response to this course baa been most gratifying, and I have profited much from some of their comments Notes taken by' Aaron Strauss and Stephen Fisher helped me greatly in the writing of the final manuscript The text contains a number of improvements which were suggested by Howard Conner, Simon Hellerstein, Marvin Knopp, and E L Stout I t is a pleasure to express my sincere thanks to them for their generous assistance Contents I Prologue The Exponential Function, Chapter I Abstract Integration, Set-theoretic notations and terminology, The concept of measurability, Simple functions, 15 Elementary properties of measures, 16 Arithmetic in [O, oo j, 18 Integration of positive functions, 19 Integration of complex functions, 24 The role played by seta of measure zero, 26 Exercises, 31 Chapter Positive Borel Measures, 33 Vector spaces, 33 Topological preliminaries, 35 The Riesz representation theorem, 40 Regularity properties of Borel measures, 47 Lebesgue measure, 49 Continuity properties of measurable functions, 53 Exercises, 56 Chapter Lp-Spaces, 60 Convex functions and inequalities, 60 The L~?-apaces, 64 vii Approximatioh by c ~ n t i n u o ufunctions, 68 Exercises, 70 Chapter I Elementary Hilbert Space Theory, 75 Inner products and linear functiods, 75 Orthonormal sets, 81 Trigonometric series, 88 Exercises, 92 Chapter Examples of Bansch Space Techniques, Banach spaces, 95 Consequences of Baire's theorem, 97 Fourier series of continuous functions, 101 Fourier coefficients of LLfunctions, 103 The Hahn-Bmmh theorem, 105 An abstract approach to the Poisson integral, 109 Exercises, 112 Chapter I Complex Measures, 117 Total variation, 117 Absolute continuity, 121 Consequences of the Radon-Nikodym theorem, 126 Bounded linear functionals on LP, 127 The Riesz representation theorem, 130 Exercises, 133 Chapter I Integration on Product Spaces, Measurability on cartesian products, 136 Product mewures, 138 The Fubini theorem, 140 Completion of product measures, 143 Convolutions, 146 Exercises, 148 Chapter I Differentiation, 151 Derivatives of measures, 151 Functions of bounded variation, 160 Differentiation of point functions, 165 136 95 Contents Differentiable transformations, 169 Exercises, 175 Chapter Fourier Transforms, 180 Formal properties, 180 The inversion theorem, 182 The Plancherel theorem, 187 The Banach algebra L1, 192 Exercises, 195 Chapter 10 Elementary Properties of Holomorphic Functions, 198 Complex differentiation, 198 Integration over paths, 202 The Cauchy theorem, 206 The power series representation, 209 The open mapping theorem, 214 Exercises, 219 Chapter 11 Harmonic Functions, 222 The Cauchy-Riemann equations, 222 The Poisson integral, 223 The mean value property, 230 Positive harmonic functions, 232 Exercises, 236 Chapter 12 The Maximum Modulus Principle, 240 Introduction, 240 The Schwarz lemma, 240 The Phragmen-Lindeliif method, 243 An interpolation theorem, 246 converse of the maximum modulus theorem, 248 Exercises, 249 Chapter 13 Approximation by Rational Functions, Preparation, 252 Runge's theorem, 255 Cauchy's theorem, 259 Simply connected regions, 262 Exercises, 265 252 I Chapter 14 Conformal Mapping, 268 Preservation of angles, 268 Linear fractional transformations, 269 Normal families, 271 The Riemann mapping theorem, 273 The claas S, 276 Continuity at the boundary, 279 Conformal mapping of an annulus, 282 Exercises, 284 Chapter 15 Zeros of Holomorphic Functions, 290 Infinite products, 290 The Weierstraas fsctorization theorem, 293 The Mittag-Leffler theorem, 296 Jensen's formula, 299 Blaachke products, 302 The MtIntz~Szaslztheorem, 304 Exercises, 307 I Chapter 16 Analytic Continuation, 312 Regular points and singular points, 312 Continuation along curves, 316 The monodromy theorem, 319 Construction of a modular function, 320 The Picard theorem, 324 Exercises, 325 Chapter 17 Hp-Spaces, 328 Subharmonic functions, 328 The spaces H9 and N, 330 The space H1, 332 The theorem of F: and M Riesz, 335 Factorisation theorems, 336 f he shift operator, 341 Conjugate functions, 345 Exercises, 347 Chapter 18 Elemenqary Theory of Banach Algebras, Introduction, 351 The invertible elements, 352 351 Contents Ideals and homamorphisms, 357 Applications, 360 Exercises, 364 Chapter 19 Holomorphic Fourier Transforms, 361 Introduction, 367 Two theorems of Paley and Wiener, 368 Quasi-analytic clsssea, 372 The Denjoy-Carleman theorem, 376 Exercises, 379 Chapter 20 Uniform Approximation by Polynomials, introduction, 382 Some lemmas, 383 Mergelym's theorem, 386 Exercises, 390 I Appendix Hausdorffs Maximali ty Theorem, 391 Notes and Comments, 393 Bibliography, 401 List of Special Symbols, 403 Index, 405 382 Notes and comments 399 Sec 18.20 This theorem is Wermer's, Proc Am Math Soc., vol 4, pp 866869, 1953 The proof of the text is due to Hoffman and Singer See j15], pp, 93-94, where an extremely short proof by P J Cohen is also given (See the reference to Sec 18.18.) Sec 18.21 This was one of the major steps in Wiener's original proof of his Tauberian theorem See [33], p 91 The painless proof given in the text was the first spectacular success of the Gelfand theory Exercise 14 The set A can be given a compact Hausdorff topology with respect to which the functipns i are continuous Thus x -,f is a homomorphism of,A into C(A) This representation of A as an algebra of continuous functions is a most important tool in the study of commutative Banach algebras Chapter 19 Secs 19.2, 19.3: [21], pp, 1-13 See also [3], where functions of exponential type are the main subject Sec 19.5 For a more detailed introduction to the classes C( M , } , see S Mandelbrojt, "S8ries de Fourier et classes quasi-analytiques," Gauthier-Villars, Paris, 1935 Sec 19.11 In [21], the proof of this theorem is based on Theorem 19.2 rather than on 19.3 Exercise The function @ is called the Borel transform off See 131, chap Exercise 12 The suggested proof is due to H Mirkil, Proc Am Math Soc., vol 7, pp 650-652, 1956 The theorem was proved by Borel in 1895 Chapter 20 See S, N Mergelyan, Uniform Approximations to Functions of a Complex Variable, Usephi Mat Nauk (N.S.) 7, no (48), 31-122, 1952; Amer Math Soc Translation No 101, 1954 Our Theorem 20.5 is Theorem 1.4 in Mergelyan's paper A functional analysis proof, based on measure-theoretic considerations, has recently been published by L Carleson in Math Scandinavica, vol 15, pp 167-175, 1964 Appendix The maximality theorem was first stated by Hausdorff on p 140 of his book "Grundziige der Mengenlehre," 1914 The proof of the text is patterned after section 16 of Halmos's book 181 The idea to choose g so that g(A) - A has at most one element appears there, as does the term "tower." The proof is similar to one of Zermelo's proofs of the well-ordering theorem; see Math Ann,, vol 65, pp 107-128, 1908 Bibliography L V Ahlfors: "CompIex Analysis," 2d ed., McGraw-Hill Book Company, York, 1966 2, S Banach: Thhorie des Opbrations linbaires, "Monografje Matematyczne," voI 1, Warsaw, 1932 R P Boas: "Entire Functions," Academic Press Inc., New York, 1954 C Carathdodory: "Theory of Functions of a Complex Variable," Chelsea Publishing Company, New York, 1954 N Dunford and J T Schwartz: "Linear Operators," Interscience PubIishers, Inc., New York, 1958 P R Halmos: "Introduction to Kilbert Space and the Theory of Spectral Multiplicity,'' Chelsea Publishing Company, New York, 1951 7, P R Halmos: "Measure Theory," D Van Nostrand Company, Inc., Princeton, N.J., 1950 P R Halmos: "Naive Set Theory," D Van Nostrand Company, Inc., Princeton, N J., 1960 G H Hardy, J E Littlewood, and G PBlya: "Inequalities," Cambridge University Press, New York, 1934 10 G H Hardy and W W Rogosinski: "Fourier Series," Cambridge Tracts no 38, Cambridge, London, and New York, 1950 11 H Helson: "Lectures on Invariant Subspaces," Academic Press Inc., New York, 1964 12 E Hewitt and K A Ross: Abstract Harmonic Analysis," Springer-Verlag OHG, Berlin, 1963 13 E HiUe: "AnaIytic Function Theory," Ginn and Company, Boston, voI I, 1959; vol 11, 1962 14 E Hi2le and R S Phillips: "Functional Analysis and Semigroups," Amer Math Sac Colloquium Publ 31, Providence, 1957 15 K Hofman: "Banach Spaces of AnaIytic Functions," Prentice-HaI1, Inc., Englewood Cliffs, N J., 1962 16 H Kestelman: "Modern Theories of Integration," Oxford University Press, New York, 1937 17 L H Loomis: "An Introduction to Abstract Harmonic Analysis," D Van Nostrand Company, Inc., Princeton, 'N.J., 1953 41 402 Real and complex analysis 18 E J McShane: "Integration,)' Princeton University Press, Princeton, N, J., 1944 19 M A Naimark: "Normed Rings," Erven P Noordhoff, NV, Groningen, Netherlands, 1959 20 Nehari: "Conformal Mapping," McGraw-Hi11 Book Company, New York, 1952 21 R E A C Paley and N Wiener: "Fourier Transforms in the Complex Domain," Amer Math Soc Colloquium Publ 19, New York, 1934 22 T E~cdo:Subharmonic F'unctions, Ergeb Math., vol 5, no 1, BerIin, 1937 23 C E, Rickart: "General Theory of Banach Algebras," D Van Nostrand Company, Inc., Princeton, N.J., 1960 24 F Riesz and lk 82.-Nagy: C%eqons d7Analyse Fonc tionnelle," Akademiai Kiad6, Budapest, 1952 25 H L R o y h : "Real Anaysis," The MacmilIan Company, New York, 1963 26 W R d i n : "Principles of Mathematical Analysis,'' 2d ed., McGraw-Hill Book Company, New York, 1964 27 W budin: "Fourier Analysis on Groups," Interscience Publishers, Inc., New York, 1962 28 Saks: "Theory of the Integral," 2d ed., "Monografje Matematyczne," vol 7, Warsaw, 1937 Reprinted by Hafner Publishing Company, Inc., New York 29 S Saks and A Zygmund: "AnaIytic Functions," "Monografje Matematyczne," voI 28, Warsaw, 1952 30 G Springer: "Introduction to Riemann Surfaces, " Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957 31 E C Titchmarsh: "The Theory of Functions," 2d ed., Oxford University Press, Fair Lawn, N.J., 1939 32 H Weyl: "The Concept of a R i e h n n Surface," 3d ed., Addison-Wesley Publishing Company, lnc., Reading, Mass., 1964 33 N Wiener: "The Fourier IntegraI and Certain of Its Applications," Cambridge University Press, New York, 1933 Reprinted by Dover PubIications, Inc., New York 34 G T Whyburn: "TopologicaI Analysis, " 2d ed., Princeton University Press, Princeton, N.J., 1964 35 J H Williamson: "Le besgue Integration," Halt , Rinehart a.nd Winst on, Inc,, New York, 1962 36 A Zygmund: "Trigonometric Series," 2d ed., Cambridge University Press, New York, 1959 List of Special Symbols a n d Abbreviationst X I lim sup lim inf The atmderd set-theoretic symbols are described on page8 and and are not listed here 403 U P,(e - t ) IPl(E) P+, CL- X