Real and complex analysis / Walter Rudin.

Cover title: Real & complex analysis.

Includes index.

Bibliography: p. 405-406.

Contents: Preface xiii Prologue: The Exponential Function i Chapter 1 Abstract Integration 5 Set-theoretic notations and terminology 6 The concept of measurability 8 Simple functions 15 Elementary properties of measures 16 Arithmetic in [0, oo] 18 Integration of positive functions 19 Integration of complex functions 24 The role played by sets of measure zero 27 Exercises 31 Chapter 2 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 55 Exercises 57 Chapter 3 Lp-Spaces 61 Convex functions and inequalities 61 The Lp~spaces 65 Approximation by continuous functions 69 Exercises 71 Chapter 4 Elementary Hilbert Space Theory 76 Inner products and linear functionals 76 Orthonormal sets 82 Trigonometric series 88 Exercises 92 Chapter 5 Examples of Banach Space Techniques 95 Banach spaces 95 Consequences of Baire's theorem 97 Fourier series of continuous functions 100 Fourier coefficients of Zj-functions 103 The Hahn-Banach theorem 104 An abstract approach to the Poisson integral 108 Exercises 112 Chapter 6 Complex Measures 116 Total variation 116 Absolute continuity 120 Consequences of the Radon-Nikodym theorem 124 Bounded linear functionals on Lp 126 The Riesz representation theorem 129 Exercises 132 Chapter 7 Differentiation 135 Derivatives of measures 135 The fundamental theorem of Calculus 144 Differentiable transformations 150 Exercises 156 Chapter 8 Integration on Product Spaces 160 Measurability on cartesian products 160 Product measures 163 The Fubini theorem 164 Completion of product measures 167 Convolutions 170 Distribution functions 172 Exercises 174 Chapter 9 Fourier Transforms 178 Formal properties 178 The inversion theorem 180 The Plancherel theorem 185 The Banach algebra Ll 190 Exercises 193 Chapter 10 Elementary Properties of Holomorphic Functions 196 Complex •differentiation 196 Integration over paths 200 The local Cauchy theorem 204 The power series representation 208 The open mapping theorem 214 The global Cauchy theorem 217 The calculus of residues 224 Exercises 227 Chapter 11 Harmonic Functions 231 The Cauchy-Riemann equations 231 The Poisson integral 233 The mean value property 237 Boundary behavior of Poisson integrals 239 Representation theorems 245 Exercises 249 Chapter 12 The Maximum Modulus Principle 253 Introduction 253 The Schwarz lemma 254 The Phragmen-Lindelof method 256 An interpolation theorem 260 A converse of the maximum modulus theorem 262 Exercises 264 Chapter 13 Approximation by Rational Functions 266 Preparation 266 Runge's theorem 270 The Mittag-Leffler theorem 273 Simply connected regions 274 Exercises 276 Chapter 14 Conformal Mapping 278 Preservation of angles 278 Linear fractional transformations 279 Normal families 281 The Riemann mapping theorem 282 The class Sf 285 Continuity at the boundary 289 Conformal mapping of an annulus 291 Exercises 293 Chapter 15 Zeros of Holomorphic Functions 298 Infinite products 298 The Weierstrass factorization theorem 301 An interpolation problem 304 Jensen's formula 307 Blaschke products 310 The Miintz-Szasz theorem 312 Exercises 315 Chapter 16 Analytic Continuation Regular points and singular points 319 Continuation along curves 323 The monodromy theorem 326 Construction of a modular function 328 The Picard theorem 331 Exercises 332 Chapter 17 Hp-Spaces 335 Subharmonic functions 335 The spaces Hp and N 337 The theorem of F. and M. Riesz 341 Factorization theorems 342 The shift operator 346 Conjugate functions 350 Exercises 352 Chapter 18 Elementary Theory of Banach Algebras 356 Introduction 356 The invertible elements 357 Ideals and homomorphisms 362 Applications 365 Exercises 369 Chapter 19 Holomorphic Fourier Transforms 371 Introduction 371 Two theorems of Paley and Wiener 372 Quasi-analytic classes 377 The Denjoy-Carleman theorem 380 Exercises 383 Chapter 20 Uniform Approximation by Polynomials 386 Introduction 386 Some lemmas 387 Mergelyan's theorem 390 Exercises 394 Appendix: Hausdorff's Maximality Theorem 395 Notes and Comments 397 Bibliography 405 List of Special Symbols 407 Index 409.

Summary : This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. 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 in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.