实分析与复分析 第3版 英文版PDF电子书下载

Prologue:The Exponential Function 1

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,∞] 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 L1-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

Derivatives of measures 135

Chapter 7 Differentiation 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 L1 190

Exercises 193

Complex differentiation 196

Chapter 10 Elementary Properties of Holomorphic Functions 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

The Cauchy-Riemann equations 231

Chapter 11 Harmonic Functions 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 Phragrnen-Lindel?f 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 ? 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 Müntz-Szasz theorem 312

Exercises 315

Regular points and singular points 319

Chapter 16 Analytic Continuation 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 s?aces 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

Introduction 386

chapter 20 Uniform Approximation by Polynomials 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