实分析与复分析 第3版 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

实分析与复分析 第3版 英文版PDF格式电子书版下载

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Prologue:The Exponential Function1

Chapter 1 Abstract Integration5

Set-theoretic notations and terminology6

The concept of measurability8

Simple functions15

Elementary properties of measures16

Arithmetic in [0,∞]18

Integration of positive functions19

Integration of complex functions24

The role played by sets of measure zero27

Exercises31

Chapter 2 Positive Borel Measures33

Vector spaces33

Topological preliminaries35

The Riesz representation theorem40

Regularity properties of Borel measures47

Lebesgue measure49

Continuity properties of measurable functions55

Exercises57

Chapter 3 LP-Spaces61

Convex functions and inequalities61

The LP-spaces65

Approximation by continuous functions69

Exercises71

Chapter 4 Elementary Hilbert Space Theory76

Inner products and linear functionals76

Orthonormal sets82

Trigonometric series88

Exercises92

Chapter 5 Examples of Banach Space Techniques95

Banach spaces95

Consequences of Baire s theorem97

Fourier series of continuous functions100

Fourier coefficients of L1-functions103

The Hahn-Banach theorem104

An abstract approach to the Poisson integral108

Exercises112

Chapter 6 Complex Measures116

Total variation116

Absolute continuity120

Consequences of the Radon-Nikodym theorem124

Bounded linear functionals on Lp126

The Riesz representation theorem129

Exercises132

Derivatives of measures135

Chapter 7 Differentiation135

The fundamental theorem of Calculus144

Differentiable transformations150

Exercises156

Chapter 8 Integration on Product Spaces160

Measurability on cartesian products160

Product measures163

The Fubini theorem164

Completion of product measures167

Convolutions170

Distribution functions172

Exercises174

Chapter 9 Fourier Transforms178

Formal properties178

The inversion theorem180

The Plancherel theorem185

The Banach algebra L1190

Exercises193

Complex differentiation196

Chapter 10 Elementary Properties of Holomorphic Functions196

Integration over paths200

The local Cauchy theorem204

The power series representation208

The open mapping theorem214

The global Cauchy theorem217

The calculus of residues224

Exercises227

The Cauchy-Riemann equations231

Chapter 11 Harmonic Functions231

The Poisson integral233

The mean value property237

Boundary behavior of Poisson integrals239

Representation theorems245

Exercises249

Chapter 12 The Maximum Modulus Principle253

Introduction253

The Schwarz lemma254

The Phragrnen-Lindel?f method256

An interpolation theorem260

A converse of the maximum modulus theorem262

Exercises264

Chapter 13 Approximation by Rational Functions266

Preparation266

Runge s theorem270

The Mittag-Leffler theorem273

Simply connected regions274

Exercises276

Chapter 14 Conformal Mapping278

Preservation of angles278

Linear fractional transformations279

Normal families281

The Riemann mapping theorem282

The class ?285

Continuity at the boundary289

Conformal mapping of an annulus291

Exercises293

Chapter 15 Zeros of Holomorphic Functions298

Infinite products298

The Weierstrass factorization theorem301

An interpolation problem304

Jensen’s formula307

Blaschke products310

The Müntz-Szasz theorem312

Exercises315

Regular points and singular points319

Chapter 16 Analytic Continuation319

Continuation along curves323

The monodromy theorem326

Construction of a modular function328

The Picard theorem331

Exercises332

Chapter 17 Hp-Spaces335

Subharmonic functions335

The s?aces Hp and N337

The theorem of F．and M．Riesz341

Factorization theorems342

The shift Operator346

Conjugate functions350

Exercises352

Chapter 18 Elementary Theory of Banach Algebras356

Introduction356

The invertible elements357

Ideals and homomorphisms362

Applications365

Exercises369

Chapter 19 Holomorphic Fourier Transforms371

Introduction371

Two theorems of Paley and Wiener372

Quasi-analytic classes377

The Denjoy-Carleman theorem380

Exercises383

Introduction386

chapter 20 Uniform Approximation by Polynomials386

Some lemmas387

Mergelyan’s theorem390

Exercises394

Appendix:Hausdorff s Maximality Theorem395

Notes and Comments397

Bibliography405

List of Special Symbols407

Index409