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