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Summary:
Publisher Summary 1
Explains real analysis and complex analysis for graduate and advanced undergraduate students who are familiar with terms such as continuity, power series, and Riemann integral. Exercises, a symbol list, and a glossary supplement chapters on integration, functional analysis, locally holomorphic functions, conformal mapping, and Thorin's theorem in convexity and complex analysis. Annotation copyright Book News, Inc. Portland, Or.
Publisher Summary 2
Modern Real and Complex Analysis
Thorough, well-written, and encyclopedic in its coverage, this text offers a lucid presentation of all the topics essential to graduate study in analysis. While maintaining the strictest standards of rigor, Professor Gelbaum's approach is designed to appeal to intuition whenever possible. Modern Real and Complex Analysis provides up-to-date treatment of such subjects as the Daniell integration, differentiation, functional analysis and Banach algebras, conformal mapping and Bergman's kernels, defective functions, Riemann surfaces and uniformization, and the role of convexity in analysis. The text supplies an abundance of exercises and illustrative examples to reinforce learning, and extensive notes and remarks to help clarify important points.
目录
Contents 11
REAL ANALYSIS 17
1 Fundamentals 19
1.1 Introduction 19
1.2 Topology and Continuity 22
1.3 Baire Category Arguments 34
1.4 Homotopy, Simplices, Fixed Points 36
1.5 Appendix 1: Filters 48
1.6 Appendix 2: Uniformity 50
1.7 Miscellaneous Exercises 54
2 Integration 60
2.1 Daniell-Lebesgue-Stone Integration 60
2.2 Measurability and Measure 71
2.3 The Riesz Representation Theorem 92
2.4 Complex-valued Functions 95
2.5 Miscellaneous Exercises 100
3 Functional Analysis 105
3.1 Introduction 105
3.2 The Spaces Lp, 1 鈮?p 鈮?鈭? }, { 108
3.3 Basic Banachology 119
3.4 Weak Topologies 127
3.5 Banach Algebras 132
3.6 Hilbert Space 144
3.7 Miscellaneous Exercises 147
4 More Measure Theory 153
4.1 Complex Measures 153
4.2 Comparison of Measures 158
4.3 LRN and Functional Analysis 163
4.4 Product Measures 167
4.5 Nonmeasurable Sets 174
4.6 Differentiation 180
4.7 Derivatives 193
4.8 Curves 201
4.9 Appendix: Haar Measure 203
4.10 Miscellaneous Exercises 208
COMPLEX ANALYSIS 217
5 Locally Holomorphic Functions 219
5.1 Introduction 219
5.2 Power Series 221
5.3 Basic Holomorphy 225
5.4 Singularities 246
5.5 Homotopy, Homology, and Holomorphy 261
5.6 The Riemann Sphere 267
5.7 Contour Integration 270
5.8 Exterior Calculus 273
5.9 Miscellaneous Exercises 281
6 Harmonic Functions 286
6.1 Basic Properties 286
6.2 Functions Harmonic in a Disc 289
6.3 Subharmonic Functions and Dirichlet's Problem 300
6.4 Appendix: Approximate Identities 312
6.5 Miscellaneous Exercises 312
7 Meromorphic and Entire Functions 317
7.1 Approximations and Representations 317
7.2 Infinite Products 329
7.3 Entire Functions 339
7.4 Miscellaneous Exercises 350
8 Conformal Mapping 352
8.1 Riemann's Mapping Theorem 352
8.2 M枚bius Transformations 358
8.3 Bergman's Kernel Functions 365
8.4 Groups and Holomorphy 373
8.5 Conformal Mapping and Green's Function 379
8.6 Miscellaneous Exercises 381
9 Defective Functions 385
9.1 Introduction 385
9.2 Bloch's Theorem 387
9.3 The Little Picard Theorem 390
9.4 The Great Picard Theorem 392
9.5 Miscellaneous Exercises 396
10 Riemann Surfaces 397
10.1 Analytic Continuation 397
10.2 Manifolds and Riemann Surfaces 407
10.3 Covering Spaces and Lifts 425
10.4 Riemann Surfaces and Analysis 435
10.5 The Uniformization Theorem 438
10.6 Miscellaneous Exercises 440
11 Convexity and Complex Analysis 447
11.1 Thorin's Theorem 447
11.2 Applications of Thorin's Theorem 450
12 Several Complex Variables 461
12.1 Survey 461
Bibliography 467
Symbol List 471
Glossary/Index 477
REAL ANALYSIS 17
1 Fundamentals 19
1.1 Introduction 19
1.2 Topology and Continuity 22
1.3 Baire Category Arguments 34
1.4 Homotopy, Simplices, Fixed Points 36
1.5 Appendix 1: Filters 48
1.6 Appendix 2: Uniformity 50
1.7 Miscellaneous Exercises 54
2 Integration 60
2.1 Daniell-Lebesgue-Stone Integration 60
2.2 Measurability and Measure 71
2.3 The Riesz Representation Theorem 92
2.4 Complex-valued Functions 95
2.5 Miscellaneous Exercises 100
3 Functional Analysis 105
3.1 Introduction 105
3.2 The Spaces Lp, 1 鈮?p 鈮?鈭? }, { 108
3.3 Basic Banachology 119
3.4 Weak Topologies 127
3.5 Banach Algebras 132
3.6 Hilbert Space 144
3.7 Miscellaneous Exercises 147
4 More Measure Theory 153
4.1 Complex Measures 153
4.2 Comparison of Measures 158
4.3 LRN and Functional Analysis 163
4.4 Product Measures 167
4.5 Nonmeasurable Sets 174
4.6 Differentiation 180
4.7 Derivatives 193
4.8 Curves 201
4.9 Appendix: Haar Measure 203
4.10 Miscellaneous Exercises 208
COMPLEX ANALYSIS 217
5 Locally Holomorphic Functions 219
5.1 Introduction 219
5.2 Power Series 221
5.3 Basic Holomorphy 225
5.4 Singularities 246
5.5 Homotopy, Homology, and Holomorphy 261
5.6 The Riemann Sphere 267
5.7 Contour Integration 270
5.8 Exterior Calculus 273
5.9 Miscellaneous Exercises 281
6 Harmonic Functions 286
6.1 Basic Properties 286
6.2 Functions Harmonic in a Disc 289
6.3 Subharmonic Functions and Dirichlet's Problem 300
6.4 Appendix: Approximate Identities 312
6.5 Miscellaneous Exercises 312
7 Meromorphic and Entire Functions 317
7.1 Approximations and Representations 317
7.2 Infinite Products 329
7.3 Entire Functions 339
7.4 Miscellaneous Exercises 350
8 Conformal Mapping 352
8.1 Riemann's Mapping Theorem 352
8.2 M枚bius Transformations 358
8.3 Bergman's Kernel Functions 365
8.4 Groups and Holomorphy 373
8.5 Conformal Mapping and Green's Function 379
8.6 Miscellaneous Exercises 381
9 Defective Functions 385
9.1 Introduction 385
9.2 Bloch's Theorem 387
9.3 The Little Picard Theorem 390
9.4 The Great Picard Theorem 392
9.5 Miscellaneous Exercises 396
10 Riemann Surfaces 397
10.1 Analytic Continuation 397
10.2 Manifolds and Riemann Surfaces 407
10.3 Covering Spaces and Lifts 425
10.4 Riemann Surfaces and Analysis 435
10.5 The Uniformization Theorem 438
10.6 Miscellaneous Exercises 440
11 Convexity and Complex Analysis 447
11.1 Thorin's Theorem 447
11.2 Applications of Thorin's Theorem 450
12 Several Complex Variables 461
12.1 Survey 461
Bibliography 467
Symbol List 471
Glossary/Index 477
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