Modern real and complex analysis

Modern Real and Complex AnalysisThorough, 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. Title......Page 3 Preface......Page 5 Contents......Page 9 Real Analysis......Page 15 1.1 Introduction......Page 17 1.2 Topology and Continuity......Page 20 1.3 Baire Category Arguments......Page 32 1.4 Homotopy, Simplices, Fixed Points......Page 34 1.5 Appendix 1: Filters......Page 46 1.6 Appendix 2: Uniformity......Page 48 1.7 Miscellaneous Exercises......Page 52 2.1 Daniell-Lebesgue-Stone Integration......Page 58 2.2 Measurability and Measure......Page 69 2.3 The Riesz Representation Theorem......Page 90 2.4 Complex-valued Functions......Page 93 2.5 Miscellaneous Exercises......Page 98 3.1 Introduction......Page 103 3.2 The Spaces L^p, 1 \leq p \leq \infty......Page 106 3.3 Basic Banachology......Page 117 3.4 Weak Topologies......Page 125 3.5 Banach Algebras......Page 130 3.6 Hilbert Space......Page 142 3.7 Miscellaneous Exercises......Page 145 4.1 Complex Measures......Page 151 4.2 Comparison of Measures......Page 156 4.3 LRN and Functional Analysis......Page 161 4.4 Product Measures......Page 165 4.5 Nonmeasurable Sets......Page 172 4.6 Differentiation......Page 178 4.7 Derivatives......Page 191 4.8 Curves......Page 199 4.9 Appendix: Haar Measure......Page 201 4.10 Miscellaneous Exercises......Page 206 Complex Analysis......Page 215 5.1 Introduction......Page 217 5.2 Power Series......Page 219 5.3 Basic Holomorphy......Page 223 5.4 Singularities......Page 244 5.5 Homotopy, Homology, and Holomorphy......Page 259 5.6 The Riemann Sphere......Page 265 5.7 Contour Integration......Page 268 5.8 Exterior Calculus......Page 271 5.9 Miscellaneous Exercises......Page 279 6.1 Basic Properties......Page 284 6.2 Functions Harmonic in a Disc......Page 287 6.3 Subharmonic Functions and Dirichlet's Problem......Page 298 6.5 Miscellaneous Exercises......Page 310 7.1 Approximations and Representations......Page 315 7.2 Infinite Products......Page 327 7.3 Entire Functions......Page 337 7.4 Miscellaneous Exercises......Page 348 8.1 Riemann's Mapping Theorem......Page 350 8.2 Möbius Transformations......Page 356 8.3 Bergman's Kernel Functions......Page 363 8.4 Groups and Holomorphy......Page 371 8.5 Conformal Mapping and Green's Function......Page 377 8.6 Miscellaneous Exercises......Page 379 9.1 Introduction......Page 383 9.2 Bloch's Theorem......Page 385 9.3 The Little Picard Theorem......Page 388 9.4 The Great Picard Theorem......Page 390 9.5 Miscellaneous Exercises......Page 394 10.1 Analytic Continuation......Page 395 10.2 Manifolds and Riemann Surfaces......Page 405 10.3 Covering Spaces and Lifts......Page 423 10.4 Riemann Surfaces and Analysis......Page 433 10.5 The Uniformization Theorem......Page 436 10.6 Miscellaneous Exercises......Page 438 11.1 Thorin's Theorem......Page 445 11.2 Applications of Thorin's Theorem......Page 448 12.1 Survey......Page 459 Bibliography......Page 465 Symbol List......Page 469 Glossary/Index......Page 475 Cover......Page 504 Modern Real and Complex Analysis Thorough, well-written, and encyclopedic in its coverage, this textoffers a lucid presentation of all the topics essential to graduatestudy in analysis. While maintaining the strictest standards ofrigor, Professor Gelbaum's approach is designed to appeal tointuition whenever possible. Modern Real and Complex Analysisprovides up-to-date treatment of such subjects as the Daniellintegration, differentiation, functional analysis and Banachalgebras, conformal mapping and Bergman's kernels, defectivefunctions, Riemann surfaces and uniformization, and the role ofconvexity in analysis. The text supplies an abundance of exercisesand illustrative examples to reinforce learning, and extensivenotes and remarks to help clarify important points. Real Analysis: Fundamentals -- Integration -- Functional Analysis -- More Measure Theory -- Complex Analysis: Locally Holomorphic Functions -- Harmonic Functions -- Meromorphic And Entire Functions -- Conformal Mapping -- Defective Functions -- Riemann Surfaces -- Convexity And Complex Analysis -- Several Complex Variables -- Bibliography -- Symbol List -- Glossary/index. Bernard R. Gelbaum. A Wiley-interscience Publication. Includes Bibliographical References (p. 451-454) And Index. 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.