Real and complex analysis
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| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
McGraw-Hill
1974
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | McGraw-Hill series in higher mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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MARC
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| 245 | 1 | 0 | |a Real and complex analysis |c Walter Rudin |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b McGraw-Hill |c 1974 | |
| 300 | |a XII, 452 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a McGraw-Hill series in higher mathematics | |
| 650 | 7 | |a Analyse (wiskunde) |2 gtt | |
| 650 | 4 | |a Analyse mathématique | |
| 650 | 4 | |a Análisis matemático | |
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Datensatz im Suchindex
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| _version_ | 1823051860261470208 |
| adam_text | CONTENTS
Preface xi
Prologue: The Exponential Function 1
1 Abstract Integration 5
Set-theoretic notations and terminology 6
The concept of measurability 8
Simple functions 16
Elementary properties of measures 17
Arithmetic in [0, oo] 19
Integration of positive functions 20
Integration of complex functions 25
The role played by sets of measure zero 28
Exercises 32
2 Positive Borel Measures 34
Vector spaces 34
Topological preliminaries 36
The Riesz representation theorem 42
Regularity properties of Borel measures 49
Lebesgue measure 51
Continuity properties of measurable functions 56
Exercises 58
VI CONTENTS
3 // -Spaces 62
Convex functions and inequalities 62
The .//-spaces 66
Approximation by continuous functions 71
Exercises 73
4 Elementary Hilbert Space Theory 79
Inner products and linear functionals 79
Orthonormal sets 86
Trigonometric series 92
Exercises 97
5 Examples of Banach Space Techniques 100
Banach spaces 100
Consequences of Baire s theorem 102
Fourier series of continuous functions 106
Fourier coefficients of L -functions 109
The Hahn-Banach theorem 111
An abstract approach to the Poisson integral 115
Exercises 119
6 Complex Measures 124
Total variation 124
Absolute continuity 128
Consequences of the Radon-Nikodym theorem 133
Bounded linear functionals on II 135
The Riesz representation theorem 138
Exercises 142
7 Integration on Product Spaces 145
Measurability on cartesian products 145
Product measures 148
The Fubini theorem 150
Completion of product measures 153
Convolutions 155
Exercises 157
8 Differentiation 162
Derivatives of measures 162
Functions of bounded variation 171
Differentiation of point functions 175
CONTENTS vii
Differentiable transformations 181
Exercises 188
9 Fourier Transforms 192
Formal properties 192
The inversion theorem 195
The Plancherel theorem 200
The Banach algebra D 205
Exercises 208
10 Elementary Properties of Holomorphic Functions 212
Complex differentiation 212
Integration over paths 217
The local Cauchy theorem 221
The power series representation 225
The open mapping theorem 231
The global Cauchy theorem 233
The calculus of residues 241
Exercises 244
11 Harmonic Functions 250
The Cauchy-Riemann equations 250
The Poisson integral 252
The mean value property 259
Positive harmonic functions 261
Exercises 266
12 The Maximum Modulus Principle 270
Introduction 270
The Schwarz lemma 271
The Phragmen-Lindelof method 273
An interpolation theorem 277
A converse of the maximum modulus theorem 279
Exercises 281
13 Approximation by Rational Functions 284
Preparation 284
Runge s theorem 288
The Mittag-Leffler theorem 291
Simply connected regions 292
Exercises 294
viii CONTENTS
14 Conformal Mapping 296
Preservation of angles 296
Linear fractional transformations 298
Normal families 300
The Riemann mapping theorem 302
The class S 304
Continuity at the boundary 308
Conformal mapping of an annulus 311
Exercises 313
15 Zeros of Holomorphic Functions 320
Infinite products 320
The Weierstrass factorization theorem 323
An interpolation problem 327
Jensen s formula 329
Blaschke products 333
The Miintz-Szasz theorem 336
Exercises 339
16 Analytic Continuation 343
Regular points and singular points 343
Continuation along curves 347
The monodromy theorem 351
Construction of a modular function 352
The Picard theorem 356
Exercises 357
17 // -Spaces 361
Subharmonic functions 361
The spaces Hp and N 363
The space H2 365
The theorem of F. and M. Riesz 369
Factorization theorems 370
The shift operator 375
Conjugate functions 379
Exercises 382
18 Elementary Theory of Banach Algebras 386
Introduction 386
The invertible elements 387
Ideals and homomorphisms 392
CONTENTS IX
Applications 396
Exercises 400
19 Holomorphic Fourier Transforms 403
Introduction 403
Two theorems of Paley and Wiener 405
Quasi-analytic classes 409
The Denjoy-Carleman theorem 412
Exercises 416
20 Uniform Approximation by Polynomials 419
Introduction 419
Some lemmas 420
Mergelyan s theorem 423
Exercises 427
Appendix: Hausdorff s Maximality Theorem 429
Notes and Comments 432
Bibliography 440
List of Special Symbols 443
Index 445
|
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| author | Rudin, Walter 1921-2010 |
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| ctrlnum | (OCoLC)704861 (DE-599)BVBBV002281259 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV002281259 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-03T16:44:36Z |
| institution | BVB |
| isbn | 0070542333 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001499112 |
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| physical | XII, 452 S. |
| publishDate | 1974 |
| publishDateSearch | 1974 |
| publishDateSort | 1974 |
| publisher | McGraw-Hill |
| record_format | marc |
| series2 | McGraw-Hill series in higher mathematics |
| spellingShingle | Rudin, Walter 1921-2010 Real and complex analysis Analyse (wiskunde) gtt Analyse mathématique Análisis matemático Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4627581-2 (DE-588)4001865-9 (DE-588)4123623-3 |
| title | Real and complex analysis |
| title_auth | Real and complex analysis |
| title_exact_search | Real and complex analysis |
| title_full | Real and complex analysis Walter Rudin |
| title_fullStr | Real and complex analysis Walter Rudin |
| title_full_unstemmed | Real and complex analysis Walter Rudin |
| title_short | Real and complex analysis |
| title_sort | real and complex analysis |
| topic | Analyse (wiskunde) gtt Analyse mathématique Análisis matemático Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analyse (wiskunde) Analyse mathématique Análisis matemático Mathematical analysis Funktionentheorie Reelle Analysis Analysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001499112&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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