Real analysis : measure theory, integration, and Hilbert spaces / Elias M. Stein
Record details
- ISBN: 0691113866
- Physical Description: xix, 402 p. : il. ; 25 cm.
- Publisher: Princeton, NJ : Princeton University Press, c2005.
Content descriptions
| Bibliography, etc. Note: | Incluye bibliografía e índice. |
| Language Note: | English |
Search for related items by subject
| Subject: | Integrales generalizadas. Teoría de la medida. Análisis funcional. |
Search for related items by series
Available copies
- 1 of 1 copy available at IPICYT.
Holds
- 0 current holds with 1 total copy.
Show Only Available Copies
| Location | Call Number / Copy Notes | Barcode | Shelving Location | Status | Due Date |
|---|---|---|---|---|---|
| Biblioteca Ipicyt | QA320 S7 R4 2005 | APL00553 | Coleccion General | Available | - |
| Foreword | vii | |
| Introduction | xv | |
| 1. | Fourier series: completion | xvi |
| 2. | Limits of continuous functions | xvi |
| 3. | Length of curves | xvii |
| 4. | Differentiation and integration | xviii |
| 5. | The problem of measure | xviii |
| Chapter 1.. | Measure Theory | 1 |
| 1. | Preliminaries | 1 |
| 2. | The exterior measure | 10 |
| 3. | Measurable sets and the Lebesgue measure | 16 |
| 4. | Measurable functions | 27 |
| 4.1. | Definition and basic properties | 27 |
| 4.2. | Approximation by simple functions or step functions | 30 |
| 4.3. | Littlewood's three principles | 33 |
| 5*. | The Brunn-Minkowski inequality | 34 |
| 6. | Exercises | 37 |
| 7. | Problems | 46 |
| Chapter 2.. | Integration Theory | 49 |
| 1. | The Lebesgue integral: basic properties and convergence theorems | 49 |
| 2. | The space L^1 of integrable functions | 68 |
| 3. | Fubini's theorem | 75 |
| 3.1. | Statement and proof of the theorem | 75 |
| 3.2. | Applications of Fubini's theorem | 80 |
| 4*. | A Fourier inversion formula | 86 |
| 5. | Exercises | 89 |
| 6. | Problems | 95 |
| Chapter 3.. | Differentiation and Integration | 98 |
| 1. | Differentiation of the integral | 99 |
| 1.1. | The Hardy-Littlewood maximal function | 100 |
| 1.2. | The Lebesgue differentiation theorem | 104 |
| 2. | Good kernels and approximations to the identity | 108 |
| 3. | Differentiability of functions | 114 |
| 3.1. | Functions of bounded variation | 115 |
| 3.2. | Absolutely continuous functions | 127 |
| 3.3. | Differentiability of jump functions | 131 |
| 4. | Rectifiable curves and the isoperimetric inequality | 134 |
| 4.1*. | Minkowski content of a curve | 136 |
| 4.2*. | Isoperimetric inequality | 143 |
| 5. | Exercises | 145 |
| 6. | Problems | 152 |
| Chapter 4.. | Hilbert Spaces: An Introduction | 156 |
| 1. | The Hilbert space L^2 | 156 |
| 2. | Hilbert spaces | 161 |
| 2.1. | Orthogonality | 164 |
| 2.2. | Unitary mappings | 168 |
| 2.3. | Pre-Hilbert spaces | 169 |
| 3. | Fourier series and Fatou's theorem | 170 |
| 3.1. | Fatou's theorem | 173 |
| 4. | Closed subspaces and orthogonal projections | 174 |
| 5. | Linear transformations | 180 |
| 5.1. | Linear functionals and the Riesz representation theorem | 181 |
| 5.2. | Adjoints | 183 |
| 5.3. | Examples | 185 |
| 6. | Compact operators | 188 |
| 7. | Exercises | 193 |
| 8. | Problems | 202 |
| Chapter 5.. | Hilbert Spaces: Several Examples | 207 |
| 1 The Fourier transform on L^2 | ||
| 207. | 2 The Hardy space of the upper half-plane | 213 |
| 3. | Constant coefficient partial differential equations | 221 |
| 3.1. | Weak solutions | 222 |
| 3.2. | The main theorem and key estimate | 224 |
| 4*. | The Dirichlet principle | 229 |
| 4.1. | Harmonic functions | 234 |
| 4.2. | The boundary value problem and Dirichlet's principle | 243 |
| 5. | Exercises | 253 |
| 6. | Problems | 259 |
| Chapter 6.. | Abstract Measure and Integration Theory | 262 |
| 1. | Abstract measure spaces | 263 |
| 1.1. | Exterior measures and Carathéodory's theorem | 264 |
| 1.2. | Metric exterior measures | 266 |
| 1.3. | The extension theorem | 270 |
| 2. | Integration on a measure space | 273 |
| 3. | Examples | 276 |
| 3.1. | Product measures and a general Fubini theorem | 276 |
| 3.2. | Integration formula for polar coordinates | 279 |
| 3.3. | Borel measures on R and the Lebesgue-Stieltjes integral | 281 |
| 4. | Absolute continuity of measures | 285 |
| 4.1. | Signed measures | 285 |
| 4.2. | Absolute continuity | 288 |
| 5*. | Ergodic theorems | 292 |
| 5.1. | Mean ergodic theorem | 294 |
| 5.2. | Maximal ergodic theorem | 296 |
| 5.3. | Pointwise ergodic theorem | 300 |
| 5.4. | Ergodic measure-preserving transformations | 302 |
| 6*. | Appendix: the spectral theorem | 306 |
| 6.1. | Statement of the theorem | 306 |
| 6.2. | Positive operators | 307 |
| 6.3. | Proof of the theorem | 309 |
| 6.4. | Spectrum | 311 |
| 7. | Exercises | 312 |
| 8. | Problems | 319 |
| Chapter 7.. | Hausdorff Measure and Fractals | 323 |
| 1. | Hausdorff measure | 324 |
| 2. | Hausdorff dimension | 329 |
| 2.1. | Examples | 330 |
| 2.2. | Self-similarity | 341 |
| 3. | Space-filling curves | 349 |
| 3.1. | Quartic intervals and dyadic squares | 351 |
| 3.2. | Dyadic correspondence | 353 |
| 3.3. | Construction of the Peano mapping | 355 |
| 4*. | Besicovitch sets and regularity | 360 |
| 4.1. | The Radon transform | 363 |
| 4.2. | Regularity of sets when d >= 3 | 370 |
| 4.3. | Besicovitch sets have dimension 2 | 371 |
| 4.4. | Construction of a Besicovitch set | 374 |
| 5. | Exercises | 380 |
| 6. | Problems | 385 |
| Notes and References | 389 | |
| Bibliography | 391 | |
| Symbol Glossary | 395 | |
| Index | 397 |
