Real Analysis
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Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, "Real Analysis" is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
작가정보
목차
Foreword p. vii Introduction p. xv Fourier series: completion p. xvi Limits of continuous functions p. xvi Length of curves p. xvii Differentiation and integration p. xviii The problem of measure p. xviii Measure Theory 1 1 Preliminaries The exterior measure p. 10 Measurable sets and the Lebesgue measure p. 16 Measurable functions p. 7 Definition and basic properties p. 27 Approximation by simple functions or step functions p. 30 Littlewood's three principles p. 33 The Brunn-Minkowski inequality p. 34 Exercises p. 37 Problems p. 46 Integration Theory p. 49 The Lebesgue integral: basic properties and convergence theorems p. 49 Thespace L 1 of integrable functions p. 68 Fubini's theorem p. 75 Statement and proof of the theorem p. 75 Applications of Fubini's theorem p. 80 A Fourier inversion formula p. 86 Exercises p. 89 Problems p. 95 Differentiation and Integration p. 98 Differentiation of the integral p. 99 The Hardy-Littlewood maximal function p. 100 The Lebesgue differentiation theorem p. 104 Good kernels and approximations to the identity p. 108 Differentiability of functions p. 114 Functions of bounded variation p. 115 Absolutely continuous functions p. 127 Differentiability of jump functions p. 131 Rectifiable curves and the isoperimetric inequality p. 134 Minkowski content of a curve p. 136 Isoperimetric inequality p. 143 Exercises p. 145 Problems p. 152 Hilbert Spaces: An Introduction p. 156 The Hilbert space L 2 p. 156 Hilbert spaces p. 161 Orthogonality p. 164 Unitary mappings p. 168 Pre-Hilbert spaces p. 169 Fourier series and Fatou's theorem p. 170 Fatou's theorem p. 173 Closed subspaces and orthogonal projections p. 174 Linear transformations p. 180 Linear functionals and the Riesz representation theorem p. 181 Adjoints p. 183 Examples p. 185 Compact operators p. 188 Exercises p. 193 Problems p. 202 Hilbert Spaces: Several Examples p. 207 The Fourier transform on L 2 p. 207 The Hardy space of the upper half-plane p. 13 Constant coefficient partial differential equations p. 221 Weaksolutions p. 222 The main theorem and key estimate p. 224 The Dirichlet principle p. 9 Harmonic functions p. 234 The boundary value problem and Dirichlet's principle p. 43 Exercises p. 253 Problems p. 259 Abstract Measure and Integration Theory p. 262 Abstract measure spaces p. 263 Exterior measures and Carathegrave;odory's theorem p. 264 Metric exterior measures p. 266 The extension theorem p. 270 Integration on a measure space p. 273 Examples p. 276 Product measures and a general Fubini theorem p. 76 Integration formula for polar coordinates p. 279 Borel measures on R and the Lebesgue-Stieltjes integral p. 281 Absolute continuity of measures p. 285 Signed measures p. 285 Absolute continuity p. 288 Ergodic theorems p. 292 Mean ergodic theorem p. 294 Maximal ergodic theorem p. 296 Pointwise ergodic theorem p. 300 Ergodic measure-preserving transformations p. 302 Appendix: the spectral theorem p. 306 Statement of the theorem p. 306 Positive operators p. 307 Proof of the theorem p. 309 Spectrum p. 311 Exercises p. 312 Problems p. 319 Hausdorff Measure and Fractals p. 323 Hausdorff measure p. 324 Hausdorff dimension p. 329 Examples p. 330 Self-similarity p. 341 Space-filling curves p. 349 Quartic intervals and dyadic squares p. 351 Dyadic correspondence p. 353 Construction of the Peano mapping p. 355 Besicovitch sets and regularity p. 360 The Radon transform p. 363 Regularity of sets whend3 p. 370 Besicovitch sets have dimension p. 371 Construction of a Besicovitch set p. 374 Exercises p. 380 Problems p. 385 Notes and References p. 389 Table of Contents provided by Publisher. All Rights Reserved.
출판사 서평
We are all fortunate that a mathematician with the experience and vision of E.M. Stein, together with his energetic young collaborator R. Shakarchi, has given us this series of four books on analysis.---Steven George Krantz, Mathematical Reviews
This series is a result of a radical rethinking of how to introduce graduate students to analysis. . . . This volume lives up to the high standard set up by the previous ones.---Fernando Q. Gouv?a, MAA Review
Elias M. Stein, Winner of the 2005 Stefan Bergman Prize, American Mathematical Society
As one would expect from these authors, the exposition is, in general, excellent. The explanations are clear and concise with many well-focused examples as well as an abundance of exercises, covering the full range of difficulty. . . . [I]t certainly must be on the instructor's bookshelf as a first-rate reference book.---William P. Ziemer, SIAM Review
기본정보
ISBN | 9780691113869 ( 0691113866 ) | ||
---|---|---|---|
발행(출시)일자 | 2008년 11월 21일 | ||
쪽수 | 424쪽 | ||
크기 |
164 * 240
* 34
mm
/ 735 g
|
||
총권수 | 1권 | ||
언어 | 영어 | ||
시리즈명 |
Princeton Lectures in Analysis
|
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