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Real Analysis : Measure Theory, Integration, and Hilbert Spaces - Elias M. Stein

Real Analysis

Measure Theory, Integration, and Hilbert Spaces


Published: 14th March 2005
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"Real Analysis" is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.

After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. 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.

Also available, the first two volumes in the Princeton Lectures in Analysis:

"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 "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

Forewordp. vii
Introductionp. xv
Fourier series: completionp. xvi
Limits of continuous functionsp. xvi
Length of curvesp. xvii
Differentiation and integrationp. xviii
The problem of measurep. xviii
Measure Theory 1 1 Preliminaries
The exterior measurep. 10
Measurable sets and the Lebesgue measurep. 16
Measurable functionsp. 7
Definition and basic propertiesp. 27
Approximation by simple functions or step functionsp. 30
Littlewood's three principlesp. 33
The Brunn-Minkowski inequalityp. 34
Exercisesp. 37
Problemsp. 46
Integration Theoryp. 49
The Lebesgue integral: basic properties and convergence theoremsp. 49
Thespace L 1 of integrable functionsp. 68
Fubini's theoremp. 75
Statement and proof of the theoremp. 75
Applications of Fubini's theoremp. 80
A Fourier inversion formulap. 86
Exercisesp. 89
Problemsp. 95
Differentiation and Integrationp. 98
Differentiation of the integralp. 99
The Hardy-Littlewood maximal functionp. 100
The Lebesgue differentiation theoremp. 104
Good kernels and approximations to the identityp. 108
Differentiability of functionsp. 114
Functions of bounded variationp. 115
Absolutely continuous functionsp. 127
Differentiability of jump functionsp. 131
Rectifiable curves and the isoperimetric inequalityp. 134
Minkowski content of a curvep. 136
Isoperimetric inequalityp. 143
Exercisesp. 145
Problemsp. 152
Hilbert Spaces: An Introductionp. 156
The Hilbert space L 2p. 156
Hilbert spacesp. 161
Orthogonalityp. 164
Unitary mappingsp. 168
Pre-Hilbert spacesp. 169
Fourier series and Fatou's theoremp. 170
Fatou's theoremp. 173
Closed subspaces and orthogonal projectionsp. 174
Linear transformationsp. 180
Linear functionals and the Riesz representation theoremp. 181
Adjointsp. 183
Examplesp. 185
Compact operatorsp. 188
Exercisesp. 193
Problemsp. 202
Hilbert Spaces: Several Examplesp. 207
The Fourier transform on L 2p. 207
The Hardy space of the upper half-planep. 13
Constant coefficient partial differential equationsp. 221
Weaksolutionsp. 222
The main theorem and key estimatep. 224
The Dirichlet principlep. 9
Harmonic functionsp. 234
The boundary value problem and Dirichlet's principlep. 43
Exercisesp. 253
Problemsp. 259
Abstract Measure and Integration Theoryp. 262
Abstract measure spacesp. 263
Exterior measures and Carathegrave;odory's theoremp. 264
Metric exterior measuresp. 266
The extension theoremp. 270
Integration on a measure spacep. 273
Examplesp. 276
Product measures and a general Fubini theoremp. 76
Integration formula for polar coordinatesp. 279
Borel measures on R and the Lebesgue-Stieltjes integralp. 281
Absolute continuity of measuresp. 285
Signed measuresp. 285
Absolute continuityp. 288
Ergodic theoremsp. 292
Mean ergodic theoremp. 294
Maximal ergodic theoremp. 296
Pointwise ergodic theoremp. 300
Ergodic measure-preserving transformationsp. 302
Appendix: the spectral theoremp. 306
Statement of the theoremp. 306
Positive operatorsp. 307
Proof of the theoremp. 309
Spectrump. 311
Exercisesp. 312
Problemsp. 319
Hausdorff Measure and Fractalsp. 323
Hausdorff measurep. 324
Hausdorff dimensionp. 329
Examplesp. 330
Self-similarityp. 341
Space-filling curvesp. 349
Quartic intervals and dyadic squaresp. 351
Dyadic correspondencep. 353
Construction of the Peano mappingp. 355
Besicovitch sets and regularityp. 360
The Radon transformp. 363
Regularity of sets whend3p. 370
Besicovitch sets have dimensionp. 371
Construction of a Besicovitch setp. 374
Exercisesp. 380
Problemsp. 385
Notes and Referencesp. 389
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691113869
ISBN-10: 0691113866
Series: Princeton Lectures in Analysis
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 424
Published: 14th March 2005
Country of Publication: US
Dimensions (cm): 24.5 x 16.0  x 3.3
Weight (kg): 0.75