An edition of Real Analysis (2005)

Real Analysis

Measure Theory, Integration, and Hilbert Spaces

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Cover of: Real Analysis by Elias M. Stein, Rami Shakarchi, Elias M. Stein, Rami Shakarchi
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An edition of Real Analysis (2005)

Real Analysis

Measure Theory, Integration, and Hilbert Spaces

by and

  • 2 Want to read
  • 1 Currently reading

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Publish Date
Publisher
Princeton University Press
Language
English
Pages
392

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Previews available in: English

Subjects

Mathematical analysis, Functional analysis, Measure theory, Generalized Integrals, Maattheorie, Hilbertruimten, Integrals, generalized
Edition Availability
Cover of: Real Analysis
Real Analysis
2010, Princeton University Press
in English
Cover of: Real Analysis
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
2009, Princeton University Press
in English
Cover of: Real Analysis
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
March 14, 2005, Princeton University Press
Hardcover in English

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Book Details

Table of Contents

Foreword
Page vii
Introduction
Page xv
1. Fourier series: completion
Page xvi
2. Limits of continuous functions
Page xvi
3. Length of curves
Page xvii
4. Differentiation and integration
Page xviii
5. The problem of measure
Page xviii
Chapter 1. Measure Theory
Page 1
1. Preliminaries
Page 1
2. The exterior measure
Page 10
3. Measurable sets and the Lebesgue measure
Page 16
4. Measurable functions
Page 27
4.1. Definition and basic properties
Page 27
4.2. Approximation by simple functions or step functions
Page 30
4.3. Littlewood's three principles
Page 33
5*. The Brunn-Minkowski inequality
Page 34
6. Exercises
Page 37
7. Problems
Page 46
Chapter 2. Integration Theory
Page 49
1. The Lebesgue integral: basic properties and convergence theorems
Page 49
2. The space L^1 of integrable functions
Page 68
3. Fubini's theorem
Page 75
3.1. Statement and proof of the theorem
Page 75
3.2. Applications of Fubini's theorem
Page 80
4*. A Fourier inversion formula
Page 86
5. Exercises
Page 89
6. Problems
Page 95
Chapter 3. Differentiation and Integration
Page 98
1. Differentiation of the integral
Page 99
1.1. The Hardy-Littlewood maximal function
Page 100
1.2. The Lebesgue differentiation theorem
Page 104
2. Good kernels and approximations to the identity
Page 108
3. Differentiability of functions
Page 114
3.1. Functions of bounded variation
Page 115
3.2. Absolutely continuous functions
Page 127
3.3. Differentiability of jump functions
Page 131
4. Rectifiable curves and the isoperimetric inequality
Page 134
4.1*. Minkowski content of a curve
Page 136
4.2*. Isoperimetric inequality
Page 143
5. Exercises
Page 145
6. Problems
Page 152
Chapter 4. Hilbert Spaces: An Introduction
Page 156
1. The Hilbert space L^2
Page 156
2. Hilbert spaces
Page 161
2.1. Orthogonality
Page 164
2.2. Unitary mappings
Page 168
2.3. Pre-Hilbert spaces
Page 169
3. Fourier series and Fatou's theorem
Page 170
3.1. Fatou's theorem
Page 173
4. Closed subspaces and orthogonal projections
Page 174
5. Linear transformations
Page 180
5.1. Linear functionals and the Riesz representation theorem
Page 181
5.2. Adjoints
Page 183
5.3. Examples
Page 185
6. Compact operators
Page 188
7. Exercises
Page 193
8. Problems
Page 202
Chapter 5. Hilbert Spaces: Several Examples
Page 207
1 The Fourier transform on L^2
207. 2 The Hardy space of the upper half-plane
Page 213
3. Constant coefficient partial differential equations
Page 221
3.1. Weak solutions
Page 222
3.2. The main theorem and key estimate
Page 224
4*. The Dirichlet principle
Page 229
4.1. Harmonic functions
Page 234
4.2. The boundary value problem and Dirichlet's principle
Page 243
5. Exercises
Page 253
6. Problems
Page 259
Chapter 6. Abstract Measure and Integration Theory
Page 262
1. Abstract measure spaces
Page 263
1.1. Exterior measures and Carathéodory's theorem
Page 264
1.2. Metric exterior measures
Page 266
1.3. The extension theorem
Page 270
2. Integration on a measure space
Page 273
3. Examples
Page 276
3.1. Product measures and a general Fubini theorem
Page 276
3.2. Integration formula for polar coordinates
Page 279
3.3. Borel measures on R and the Lebesgue-Stieltjes integral
Page 281
4. Absolute continuity of measures
Page 285
4.1. Signed measures
Page 285
4.2. Absolute continuity
Page 288
5*. Ergodic theorems
Page 292
5.1. Mean ergodic theorem
Page 294
5.2. Maximal ergodic theorem
Page 296
5.3. Pointwise ergodic theorem
Page 300
5.4. Ergodic measure-preserving transformations
Page 302
6*. Appendix: the spectral theorem
Page 306
6.1. Statement of the theorem
Page 306
6.2. Positive operators
Page 307
6.3. Proof of the theorem
Page 309
6.4. Spectrum
Page 311
7. Exercises
Page 312
8. Problems
Page 319
Chapter 7. Hausdorff Measure and Fractals
Page 323
1. Hausdorff measure
Page 324
2. Hausdorff dimension
Page 329
2.1. Examples
Page 330
2.2. Self-similarity
Page 341
3. Space-filling curves
Page 349
3.1. Quartic intervals and dyadic squares
Page 351
3.2. Dyadic correspondence
Page 353
3.3. Construction of the Peano mapping
Page 355
4*. Besicovitch sets and regularity
Page 360
4.1. The Radon transform
Page 363
4.2. Regularity of sets when d >= 3
Page 370
4.3. Besicovitch sets have dimension 2
Page 371
4.4. Construction of a Besicovitch set
Page 374
5. Exercises
Page 380
6. Problems
Page 385
Notes and References
Page 389
Bibliography
Page 391
Symbol Glossary
Page 395
Index
Page 397

Edition Notes

Series
Princeton Lectures in Analysis

Classifications

Library of Congress
QA300, QA320 .S84 2005

The Physical Object

Format
Hardcover
Number of pages
392
Dimensions
9.3 x 6.5 x 1.5 inches
Weight
1.6 pounds

Edition Identifiers

Open Library
OL7758609M
Internet Archive
realanalysismeas00stei
ISBN 10
0691113866
ISBN 13
9780691113869
LCCN
2004114065
OCLC/WorldCat
57750299
Library Thing
175598
Goodreads
218523

Work Identifiers

Work ID
OL15624842W

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Library of Congress MARC record
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August 17, 2024 Edited by ImportBot import existing book
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January 6, 2011 Created by Yuri Sulyma separated series out of subtitle