Real Analysis
Subtitle: Measure Theory, Integration, and Hilbert Spaces
Series: Princeton Lectures in Analysis
Publisher: Princeton University Press
Published: March 14, 2005
Language: eng
Format: Hardcover
Number of pages: 392
ISBN-13: 9780691113869
Subjects: Real analysis; Mathematics; Science/Mathematics; Calculus; Mathematical Analysis; Mathematics / Mathematical Analysis
Table of contents: title: Foreword
pagenum: vii; title: Introduction
pagenum: xv; label: 1
title: Fourier series: completion
pagenum: xvi; label: 2
title: Limits of continuous functions
pagenum: xvi; label: 3
title: Length of curves
pagenum: xvii; label: 4
title: Differentiation and integration
pagenum: xviii; label: 5
title: The problem of measure
pagenum: xviii; label: Chapter 1.
title: Measure Theory
pagenum: 1; label: 1
title: Preliminaries
pagenum: 1; label: 2
title: The exterior measure
pagenum: 10; label: 3
title: Measurable sets and the Lebesgue measure
pagenum: 16; label: 4
title: Measurable functions
pagenum: 27; label: 4.1
title: Definition and basic properties
pagenum: 27; label: 4.2
title: Approximation by simple functions or step functions
pagenum: 30; label: 4.3
title: Littlewood's three principles
pagenum: 33; label: 5*
title: The Brunn-Minkowski inequality
pagenum: 34; label: 6
title: Exercises
pagenum: 37; label: 7
title: Problems
pagenum: 46; label: Chapter 2.
title: Integration Theory
pagenum: 49; label: 1
title: The Lebesgue integral: basic properties and convergence theorems
pagenum: 49; label: 2
title: The space L^1 of integrable functions
pagenum: 68; label: 3
title: Fubini's theorem
pagenum: 75; label: 3.1
title: Statement and proof of the theorem
pagenum: 75; label: 3.2
title: Applications of Fubini's theorem
pagenum: 80; label: 4*
title: A Fourier inversion formula
pagenum: 86; label: 5
title: Exercises
pagenum: 89; label: 6
title: Problems
pagenum: 95; label: Chapter 3.
title: Differentiation and Integration
pagenum: 98; label: 1
title: Differentiation of the integral
pagenum: 99; label: 1.1
title: The Hardy-Littlewood maximal function
pagenum: 100; label: 1.2
title: The Lebesgue differentiation theorem
pagenum: 104; label: 2
title: Good kernels and approximations to the identity
pagenum: 108; label: 3
title: Differentiability of functions
pagenum: 114; label: 3.1
title: Functions of bounded variation
pagenum: 115; label: 3.2
title: Absolutely continuous functions
pagenum: 127; label: 3.3
title: Differentiability of jump functions
pagenum: 131; label: 4
title: Rectifiable curves and the isoperimetric inequality
pagenum: 134; label: 4.1*
title: Minkowski content of a curve
pagenum: 136; label: 4.2*
title: Isoperimetric inequality
pagenum: 143; label: 5
title: Exercises
pagenum: 145; label: 6
title: Problems
pagenum: 152; label: Chapter 4.
title: Hilbert Spaces: An Introduction
pagenum: 156; label: 1
title: The Hilbert space L^2
pagenum: 156; label: 2
title: Hilbert spaces
pagenum: 161; label: 2.1
title: Orthogonality
pagenum: 164; label: 2.2
title: Unitary mappings
pagenum: 168; label: 2.3
title: Pre-Hilbert spaces
pagenum: 169; label: 3
title: Fourier series and Fatou's theorem
pagenum: 170; label: 3.1
title: Fatou's theorem
pagenum: 173; label: 4
title: Closed subspaces and orthogonal projections
pagenum: 174; label: 5
title: Linear transformations
pagenum: 180; label: 5.1
title: Linear functionals and the Riesz representation theorem
pagenum: 181; label: 5.2
title: Adjoints
pagenum: 183; label: 5.3
title: Examples
pagenum: 185; label: 6
title: Compact operators
pagenum: 188; label: 7
title: Exercises
pagenum: 193; label: 8
title: Problems
pagenum: 202; label: Chapter 5.
title: Hilbert Spaces: Several Examples
pagenum: 207; title: 1 The Fourier transform on L^2; label: 207
title: 2 The Hardy space of the upper half-plane
pagenum: 213; label: 3
title: Constant coefficient partial differential equations
pagenum: 221; label: 3.1
title: Weak solutions
pagenum: 222; label: 3.2
title: The main theorem and key estimate
pagenum: 224; label: 4*
title: The Dirichlet principle
pagenum: 229; label: 4.1
title: Harmonic functions
pagenum: 234; label: 4.2
title: The boundary value problem and Dirichlet's principle
pagenum: 243; label: 5
title: Exercises
pagenum: 253; label: 6
title: Problems
pagenum: 259; label: Chapter 6.
title: Abstract Measure and Integration Theory
pagenum: 262; label: 1
title: Abstract measure spaces
pagenum: 263; label: 1.1
title: Exterior measures and Carathéodory's theorem
pagenum: 264; label: 1.2
title: Metric exterior measures
pagenum: 266; label: 1.3
title: The extension theorem
pagenum: 270; label: 2
title: Integration on a measure space
pagenum: 273; label: 3
title: Examples
pagenum: 276; label: 3.1
title: Product measures and a general Fubini theorem
pagenum: 276; label: 3.2
title: Integration formula for polar coordinates
pagenum: 279; label: 3.3
title: Borel measures on R and the Lebesgue-Stieltjes integral
pagenum: 281; label: 4
title: Absolute continuity of measures
pagenum: 285; label: 4.1
title: Signed measures
pagenum: 285; label: 4.2
title: Absolute continuity
pagenum: 288; label: 5*
title: Ergodic theorems
pagenum: 292; label: 5.1
title: Mean ergodic theorem
pagenum: 294; label: 5.2
title: Maximal ergodic theorem
pagenum: 296; label: 5.3
title: Pointwise ergodic theorem
pagenum: 300; label: 5.4
title: Ergodic measure-preserving transformations
pagenum: 302; label: 6*
title: Appendix: the spectral theorem
pagenum: 306; label: 6.1
title: Statement of the theorem
pagenum: 306; label: 6.2
title: Positive operators
pagenum: 307; label: 6.3
title: Proof of the theorem
pagenum: 309; label: 6.4
title: Spectrum
pagenum: 311; label: 7
title: Exercises
pagenum: 312; label: 8
title: Problems
pagenum: 319; label: Chapter 7.
title: Hausdorff Measure and Fractals
pagenum: 323; label: 1
title: Hausdorff measure
pagenum: 324; label: 2
title: Hausdorff dimension
pagenum: 329; label: 2.1
title: Examples
pagenum: 330; label: 2.2
title: Self-similarity
pagenum: 341; label: 3
title: Space-filling curves
pagenum: 349; label: 3.1
title: Quartic intervals and dyadic squares
pagenum: 351; label: 3.2
title: Dyadic correspondence
pagenum: 353; label: 3.3
title: Construction of the Peano mapping
pagenum: 355; label: 4*
title: Besicovitch sets and regularity
pagenum: 360; label: 4.1
title: The Radon transform
pagenum: 363; label: 4.2
title: Regularity of sets when d >= 3
pagenum: 370; label: 4.3
title: Besicovitch sets have dimension 2
pagenum: 371; label: 4.4
title: Construction of a Besicovitch set
pagenum: 374; label: 5
title: Exercises
pagenum: 380; label: 6
title: Problems
pagenum: 385; title: Notes and References
pagenum: 389; title: Bibliography
pagenum: 391; title: Symbol Glossary
pagenum: 395; title: Index
pagenum: 397