实分析.pdf

实分析.pdf
 

书籍描述

编辑推荐
《实分析(英文版)》由世界图书出版公司北京公司出版。

作者简介
作者:(美国)斯坦恩(Elias M.Stein) (美国)Rami Shakarchi

目录
Foreword
Introduction
1 P0urier series:completion
2 Limits of continuous functions
3 Length of curves
4 Dliferentiation and integration
5 The problem of measure
Chapter l.Measure Theory
1 Preliminaries
2 The exterior measure
3 Measurable sets and the Lebesgue measure
4 Measurable functions
4.1 Definition and basic properties
4.2 Approximation by simple functions or step functions
4.3 Littlewood?S three principles
5 The Brunn-Minkowski inequality
6 Exercises
7 Problems
Chapter 2.Integration Theory
1 The Lebesgue integral:basic properties and convergence
theorems
2 The space L1 of integrable functions
3 Fubini's theorem
3.1 Statement and proof of the theorem
3.2 Applications of Fubini's theorem
4 A Fourier inversion formula
5 Exercises
6 Problems
Chapter 3.Differentiation and Integration
1 Differentiation of the integral
1.1 The Hardy-Littlewood maximal function
1.2 The Lebesgue differentiation theorem
2 Good kernels and approximations to the identity
3 Differentiability of functions
3.1 Functions of bounded variation
3.2 Absolutely continuous functions
3.3 Differentiability of jump functions
4 Rectifiable curves and the isoperimetric inequality
4.1 Minkowski content of a curve
4.2 Isoperimetric inequality
5 Exercises
6 Problems
Chapter 4. Hilbert Spaces:An Introduction
1 The Hilbert space L2
2 Hilbert spaces
2.1 Orthogonality
2.2 Unitary mappings
2.3 Pre-Hilbert spaces
3 Fourier series and Fatou's theorem
3.1 Fatou's theorem
4 Closed subspaces and orthogonal projections
5 Linear transformations
5.1 Linear functionals and the Riesz representation the-orem
5.2 Adjoints
5.3 Examples
6 Compact operators
7 Exercises
8 Problems
Chapter 5. Hilbert Spaces:Several Examples
1 The Fourier transform on L2
2 The Hardy space of the upper half-plane
3 Constant coefficient partial differential equations
3.1 Weak solutions
3.2 The main theorem and key estimate
4* The Dirichlet principle
4.1 Harmonic functions
4.2 The boundary value problem and Dirichlet's principle
5 Exercises
6 Problems
Chapter 6. Abstract Measure and Integration Theory
1 Abstract measure spaces
1.1 Exterior measures and Carath6odory's theorem
1.2 Metric exterior measures
1.3 The extension theorem
2 Integration on a measure space
3 Examples
3.1 Product measures and a general Fubini theorem
3.2 Integration formula for polar coordinates
3.3 Borel measures on R and the Lebesgue-Stieltjes in-tegral
4 Absolute continuity of measures
4.1 Signed measures
4.2 Absolute continuity
5 Ergodic theorems
5.1 Mean ergodic theorem
5.2 Maximal ergodic theorem
5.3 Pointwise ergodic theorem
5.4 Ergodic measure-preserving transformations
6 Appendix:the spectral theorem
6.1 Statement of the theorem
6.2 Positive operators
6.3 Proof of the theorem
6.4 Spectrum
7 Exercises
8 Problems
Chapter 7. Hausdorff Measure and Fractals
1 Hausdorff measure
2 Hausdorff dimension
2.1 Examples
2.2 Self-similarity
3 Space-filling curves
3.1 Quartic intervals and dyadic squares
3.2 Dyadic correspondence
3.3 Construction of the 15eano mapping
4 Besicovitch sets and regularity
4.1 The Radon transform
4.2 Regularity of sets when d≥3
4.3 Besicovitch sets have dimension 2
4.4 Construction of a Besicovitch set
5 Exercises
6 Problems
Notes and References
Bibliography
Symbol Glossary
Index

文摘
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实分析

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实分析

内容简介
《实分析(英文版)》是一部为数学及相关专业大学二年级和三年级学生编写的教材,理论与实践并重。为了便于非数学专业的学生学习,《实分析(英文版)》内容简明、易懂,读者只需掌握微积分和线性代数知识。

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