实分析

内容概要

　　《实分析（英文）》在Princeton大学使用，同时在其它学校，比如UCLA等名校也在本科生教学中得到使用。其教学目的是，用统一的、联系的观点来把现代分析的“核心”内容教给本科生，力图使本科生的分析学课程能接上现代数学研究的脉络。

书籍目录

Foreword
Introduction
  1 Fourier series： completion
  2 Limits of continuous functio
  3 Length of curves
  4 Differentiation and integration
  5 The problem of measure
Chapter  1. Measure Theory
  1 Prelhninaries
  2 The exterior measure
  3 Measurable sets and the Lebesgue measure
  4 Measurable functio
    4.1 Definition and basic properties
    4.2 Approximation by simple functio or step functio
    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 ofintegrable functio
  3 Fubini's theorem
    3.1 Statement and proof of the theorem
    3.2 Applicatio of Fubini's theorem
  4* A Fourier inveion 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 approximatio to the identity
  3 Differentiability of functio
    3.1 Functio of bounded variation
    3.2 Absolutely continuous functio
    3.3 Differentiability ofjump functio
  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 projectio
  5 Linear traformatio
    5.1 Linear functionals and the Riesz representation theorem
    5.2 Adjoints
    5.3 Examples
  6 Compact operato
  7 Exercises
  8 Problems
Chapter  5. Hilbert Spaces： Several Examples
  1 The Fourier traform on L2
  2 The Hardy space of the upper half-plane
  3 Cotant coefficient partial differential equatio
    3.1 Weaak solutio
    3.2 The main theorem and key estimate
  4 The Dirichlet principle
    4.1 Harmonic functio
    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 Carathodory's theorem
    1.2  Metric exterior measures
    1.3  The exteion theorem
  2  Integration o 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  and the Lebesgue-Stieltjes integral
  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 traformatio
  6* Appendix: the spectral theorem
    6.1  Statement of the theorem
    6.2  Positive operato
    6.3  Proof of the theorem
    6.4  Spectrum
  7  Exercises
  8  Problems
Chapter 7. Hausdorff Measure and Fractals
  1  Hausdorff measure
  2  Hausdorff dimeion
    2.1  Examples
    2.2  Self-similarity
  3  Space-filling curves
    3.1  Quartic intervals and dyadic squares
    3.2  Dyadic correspondence
    3.3  Cotruction of the Peano mapping
  4* Besicovitch sets and regularity
    4.1  The Radon traform
    4.2  Regularity of sets when d ≥ 3
    4.3  Besicovitch sets have dimeion 2
    4.4  Cotruction of a Besicovitch set
  5  Exercises
  6  Problems
Notes and References
Bibliography
Symbol Glossary
Index

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