实分析

Foreword

Introduction

1 Fourier series： completion

2 Limits of continuous functions

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 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 ofintegrable 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 ofjump 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 theorem

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 Weaak 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

Chapter 7. Hausdorff Measure and Fractals

Notes and References

Bibliography

Symbol Glossary

Index

《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。