泛函分析

Foreword

Preface

Chapter 1.Lp Spaces and Banach Spaces

1 Lp spaces

1.1 The Holder and Minkowski inequalities

1.2 Corupleteness of Lp

1.3 Further remarks

2 The case p = ∞

3 Banach spaces

3.1 Examples

3.2 Linear functionals and the dual of a Banach space

4 The dual space of Lp when l < p < ∞

5 More about linear functionals

5.1 Separation of convex sets

5.2 The Hahn-Banach Theorem

5.3 Some consequences

5.4 The Droblem of measure

6 Complex Lp and Banach spaces

7 Appendix： The dual of C（X）

7.1 The case of positive linear functionals

7.2 The main result

7.3 An extension

8 Exercises

9 Problems

Chapter 2.Lp Spaces in Harmonic Analysis

1 Early Motivations

2 The Riesz interpolation theorem

2.1 Some examples

3 The Lp t.heory of the Hilbert t.ransform

3.1 The L2 formalism

3.2 The Lp theorem

3.3 Proof of Theorem 3.2

4 The maximal function and weak-type estimates

4.1 The Lp inequality

……

Chapter 3.Distributions： Generalized Functions

Chapter 4.Applications of the Baire Category Theorem

Chapter 5.Rudiments of Probability Theory

Chapter 6.An Introduction to Brownian Motion

Chapter 7.A Glimpse into Several Complex Variables

Chapter 8.Oscillatory Integrals in Fourier Analysis

Notes and References

Bibliography

Svmbol Glossary

Index

《泛函分析（英文版）》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。