Elias M.Stein、Rami Shakarchi所著的《泛函分析》旨在全面剖析分析的核心，从泛函分析的基础开始，讲述巴纳赫空间、LP空间和分布理论，强调了它们在调和分析中的核心地位。接着应用Baire范畴定理详解了一些重点，包括Besicovitch集合的存在性；本书的第二部分引导读者进入概率论和布朗运动等分析的其他核心话题，以Dirichlet问题作为结束；最后几章讲述了多复变量和傅里叶分析中的振荡积分，并简述了在非线性色散方程中的计数网格点问题中的应用。作者通篇紧紧围绕这个理论诸领域的核心思想，使得本课题的各个有机部分更加紧凑，层次分明，清晰易懂

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

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