本书是美国科学院院士Peter D．Lax在CotJrant数学所长期讲授泛函分析课程的教学经验基础上编写的。

本书包括泛函分析的基本内容：Barlach空间、Hilbert空间和线性拓扑空间的基本概念和性质，线性拓扑空间中的凸集及其端点集的性质，有界线性算子的性质等。可作为本科生泛函分析课的教学内容；还包括泛函分析较深的内容：自伴算子的谱分解理论。紧算子的理论，交换Barlach代数的Gelfand理论，不变子空间的理论等。可作为研究生泛函分析课的教学内容；另外。本书还包括泛函分析更为专门的内容，可供讨论班或学生阅读用。

本书特别强调泛函分析与其他数学分支的联系及泛函分析理论的应用，可以使读者深刻地理解到：抽象的泛函分析理论有着丰富的数学背景。

Foreword

1.Linear Spaces

2.Linear Maps

3.The Hahn-Banach Theorem

4.Applications of the Hahn-Banach theorem

5.Normed Linear Spaces

6.Hilbert Space

7.Applications of Hilbert Space Results

8.Duals of Normed Linear Speaces

9.Applications of Duality

10.Weak Convergence

11.Applications of Weak Convergence

12.The Weak and Weak Topologies

13.Locally Convex Topologies and the Krein-Milman Theorem

14.Examples of Convex Sets and Their Extreme Points

15.Bounded Linear Maps

16.Examples of Bounded Linear Maps

17.Banach Algebras and their Elementary Spectral Theory

18.Gelfand's Theory of Commutative Banach Algebras

19.Applications of Gelfand's Theory of Commutative Banach Algebras

20.Examples of Operators and Their Spectra

21.Compact Maps

22.Examples of Compact Operators

23.Positive compact operators

24.Fredholm's Theory of Integral Equations

25.Invariant Subspaces

26.Harmonic Analysis on a Halfline

27.Index Theory

28.Compact Symmetric Operators in Hilbert Space

29.Examples of Compact Sysmmetric Operators

30.Trace Class and Trace Formula

31.Spectral Theory of Symmetric,Normal,and Unitary Operators

32.Spectral Theory of Self-Adjoint Operators

33.Examples of Self-Adjoint Operators

34.Semigroups of Operators

35.Groups of Unitary Operators

36.Examples of Strongly Continuous Semigroups

37.Scattering Theory

38.A Theorem of Beurling

A.Riesz-Kakutani representation theorem

B.Theory of distrbutions

C.Zorn's Lemma

Author Index

Subject Index