本书以简洁的方式介绍了泛函分析的所有基本概念和结果，略去了更专的主题。作者根据需要介绍了足够的Sobolev空间和线性算子半群的理论，用以发展泛函分析在椭圆型、抛物型和双曲型偏微分方程中的重要应用。贯穿全书，作者详尽解释了泛函分析中的定理与有限维线性代数中的熟知结果之间的联系。
本书使用了大量插图来解释证明中用到的主要概念和思想，大部分章节末尾都包含了数量可观的习题。
本书可作为一学期研究生课程的教材，也可供对泛函分析和偏微分方程感兴趣的相关专业的研究生阅读参考。

Preface
Chapter 1.Introduction
§1.1.Linear equations
§1.2.Evolution equations
§1.3.Function spaces
§1.4.Compactness
Chapter 2.Banach Spaces
§2.1.Basic definitions
§2.2.Linear operators
§2.3.Finite-dimensional spaces
§2.4.Seminorms and Fréchet spaces
§2.5.Extension theorems
§2.6.Separation of convex sets
§2.7.Dual spaces and weak convergence
§2.8.Problems
Chapter 3.Spaces of Continuous Functions
§3.1.Bounded continuous functions
§3.2.The Stone-Weierstrass approximation theorem
§3.3.Ascoli's compactness theorem
§3.4.Spaces of Holder continuous functions
§3.5.Problems
Chapter 4.Bounded Linear Operators
§4.1.The uniform boundedness principle
§4.2.The open mapping theorem
§4.3.The closed graph theorem
§4.4.Adjoint operators
§4.5.Compact operators
§4.6.Problems
Chapter 5.Hilbert Spaces
§5.1.Spaces with an inner product
§5.2.Orthogonal projections
§5.3.Linear functionals on a Hilbert space
§5.4.Gram-Schmidt orthogonalization
§5.5.Orthonormal sets
§5.6.Positive definite operators
§5.7.Weak convergence
§5.8.Problems
Chapter 6.Compact Operators on a Hilbert Space
§6.1.Fredholm theory
§6.2.Spectrum of a compact operator
§6.3.Selfadjoint operators
§6.4.Problems
Chapter 7.Semigroups of Linear Operators
§7.1.Ordinary differential equations in a Banach space
§7.2.Semigroups of linear operators
§7.3.Resolvents
§7.4.Generation of a semigroup
§7.5.Problems
Chapter 8.Sobolev Spaces
§8.1.Distributions and weak derivatives
§8.2.Mollifications
§8.3.Sobolev spaces
§8.4.Approximations of Sobolev functions
§8.5.Extension operators
§8.6.Embedding theorems
§8.7.Compact embeddings
§8.8.Differentiability properties
§8.9.Problems
Chapter 9.Linear Partial Differential Equations
§9.1.Elliptic equations
§9.2.Parabolic equations
§9.3.Hyperbolic equations
§9.4.Problems
Appendix.Background Material
§A.1.Partially ordered sets
§A.2.Metric and topological spaces
§A.3.Review of Lebesgue measure theory
§A.4.Integrals of functions taking values in a Banach space
§A.5.Mollifications
§A.6.Inequalities
§A.7.Problems
Summary of Notation
Bibliography
Index