布勒齐著的《泛函分析索伯列夫空间和偏微分方程(英文版)》提出了一个连贯的、确切的、统一的方法将两个来自不同领域的元素——泛函分析和偏微分方程，结合在一起，旨在为具有良好实分析背景的学生提供帮助。通过详细地分析一维PDEs的简单案例，即ODEs，一个对初学者来说比较简单的方法，该书展示了从泛函分析到偏微分方程的平滑过渡。

Preface

The Hahn-Banach Theorems. Introduction to the Theory of

Conjugate Convex Functions

1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of

Linear Functionals

1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation

of Convex Sets

1.3 The Bidual E. Orthogonality Relations

1.4 A Quick Introduction to the Theory of Conjugate Convex Functions

Comments on Chapter 1

Exercises for Chapter 1

2 The Uniform Boundedness Principle and the Closed Graph Theorem

2.1 The Baire Category Theorem

2.2 The Uniform Boundedness Principle

2.3 The Open Mapping Theorem and the Closed Graph Theorem

2.4 Complementary Subspaces. Right and Left inve.rtibility of Linear

Operators

2.5 Orthogonality Revisited

2.6 An Introduction to Unbounded Linear Operators. Definition of the

Adjoint

2.7 A Characterization of Operators with Closed Range.

A Characterization of Surjective Operators

Comments on Chapter 2

Exercises for Chapter 2

Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform

Convexity

3.1 The Coarsest Topology for Which a Collection of Maps Becomes

Continuous

3.2 Definition and Elementary Properties of the Weak Topology

a(E, E*)

3.3 Weak Topology, Convex Sets, and Linear Operators

3.4- The Weak* Topology tr (E', E)

3.5 Reflexive Spaces

3.6 Separable Spaces

3.7 Uniformly Convex Spaces

Comments on Chapter 3

Exercises for Chapter 3

4 Lp Spaces

4.1 Some Results about Integration That Everyone Must Know

4.2 Definition and Elementary Properties of Lp Spaces

4.3 Reflexivity. Separability. Dual of Lp

4.4 Convolution and regularization

4.5 Criterion for Strong Compactness in Lp

Comments on Chapter 4

Exercises for Chapter 4

5 Hilbert Spaces

5.1 Definitions and Elementary Properties. Projection onto a Closed

Convex Set

5.2 The Dual Space of a Hilbert Space

5.3 The Theorems of Stampacchia and Lax-Milgram

5.4 Hilbert Sums. Orthonormal Bases

Comments on Chapter 5

Exercises for Chapter 5

Compact Operators. Spectral Decomposition of Self-Adjoint

Compact Operators

6.1 Definitions. Elementary Properties. Adjoint

6.2 The Riesz-Fredholm Theory

6.3 The Spectrum of a Compact Operator

6.4 Spectral Decomposition of Self-Adjoint Compact Operators

Comments on Chapter 6

Exercises for Chapter 6

The Hille--Yosida Theorem

7.1 Definition and Elementary Properties of Maximal Monotone

Operators

7.2 Solution of the Evolution Problem du

"37 + Au = 0 on [0, +cx),

u(0) = u0. Existence and uniqueness

7.3 Regularity

7.4 The Self-Adjoint Case

Comments on Chapter 7

8 Sobolev Spaces and the Variational Formulation of Boundary Value

Problems in One Dimension

8.1 Motivation

8.2 The Sobolev Space Wl'P(l)

8.3 The Space W 'p

8.4 Some Examples of Boundary Value Problems

8.5 The Maximum Principle

8.6 Eigenfunctions and Spectral Decomposition

Comments on Chapter 8

Exercises for Chapter 8

9 Sobolev Spaces and the Variational Formulation of Elliptic

Boundary Value Problems in N Dimensions

9.1 Definition and Elementary Properties of the Sobolev Spaces

WI,P()

9.2 Extension Operators

9.3 Sobolev Inequalities

9.4 The Space W'P(f2)

9.5 Variational Formulation of Some Boundary Value Problems

9.6 Regularity of Weak Solutions

9.7 The Maximum Principle

9.8 Eigenfunctions and Spectral Decomposition

Comments on Chapter 9 .

10 Evolution Problems: The Heat Equation and the Wave Equation ..

I0.1 The Heat Equation: Existence, Uniqueness, and Regularity

10.2 The Maximum Principle

10.3 The Wave Equation

Comments on Chapter 10

11 Miscellaneous Complements

11.1 Finite-Dimensional and Finite-Codimensional Spaces

11.2 Quotient Spaces

11.3 Some Classical Spaces of Sequences

11.4 Banach Spaces over C: What Is Similar and What Is Different?..

Solutions of Some Exercises

Problems

Partial Solutions of the Problems

Notation

References

Index