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内勒、塞尔编写的《工程与科学中的线性算子理论(英文版)》旨在为工程师、科研工作者和应用数学工作者提供适用于他们的泛函分析的基础知识。尽管书中采取的是定义-定理-证明的数学模式，但是该书在所涵盖知识点的选取和解释说明方面还是下了很大的功夫。该书也可以被用作高级教程，为了便于不同知识背景的学生学习，书中附录部分涵盖了许多有益的数学课题。

Preface

Chapter 1 Introduction

1. Black Boxes

2. Structure of the Plane

3. Mathematical Modeling

4. The Axiomatic Method. The

Process of Abstraction

5. Proofs of Theorems

Chapter 2 Set-Theoretic Structure

1. Introduction

2. Basic Set Operations

3. Cartesian Products

4. Sets of Numbers

5. Equivalence Relations and

Partitions

6. Functions

7. Inverses

8. Systems Types

Chapter 3 Topological Structure

1. Introduction

Port A Introduction to Metric Spaces

2. Metric Spaces: Definition

3. Examples of Metric Spaces

4. Subspaces and Product Spaces

5. Continuous Functions

6. Convergent Sequences

7. A Connection Between

Continuity and Convergence

Part B Some Deeper Metric

Space Concepts

8. Local Neighborhoods

9. Open Sets

10. More on Open Sets

11. Examples of Homeomorphic

Metric Spaces

12. Closed Sets and the Closure

Operation

13. Completeness

14. Completion of Metric Spaces

15. Contraction Mapping

16. Total Boundexlness and

Approximations

17. Compactness

Chapter 4 Algebraic Structure

1. Introduction

Part A Introduction to Linear Spaces

2. Linear Spaces and Linear

Subspaces

3. Linear Transformations

4. Inverse Transformations

5. Isomorphisms

6. Linear Independence and

Dependence

7. Hamel Bases and Dimension

8. The Use of Matrices to Represent

Linear Transformations

9. Equivalent Linear

Transformations

Part B Further Topics

10. Direct Sums and Sums

11. Projections

12. Linear Functionals and the Alge-

braic Conjugate of a Linear Space

13. Transpose of a Linear

Transformation

Chapter 5 Combined Topological

and Algebraic Structure

1. Introduction

Part A Banach Spaces

2. Definitions

3. Examples of Normal Linear

Spaces

4. Sequences and Series

5. Linear Subspaces

6. Continuous Linear

Transformations

7. Inverses and Continuous Inverses

8. Operator Topologies

9. Equivalence of Normed Linear

Spaces

10. Finite-Dimensional Spaces

11. Normed Conjugate Space and

Conjugate Operator

Part B Hilbert Spaces

12. Inner Product and HUbert Spaces

13. Examples

14. Orthogonality

15. Orthogonal Complements and the

Projection Theorem

16. Orthogonal Projections

17. Orthogonal Sets and Bases:

Generalized Fourier Series

18. Examples of Orthonormal Bases

19. Unitary Operators and Equiv-

alent Inner Product Spaces

20. Sums and Direct Sums of

Hilbert Spaces

21. Continuous Linear Functionals

Part C Special Operators

22. The Adjoint Operator

23. Normal and Self-Adjoint

Operators

24. Compact Operators

25. Foundations of Quantum

Mechanics

Chapter 6 Analysis of Linear Oper-

ators (Compact Case)

1. Introductioa

Part A An Illustrative Example

2. Geometric Analysis of Operators

3. Geometric Analysis. The Eigen-

value-Eigenvector Problem

4. A Finite-Dimensional Problem

Part B The Spectrum

5. The Spectrum of Linear

Transformations

6. Examples of Spectra

7. Properties of the Spectrum

Part C Spectral Analysis

8. Resolutions of the Identity

9. Weighted Sums of Projections

10. Spectral Properties of Compact,

Normal, and Self-Adjoint

Operators

11. The Spectral Theorem

12. Functions of Operators

(Operational Calculus)

13. Applications of the Spectral

Theorem

14. Nonnormal Operators

Chapter 7 Analysis of Unbounded

Operators

1. Introduction

2. Green's Functions

3. Symmetric Operators

4. Examples of Symmetric

Operators

5. Sturmiouville Operators

6. Ghrding's Inequality

7. EUiptie Partial Differential

Operators

8. The Dirichlet Problem

9. The Heat Equation and Wave

Equation

10. Self-Adjoint Operators

11. The Cayley Transform

12. Quantum Mechanics, Revisited

13. Heisenberg Uncertainty Principle

14. The Harmonic Oscillator

Appendix ,4 The H61der, Schwartz,

and Minkowski

Inequalities

Appendix B Cardinality

Appendix C Zom's temnm

Appendix D Integration and

Measure Theory

1. Introduction

2. The Riemann Integral

3. A Problem with the Riemann

Integral

4. The Space Co

5. Null Sets

6. Convergence Almost Everywhere

7. The Lebesgue Integral

8. Limit Theorems

9. Miscellany

10. Other Definitions of the Integral

11. The Lebesgue Spaces,

12. Dense Subspaees of

13. Differentiation

14. The Radon-Nikodym Theorem

15. Fubini Theorem

Appendix E Probability Spaces and

Stochastic Processes

1. Probability Spaces

2. Random Variables and

Probability Distributions

3. Expectation

4. Stochastic Independence

5. Conditional Expectation Operator

6. Stochastic Processes

Index of Symbols

Index