《希尔伯特空间及应用导论（第3版）》是一部深入介绍希尔伯特空间理论及其广泛应用的教材。书中内容从内积空间和希尔伯特空间的基本概念出发，详细阐述了这些空间的几何性质和重要定理。同时，本书还通过丰富的实例和详尽的解释，展示了希尔伯特空间在傅里叶分析、积分方程、微分方程和量子力学等多个领域的实际应用。内容组织严谨，语言简洁明了，适合数学、物理和工程领域的研究生和研究人员阅读。通过阅读本书，读者不仅能够系统地掌握希尔伯特空间的理论知识，还能将其灵活应用于实际问题的解决中。

理论与实践的深度结合
本书以清晰的逻辑构建希尔伯特空间理论体系，从基本概念到重要定理层层递进，同时通过傅里叶分析、量子力学等领域的实际案例，展现理论在多个学科中的具体应用，帮助读者建立完整的知识框架。
内容编排科学合理
第三版在保持前版严谨结构的基础上，增加了更多直观的实例说明和详细的推导过程，使抽象的泛函分析概念变得易于理解。特别适合作为研究生阶段系统学习泛函分析的教材。
跨学科视角的独特呈现
由资深数学家德布纳斯（国际知名数学期刊创始人）和米库辛斯基（广义函数理论专家）共同编写，融合了理论研究与工程应用的多元视角，为读者提供了既有理论深度又具实用价值的学习资源。

洛肯纳斯·德布纳斯（Lokenath Debnath），一位卓越的印度裔美国数学家，现任德克萨斯里奥格兰德河谷大学（原德克萨斯泛美大学）数学系主任。德布纳斯博士在数学领域深耕细作，发表了数百篇高质量的研究论文和文章，同时撰写了多部涵盖广泛数学课题的教科书。此外，他还是国际知名的数学期刊《国际数学与数学科学杂志》的创办者，为推动数学研究的发展做出了杰出贡献。

皮奥特·米库辛斯基（Piotr Mikusiński）毕业于波兰科学学院数学研究所，获得数学博士学位后，曾在加州大学圣巴巴拉分校担任客座讲师。目前，米库辛斯基教授是佛罗里达大学数学系的教授，主要研究方向为广义函数理论和实分析。

Preface to the Third Edition xi

Preface to the Second Edition xiii

Preface to the First Edition xv

CHAPTER 1 Normed Vector Spaces 1

1.1 Introduction 1

1.2 Vector Spaces 2

1.3 Normed Spaces 8

1.4 Banach Spaces 19

1.5 Linear Mappings 25

1.6 Contraction Mappings and the Banach Fixed Point Theorem 32

1.7 Exercises 34

CHAPTER 2 The Lebesgue Integral 39

2.1 Introduction 39

2.2 Step Functions 40

2.3 Lebesgue Integrable Functions 45

2.4 The Absolute Value of an Integrable Function 48

2.5 Series of Integrable Functions 50

2.6 Norm in 52

2.7 Convergence Almost Everywhere 55

2.8 Fundamental Convergence Theorems 58

2.9 Locally Integrable Functions 62

2.10 The Lebesgue Integral and the Riemann Integral 64

2.11 Lebesgue Measure on 67

2.12 Complex-Valued Lebesgue Integrable Functions 71

2.13 The Spaces 74

2.14 Lebesgue Integrable Functions on78

2.15 Convolution 82

2.16 Exercises 84

CHAPTER 3 Hilbert Spaces and Orthonormal Systems 93

3.1 Introduction 93

3.2 Inner Product Spaces 94

3.3 Hilbert Spaces 99

3.4 Orthogonal and Orthonormal Systems 105

3.5 Trigonometric Fourier Series 122

3.6 Orthogonal Complements and Projections 127

3.7 Linear Functionals and the Riesz Representation Theorem 132

3.8 Exercises 135

CHAPTER 4 Linear Operators on Hilbert Spaces 145

4.1 Introduction 145

4.2 Examples of Operators 146

4.3 Bilinear Functionals and Quadratic Forms 151

4.4 Adjoint and Self-Adjoint Operators 158

4.5 Invertible, Normal, Isometric, and Unitary Operators 163

4.6 Positive Operators 168

4.7 Projection Operators 175

4.8 Compact Operators 180

4.9 Eigenvalues and Eigenvectors 186

4.10 Spectral Decomposition 196

4.11 Unbounded Operators 201

4.12 Exercises 211

CHAPTER 5 Applications to Integral and Differential Equations 217

5.1 Introduction 217

5.2 Basic Existence Theorems 218

5.3 Fredholm Integral Equations 224

5.4 Method of Successive Approximations 226

5.5 Volterra Integral Equations 228

5.6 Method of Solution for a Separable Kernel 233

5.7 Volterra Integral Equations of the First Kind and

Abels Integral Equation 236

5.8 Ordinary Differential Equations and Differential Operators 239

5.9 SturmLiouville Systems 247

5.10 Inverse Differential Operators and Greens Functions 253

5.11 The Fourier Transform 258

5.12 Applications of the Fourier Transform to Ordinary

Differential Equations and Integral Equations 271

5.13 Exercises 279

CHAPTER 6 Generalized Functions and Partial Differential

Equations 287

6.1 Introduction 287

6.2 Distributions 288

6.3 Sobolev Spaces 300

6.4 Fundamental Solutions and Greens Functions for

Partial Differential Equations 303

6.5 Weak Solutions of Elliptic Boundary Value Problems 323

6.6 Examples of Applications of the Fourier Transform to

Partial Differential Equations 329

6.7 Exercises 343

CHAPTER 7 Mathematical Foundations of Quantum Mechanics 351

7.1 Introduction 351

7.2 Basic Concepts and Equations of Classical Mechanics 352

Poissons Brackets in Mechanics 361

7.3 Basic Concepts and Postulates of Quantum Mechanics 363

7.4 The Heisenberg Uncertainty Principle 377

7.5 The Schrdinger Equation of Motion 379

7.6 The Schrdinger Picture 395

7.7 The Heisenberg Picture and the Heisenberg Equation

of Motion 401

7.8 The Interaction Picture 405

7.9 The Linear Harmonic Oscillator 407

7.10 Angular Momentum Operators 412

7.11 The Dirac Relativistic Wave Equation 420

7.12 Exercises 423

CHAPTER 8 Wavelets and Wavelet Transforms 433

8.1 Brief Historical Remarks 433

8.2 Continuous Wavelet Transforms 436

8.3 The Discrete Wavelet Transform 444

8.4 Multiresolution Analysis and Orthonormal Bases of

Wavelets 452

8.5 Examples of Orthonormal Wavelets 462

8.6 Exercises 473

CHAPTER 9 Optimization Problems and Other Miscellaneous

Applications 477

9.1 Introduction 477

9.2 The Gateaux and Fréchet Differentials 478

9.3 Optimization Problems and the EulerLagrange

Equations 490

9.4 Minimization of Quadratic Functionals 505

9.5 Variational Inequalities 507

9.6 Optimal Control Problems for Dynamical Systems 510

9.7 Approximation Theory 517

9.8 The Shannon Sampling Theorem 522

9.9 Linear and Nonlinear Stability 526

9.10 Bifurcation Theory 530

9.11 Exercises 535

Hints and Answers to Selected Exercises 547

Bibliography 565

Index 571