希尔伯特空间及其应用导论 : 第3版

preface to the third edition

preface to the second edition

preface to the first edition

chapter1 nermed vector spaces

1.1 introduction

1.2 vector spaces

1.3 normed spaces

1.4 knach spaces

1.s linear mappings

1.6 contraction mappings and the banach fixed point theorem

1.7 exercises

chapter2 the lebesgue integral

2.1 introduction

2.2 step functions

2.3 lebesl~e intelfable functions

2.4 the absolute value of on intei fable function

2.5 series of intelqble functions so

2.6 norm in l1（r）

2.7 convergence almost everywhere ss

2.8 fundamentol convergence theorems

2.9 locally integmble functions

2.10 the lebesgue integral and the riemann integral

2.11 lebesgue measure on r

2.12 complex-valued lebesgue integrable functions

2.13 the spaces lp（r）

2.14 lebesgue integrable functions on rn

2.15 convolution

2.16 exercises

chapter3 hilbert spaces and orthonormal systems

3.1 introduction

3.2 inner product spaces

3.3 hilbert spaces

3.4 orthogonal and orthonormal systems

3.5 trigonometric fourier series

3.6 orthogonal complements and projections

3.7 linear functionals and the riesz representation theorem

3.8 exercises

chapter4 linear operators on hilbert spaces

4.1 introduction

4.2 examples of operators

4.3 bilinear functionals and quadratic forms

4.4 adjoint and seif-adjoint operators

4.5 invertible, normal, isometric, and unitary operators

4.6 positive operators

4.7 projection operators

4.8 compact operators

4.9 eigenvalues and eigenvectors

4.10 spectral decomposition

4.11 unbounded operators

4.12 exercises

chapter5 applications to integral and differential equations

5.1 introduction

5.2 basic existence theorems

5.3 fredholm integral equations

5.4 method of successive approximations

5.5 volterra integral equations

5.6 method of solution for a separable kernel

5.7 volterra integral equations of the first kind and abel's integral equation

5.8 ordinary differential equations and differential operators

5.9 sturm-liouville systems

5.10 inverse differential operators and green's functions

5.11 the fourier transform

5.12 applications of the fourier transform to ordinary differential equations and integral equations

6.13 exercises

chapter6 generalized functions and partial differential equations

6.1 introduction

6.2 distributions

6.3* sobolevspaces

6.4 fundamental solutions and green's functions for partial differential equations

6.5 weak solutions of elliptic boundary value problems

6.6 examples of applications of the fourier transform to partial differential equations

6.7 exercises

chapter7 mathematical foundations of @uantum mechanics

7.1 introduction

7.2 basic concepts and equations of classical mechanics poisson's brackets in mechanics

7.3 basic concepts and postulates of quantum mechanics

7.4 the heisenberg uncertainty principle

7.5 the schrodinger equation of motion

7.6 the schrodinger picture

7.7 the heisenberg picture and the heisenberg equation of motion

7.8 the interaction picture

7.9 the linear harmonic oscillator

7.10 angular momentum operators

7.11 the dirac relativistic wave equation

7.12 exercises

chapter8 wavelets and wavelet transforms

8.1 brief historical remarks

8.2 continuous wavelet transforms

8.3 the discrete wavelet transform

8.4 multirosolution analysis and orthonormal bases of wavelets

8.5 examples of orthonormal wavelets

8.6 exercises

chapter9 optimization problems and other miscellaneous applications

9.1 introduction

9.2 the gateaux and frechet differentials

9.3 optimization problems and the euler-lagrange equations

9.4 minimization of quadratic functionals s0s

9.5 variational inequalities s07

9.6 optimal control problems for dynamical systems

9.7 approximation theory

9.8 the shannon samplingtheorem

9.9 linear and nonlinear stability

9.10 bifurcation theory

9.11 exercises

hints and answers to selected exercises

bibliography

index

《希尔伯特空间及其空间导论（英文版）（第3版）》是一部学习希尔伯特空间的入门级教程。无论是学生还是科研人员，都将从本书的特别表达中受益。本书在原来版本的基础上做了不少改动，新增加了一部分讲述Sobolev空间，展开讲述了有限维赋范空间，有关小波的一章做了全面更新。并且包括了积分和微分方程、量子力学、最优化、变分和控制问题、逼近理论问题、非线性不稳定性和分岔理论的多种应用。在众多希尔伯特空间的书中，本书在讲述勒贝格积分方面独具特色。学习泛函分析和希尔伯特理论的老师和学生都十分推崇这本书作为教材或者参考书。