德布納斯編著的《希爾伯特空間及其應用導論（第3版）（英文影印版）》是一部學習希爾伯特空間的入門級教程。無論是學生還是科研人員，都將從本書的特別表達中受益。本書在原來版本的基礎上做了不少改動，新增加了一部分講述Sobolev空間，展開講述了有限維賦範空間，有關小波的一章做了全面*新。並且包括了積分和微分方程、量子力學、*優化、變分和控制問題、逼近理論問題、非線性不穩定性和分岔理論的多種應用。在眾多希爾伯特空間的書中，本書在講述勒貝格積分方面**特色。學習泛函分析和希爾伯特理論的老師和學生都十分推崇這本書作為教材或者參考書。

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