希尔伯特空间导论（英文版）
出版时间：2012年版
内容简介
在数学领域，希尔伯特空间是欧几里德空间的一个推广，其不再局限于有限维的情形。与欧几里德空间相仿，希尔伯特空间也是一个内积空间，其上有距离和角的概念。此外，希尔伯特空间还是一个完备的空间，其上所有的柯西列等价于收敛列，从而微积分中的大部分概念都可以无障碍地推广到希尔伯特空间中。希尔伯特空间为基于任意正交系上的多项式表示的傅立叶级数和傅立叶变换提供了一种有效的表述方式，而这也是泛函分析的核心概念之一。希尔伯特空间是公式化数学和量子力学的关键性概念之一。这本《希尔伯特空间导论》（作者勇）是英文导论本。
目录
Introduction
1 Inner product spaces
1.1 Inner product spaces as metric spaces
1.2 Problems
2 Normed spaces
2.1 Closed linear subspaces
2.2 Problems
3 Hiibert and Banach spaces
3.1 The space L2(a, b)
3.2 The closest point property
3.3 Problems
4 Orthogonal expa io
4.1 Bessel's inequality
4.2 Pointwise and L2 convergence
4.3 Complete orthonormai sequences
4.4 Orthogonal complements
4.5 Problems
5 Classical Fourier series
5.1 The Fejer kernel
5.2 Fejer's theorem
5.3 Pa eval's formula
5.4 Weie trass' approximation theorem
5.5 Problems
6 Dual spaces
6.1 I The Riesz-Frechet theorem
6.2 Problems
7 Linear operato
7.1 The Banach space .~(E, F)
7.2 Inve es of operato
7.3 Adjoint operato
7.4 Hermitian operato
7.5 The spectrum
7.6 Infinite matrices
7.7 Problems
8 Compact operato
8.1 Hilbert-Schmidt operato
8.2 The spectral theorem for compact Hermitian operato
8.3 Problems
9 Storm-Liouville systems
9.1 Small oscillatio of a hanging chain
9.2 Eigenfunctio and eigenvalues
9.3 Orthogonality of eigenfunctio
9.4 Problems
10 Green's functio
10.1 Compactness of the inve e of a Sturm-Liouville operator
10.2 Problems
11 Eigenfunction expa io
11.1 Solution of the hanging chain problem
11.2 Problems
12 Positive operato and contractio
12.1 Operator matrices
12.2 M6bius tra formatio
12.3 Completing matrix contractio
12.4 Problems
13 Hardy spaces
13.1 Poisson's kernel
13.2 Fatou's theorem
13.3 Zero sets of H2 functio
13.4 Multiplication operato and infinite Toeplitz and Hankelmatrices
13.5 Problems
14 Interlude: complex analysis and operato inengineering
15 Approximation by analytic functio
15.1 The Nehari problem
15.2 Hankel operato
15.3 Solution of Nehari's problem
15.4 Problems
16 Appmximatioa by meromorphie functio
16.1 The singular values of an operator
16.2 Schmidt pai and singular vecto
16.3 The Adamyan-Arov-Krein theorem
16.4 Problems
Appendix: square roots of positive operato
References
A we to selected problems
Afierword
Index of notation
Subject index
