希尔伯特空间导论

出版时间：2012-3  出版社：世界图书出版公司  作者：N. Young  页数：239  
Tag标签：无  

内容概要

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。

书籍目录

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 Hankel
matrices
　13.5 Problems
14 Interlude: complex analysis and operato in
engineering
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

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