希爾伯特空間導(dǎo)論

內(nèi)容概要

在數(shù)學(xué)領(lǐng)域，希爾伯特空間是歐幾里德空間的一個推廣，其不再局限于有限維的情形。與歐幾里德空間相仿，希爾伯特空間也是一個內(nèi)積空間，其上有距離和角的概念。此外，希爾伯特空間還是一個完備的空間，其上所有的柯西列等價于收斂列，從而微積分中的大部分概念都可以無障礙地推廣到希爾伯特空間中。希爾伯特空間為基于任意正交系上的多項式表示的傅立葉級數(shù)和傅立葉變換提供了一種有效的表述方式，而這也是泛函分析的核心概念之一。希爾伯特空間是公式化數(shù)學(xué)和量子力學(xué)的關(guān)鍵性概念之一。
這本《希爾伯特空間導(dǎo)論》(作者勇)是英文導(dǎo)論本。

書籍目錄

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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