希爾伯特空間導論 英文

Foreword

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 expansions
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 Parseval’’s formula
5.4 Weierstrass’’ approximation theorem
5.5 Problems

6 Dual spaces
6.1 The Riesz—Frechet theorem
6.2 Problems

7 Linear operators
7.1 The Banach space y（E， F）
7.2 Inverses of operators
7.3 Adjoint operators
7.4 Hermitian operators
7.5 The spectrum
7.6 Infinite matrices
7.7 Problems

8 Compact operators
8.1 Hilbert—Schmidt operators
8.2 The spectral theorem for compact Hermitian operators
8.3 Problems
9 Storm—Liouville systems
9.1 Small oscillations of a hanging chain
9.2 Eigenfunctions and eigenvalues
9.3 Orthogonality of eigenfunctions
9.4 Problems

10 Green’’s functions
10.1 Compactness of the inverse of a Sturm—Liouville operator
10.2 Problems

11 Eigenfunction expansions
11.1 Solution of the hanging chain problem
11.2 Problems

12 Positive operators and contractions
12.1 Operator matrices
12.2 M6bius transformations
12.3 Completing matrix contractions
12.4 Problems

13 Hardy spaces
13.1 Poisson’’s kernel
13.2 Fatou’’s theorem
13.3 Zero sets of H2 functions
13.4 Multiplication operators and infinite Toeplitz and Hankel matrices
13.5 Problems

14 Interlude： complex analysis and operators in engineering
15 Approximation by analytic functions
15.1 The Nehari problem
15.2 Hankel operators
15.3 Solution of Nehari’’s problem

15.4 Problems

16 Appmximatioa by meromorphie functions
16.1 The singular values of an operator
16.2 Schmidt pairs and singular vectors
16.3 The Adamyan—Arov—Krein theorem
16.4 Problems

Appendix： square roots of positive operators
References
Answers to selected problems
Afierword
Index of notation
Subject index～