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内容简介
Introduction
1 Inner prodcut spaces
1.1 Inner product spaces as metric spaces
1.2 Problems
2 Normed spaces
2.1 Closed linear subspaces
2.2 Problems
3 Hilbert 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 orthonormal sequences
4.4 Orthogonal complements
4.5 Problems
5 Classical Fourier series
5.1 The Fejér kernel
5.2 Fejér's theorem
5.3 Parseval's formula
5.4 Weierstrass' approximation theorem
5.5 Problems
6 Dual spaces
6.1 The Riesz-Fréchet theorem
6.2 Problems
7 Linear operators
7.1 The Banach space ？(E,F)
7.2 Inverses of operators
7.3 Adioint 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 Sturm-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 M？bius 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 fuctions
15.1 The Nehari problem
15.2 Hankel operators
15.3 Solution of Nehari's problem
15.4 Problems
16 Approximation by meromorphic 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
Afterword
Index of notation
Subject index
