希尔伯特空间导论

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. I The Riesz-Frechet theorem 6.2 Problems 7 Linear operators 7.1 The Banach space .~(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