ForewordIntroduction1 Inner product spaces1.1 Inner product spaces as metric spaces1.2 Problems2 Normed spaces2.1 Closed linear subspaces2.2 Problems3 Hiibert and Banach spaces3.1 The space L2(a, b)3.2 The closest point property3.3 Problems4 Orthogonal expansions4.1 Bessel's inequality4.2 Pointwise and L2 convergence4.3 Complete orthonormai sequences4.4 Orthogonal complements4.5 Problems5 Classical Fourier series5.1 The Fejer kernel5.2 Fejer's theorem5.3 Parseval's formula5.4 Weierstrass' approximation theorem5.5 Problems6 Dual spaces6. I The Riesz-Frechet theorem6.2 Problems7 Linear operators7.1 The Banach space .~(E, F)7.2 Inverses of operators7.3 Adjoint operators7.4 Hermitian operators7.5 The spectrum7.6 Infinite matrices7.7 Problems8 Compact operators8.1 Hilbert-Schmidt operators8.2 The spectral theorem for compact Hermitian operators8.3 Problems9 Storm-Liouville systems9.1 Small oscillations of a hanging chain9.2 Eigenfunctions and eigenvalues9.3 Orthogonality of eigenfunctions9.4 Problems10 Green's functions10.1 Compactness of the inverse of a Sturm-Liouville operator10.2 Problems11 Eigenfunction expansions11.1 Solution of the hanging chain problem11.2 Problems12 Positive operators and contractions12.1 Operator matrices12.2 M6bius transformations12.3 Completing matrix contractions12.4 Problems13 Hardy spaces13.1 Poisson's kernel13.2 Fatou's theorem13.3 Zero sets of H2 functions13.4 Multiplication operators and infinite Toeplitz and Hankel matrices13.5 Problems14 Interlude: complex analysis and operators in engineering15 Approximation by analytic functions15.1 The Nehari problem15.2 Hankel operators15.3 Solution of Nehari's problem15.4 Problems16 Appmximatioa by meromorphie functions16.1 The singular values of an operator16.2 Schmidt pairs and singular vectors16.3 The Adamyan-Arov-Krein theorem16.4 ProblemsAppendix: square roots of positive operatorsReferencesAnswers to selected problemsAfierwordIndex of notationSubject index