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 L2a, 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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