Introduction 1
1 Inner prodcut spaces 4
1.1 Inner product spaces as metric spaces 6
1.2 Problems 11
2 Normed spaces 13
2.1 Closed linear subspaces 15
2.2 Problems 18
3 Hilbert and Banach spaces 21
3.1 The space L2(a,b) 23
3.2 The closest point property 26
3.3 Problems 28
4 Orthogonal expansions 31
4.1 Bessel's inequality 34
4.2 Pointwise and L2 convergence 35
4.3 Complete orthonormal sequences 36
4.4 Orthogonal complements 39
4.5 Problems 42
5 Classical Fourier series 45
5.1 The Fejér kernel 46
5.2 Fejér's theorem 52
5.3 Parseval's formula 54
5.4 Weierstrass' approximation theorem 54
5.5 Problems 55
6 Dual spaces 59
6.1 The Riesz-Fréchet theorem 62
6.2 Problems 64
7 Linear operators 67
7.1 The Banach space ？(E,F) 71
7.2 Inverses of operators 72
7.3 Adioint operators 75
7.4 Hermitian operators 78
7.5 The spectrum 80
7.6 Infinite matrices 82
7.7 Problems 83
8 Compact operators 89
8.1 Hilbert-Schmidt operators 92
8.2 The spectral theorem for compact Hermitian operators 96
8.3 Problems 102
9 Sturm-Liouville systems 105
9.1 Small oscillations of a hanging chain 105
9.2 Eigenfunctions and eigenvalues 111
9.3 Orthogonality of eigenfunctions 114
9.4 Problems 115
10 Green's functions 119
10.1 Compactness of the inverse of a Sturm-Liouville operator 124
10.2 Problems 128
11 Eigenfunction expansions 131
11.1 Solution of the hanging chain problem 134
11.2 Problems 138
12 Positive operators and contractions 141
12.1 Operator matrices 144
12.2 M？bius transformations 146
12.3 Completing matrix contractions 149
12.4 Problems 152
13 Hardy spaces 157
13.1 Poisson's kernel 161
13.2 Fatou's theorem 164
13.3 Zero sets of H2 functions 169
13.4 Multiplication operators and infinite Toeplitz and Hankel matrices 171
13.5 Problems 174
14 Interlude:complex analysis and operators in engineering 177
15 Approximation by analytic fuctions 187
15.1 The Nehari problem 189
15.2 Hankel operators 190
15.3 Solution of Nehari's problem 196
15.4 Problems 200
16 Approximation by meromorphic functions 203
16.1 The singular values of an operator 204
16.2 Schmidt pairs and singular vectors 206
16.3 The Adamyan-Arov-Krein theorem 210
16.4 Problems 219
Appendix:square roots of positive operators 221
References 225
Answers to selected problems 226
Afterword 230
Index of notation 236
Subject index 238