希尔伯特空间导论 英文版PDF电子书下载

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