1．Preliminaries 1
1.1 Linear Algebra 2
1.2 Metric Spaces 11
1.3 Lebesgue Integration 20
2．Normed Spaces 31
2.1 Examples of Normed Spaces 31
2.2 Finite-dimensional Normed Spaces 39
2.3 Banach Spaces 45
3．Inner Product Spaces,Hilbert Spaces 51
3.1 Inner Products 51
3.2 Orthogonality 60
3.3 Orthogonal Complements 65
3.4 Orthonormal Bases in Infinite Dimensions 72
3.5 Fourier Series 82
4．Linear Operators 87
4.1 Continuous Linear Transformations 87
4.2 The Norm of a Bounded Linear Operator 96
4.3 The Space B(X,Y)and Dual Spaces 104
4.4 Inverses of Operators 111
5．Linear Operators on Hilbert Spaces 123
5.1 The Adjoint of an Operator 123
5.2 Normal,Self-adjoint and Unitary Operators 132
5.3 The Spectrum of an Operator 139
5.4 Positive Operators and Projections 148
6．Compact Operators 161
6.1 Compact Operators 161
6.2 Spectral Theory of Compact Operators 172
6.3 Self-adjoint Compact Operators 182
7．Integral and Differential Equations 191
7.1 Fredholm Integral Equations 191
7.2 Volterra Integral Equations 201
7.3 Differential Equations 203
7.4 Eigenvalue Problems and Green's Functions 208
8．Solutions to Exercises 221
Further Reading 265
References 267
Notation Index 269
Index 271