Contents
Preface i
Introduction iii
List of Symbols vii
Chapter 1 Metric Spaces 1
1.1 Preliminaries 1
1.2 Definitions and Examples 6
1.3 Convergence of Sequences in Metric Spaces 12
1.4 Sets in a Metric Space 17
1.5 Complete Metric Spaces 25
1.6 Continuous Mappings on Metric Spaces 33
1.7 Compact Metric Spaces 38
1.8 Banach Fixed Point Theorem 46
Chapter 2 Normed Linear Spaces. Banach Spaces 57
2.1 Review of Linear Spaces 57
2.2 Norms in Linear Spaces 59
2.3 Examples of Normed Linear Spaces 65
2.4 Finite-Dimensional Normed Linear Spaces 77
2.5 Linear Subspaces of Normed Linear Spaces 83
2.6 Quotient Spaces 90
2.7 Weierstrass Approximation Theorem 94
Chapter 3 Inner Product Spaces. Hilbert Spaces 101
3.1 Inner Products 101
3.2 Orthogonality 114
3.3 Orthonormal Systems 123
3.4 Fourier Series 138
Chapter 4 Linear Operators. Fundamental Theorems 145
4.1 Bounded Linear Operators and Functionals 145
4.2 Spaces of Bounded Linear Operators and Dual Spaces 162
4.3 Banach-Steinhaus Theorem 173
4.4 Inverses of Operators. Banach's Theorem 180
4.5 Hahn-Banach Theorem 190
4.6 Strong and Weak Convergence 203
Chapter 5 Linear Operators on Hilbert Spaces 215
5.1 Adjoint Operators. Lax-Milgram Theorem 215
5.2 Spectral Theorem for Self-adjoint Compact Operators 229
Chapter 6 Differential Calculus in Normed Linear Spaces 257
6.1 Gateaux and Frechet Derivatives 257
6.2 Taylor's Formula, Implicit and Inverse Function Theorems 270
Bibliography 279
Index 283