Contents Preface List of Symbols Chapter 1 Metric Spaces 1 1.1 Preliminaries 1 1.2 Definitions and Examples 5 1.3 Convergence in a Metric Space 11 1.4 Sets in a Metric Space 16 1.5 Complete Metric Spaces 23 1.6 Continuous Mappings on Metric Spaces 30 1.7 Compact Metric Spaces 35 1.8 The Contraction Mapping Principle 42 Chapter 2 Normed Linear Spaces Banach Spaces 50 2.1 Review of Linear Spaces 50 2.2 Norm in a Linear Space 52 2.3 Examples of Normed Linear Spaces 58 2.4 Finite Dimensional Normed Linear Spaces 66 2.5 Linear Subspaces of Normed Linear Spaces 72 2.6 Quotient Spaces 78 2.7 The Weierstrass Approximation Theorem 81 Chapter 3 Inner Product Spaces Hilbert Spaces 88 3.1 Inner Products 88 3.2 Orthogonality 97 3.3 Orthonormal Systems 105 3.4 Fourier Series 118 Chapter 4 Linear Operators Fundamental Theorems 123 4.1 Continuous Linear Operators and Functionals 123 4.2 Spaces of Bounded Linear Operators and Dual Spaces 138 4.3 The Banach-Steinhaus Theorem 149 4.4 Inverses of Operators The Banach Theorem 155 4.5 The Hahn-Banach Theorem 165 4.6 Strong and Weak Convergence 177 Chapter 5 Linear Operators on Hilbert Spaces 187 5.1 Adjoint Operators The Lax-Milgram Theorem 187 5.2 Spectral Theorem for Self-adjoint Compact Operators 201 Bibliography 224 Index 226