First Course in Functional Analysis

✍ Scribed by Casper Goffman, George Pedrick

• Publisher: Prentice Hall
• Year: 1965
• Tongue: English
• Leaves: 299
• Edition: 1st
• Category: Library

✦ Synopsis

This second edition includes exercises at the end of each chapter, revised bibliographies, References and an Index.

✦ Table of Contents

CONTENTS

Chapter I - Metric Space 1
1.1 Definitions and Examples 1
1.2 Inequalities of Holder and Minkowski 2
1.3 Examples Continued; $l_p$ Spaces 4
1.4 Examples Continued; Function Spaces 6
1.5 Convergence and Related Notions 8
1.6 Separable Space, Examples 9
1.7 Complete Space, Examples 11
1.8 Contractions, Applications to Differential and Integral Equations 16
1.9 Completion 20
1.10 Category, Nowhere Differentiable Continuous Functions 21
1.11 Compactness, Continuity 24
1.12 Equicontinuity, Application to Differential Equations 28
1.13 Stone-Weierstrass Theorems 32
1.14 Normal Families 35
1.15 Semi-continuity, Application to Arc Length 38
1.16 Space of Compact, Convex Sets 41
Exercises 43

Chapter 2 - Banach Spaces 50
2.1 Vector Space 50
2.2 Subspace 52
2.3 Quotient Space 54
2.4 Dimension, Hamel Basis 55
2.5 Algebraic Dual, Second Dual 57
2.6 Convex Sets 58
2.7 Ordered Groups 59
2.8 Hahn-Banach Theorem, Separation Form 60
2.9 Hahn-Banach Theorem, Extension Form 62
2.10 Applications, Banach Limits, Invariant Measure 64
2.11 Banach Space, Dual Space 71
2.12 Hahn-Banach Theorem in Normed Space 73
2.13 Uniform Boundedness Principle, Applications 76
2.14 Lemma of F. Riesz, Applications 81
2.15 Application to Compact Transformations 83
2.16 Applications, Weak Convergence, Summability Methods, Approximate Integration 86
2.17 Second Dual Space 91
2.18 Dual of $l_p$ 93
2.19 Dual of $C[a,b]$, Riesz Representation Theorem 94
2.20 Open Mapping and Closed Graph Theorems 97
2.21 Application, Projections 100
2.22 Application, Schauder Expansion 101
2.23 A Theorem on Operators in $C[O,1]$ 102
Exercises 103

Chapter 3 - Measure and Integration, $L_p$ Spaces 109
3.1 Lebesgue Measure for Bounded Sets in $E_n$ 109
3.2 Lebesgue Measure for Unbounded Sets 113
3.3 Totally a Finite Measures 114
3.4 Measurable Functions, Egoroff Theorem 115
3.5 Convergence in Measure 118
3.6 Summable Functions 120
3.7 Fatou and Lebesgue Dominated Convergence Theorems 125
3.8 Integral as a Set Function 128
3.9 Signed Measure, Decomposition into Measures 129
3.10 Absolute Continuity and Singularity of Measures 132
3.11 The $L_p$ Spaces, Completeness 135
3.12 Approximation and Smoothing Operations 138
3.13 The Dual of $L_p$, $p>1$ 142
3.14 The Dual of $L_1$ 146
3.15 The Individual Ergodic Theorem 148
3.16 $L_p$ Convergence of Fourier Series 150
3.17 Functions Whose Fourier Series Diverge Almost Everywhere 153
3.18 Continuous Functions Which Differ from All Those Having a Given Modulus 155
Exercises 159

Chapter 4 - Hilbert Space 165
4.1 Inner Product, Hilbert Space 165
4.2 Basic Lemma, Projection Theorem, Dual 168
4.3 Application, Mean Ergodic Theorem 171
4.4 Orthonormal Sets, Fourier Expansion 173
4.5 Application, Isoperimetric Theorem 178
4.6 Muntz Theorem 179
4.7 Dimension, Riesz-Fischer Theorem 182
4.8 Reproducing Kernel 184
4.9 Application, Bergman Kernel 187
4.10 Examples of Complete Orthonormal Sets 191
4.11 Systems of Haar, Rademacher, Walsh; Applications 194
Exercises 201

Chapter 5 - Topological Vector Spaces 206
5.1 Topology 206
5.2 Tychonoff Theorem, Application in Banach Space 207
5.3 Topological Vector Space 210
5.4 Normable Space 211
5.5 Space of Measurable Functions 212
5.6 Locally Convex Space 217
5.7 Metrizable Space, Space of Entire Functions 219
5.8 FK Spaces 224
5.9 Application to Summability Methods 228
5.10 Ordered Vector Spaces 232
5.11 Banach Lattice 235
5.12 Kothe Spaces 238
Exercises 242

Chapter 6 - Banach Algebras 248
6.1 Definition and Examples 248
6.2 Adjunction of Identity 250
6.3 Haar Measure 250
6.4 Commutative Banach Algebras, Maximal Ideals 254
6.5 The Set $C({\mathcal M})$ 258
6.6 Gelfand Representation for Algebras with Identity 260
6.7 Analytic Functions 261
6.8 Isomorphism Theorem for Algebras with Identity 263
6.9 Algebras without Identity 266
6.10 Application to $L_1(G)$ 269

Exercises 272

References 276

Index 279