The second edition of this successful textbook, first published in 1970, retains the aims of the first, namely to provide a truly introductory course in functional analysis, but the opportunity has been taken to add more detail and worked examples. The main changes are complete revisons of the work on convex sets, metric and topological linear spaces, reflexivity and weak convergence. Additional material on the Weiner algebra of absolutely convergent Fourier series and on weak topologies is included. A final chapter includes elementary applications of functional analysis to differential and integral equations.
Author(s): I. J. Maddox
Publisher: CUP
Year: 1970
Language: English
Pages: 214
Title page......Page 1
Copyright page......Page 2
CONTENTS......Page 3
Preface......Page 5
1 Sets and functions......Page 7
2 Real and complex numbers......Page 18
3 Sequences of functions, continuity, differentiability......Page 23
4 Inequalities......Page 27
1 Metric and semimetric spaces......Page 30
2 Complete metric spaces......Page 39
3 Some metric and topological concepts......Page 45
4 Continuous functions on metric and topological spaces......Page 55
5 Compact sets......Page 66
6 Category and uniform boundedness......Page 72
1 Linear spaces......Page 76
2 Subspaces, dimensionality, factorspaces, convex sets......Page 80
3 Linear metric spaces, paranorms, seminorms and norms......Page 88
4 Basis......Page 92
5 Distributions......Page 95
1 Convergence and completeness......Page 100
2 Linear operators and functionals......Page 108
3 The Banach-Steinhaus theorem......Page 120
4 The open mapping and closed graph theorems......Page 124
5 The Hahn-Banach extension theorem......Page 127
6 Weak convergence......Page 134
1 Algebras and Banach algebras......Page 138
2 Homomorphisms and isomorphisms......Page 142
3 The spectrum and the Gelfand-Mazur theorem......Page 146
4 The Gelfand representation theorem......Page 152
1 Inner product and Hilbert spaces......Page 155
2 Orthonormal sets......Page 159
3 The dual space of a Hilbert space......Page 163
1 Matrix and linear transformations......Page 167
2 Algebras of matrices......Page 184
3 Summability......Page 191
4 Tauberian theorems......Page 199
5 Some problems for further study......Page 203
Bibliography......Page 209
Index......Page 211
