These notes present a theorem on infinite matrices with values in a topological group due to P. Antosik and J. Mikusinski. Using the matrix theorem and classical gliding hump techniques, a number of applications to various topics in functional analysis, measure theory and sequence spaces are given. There are a number of generalizations of the classical Uniform Boundedness Principle given; in particular, using stronger notions of sequential convergence and boundedness due to Antosik and Mikusinki, versions of the Uniform Boundedness Principle and the Banach-Steinhaus Theorem are given which, in contrast to the usual versions, require no completeness or barrelledness assumptions on the domain space. Versions of Nikoym Boundedness and Convergence Theorems of measure theory, the Orlicz-Pettis Theorem on subseries convergence, generalizations of the Schur Lemma on the equivalence of weak and norm convergence in 1i and the Mazur-Orlicz Theorem on the continuity of separately continuous bilinear mappings are also given. Finally, the matrix theorems are also employed to treat a number of topics in sequence spaces.
Author(s): Swartz C.
Publisher: WS
Year: 1996
Language: English
Pages: 222
Tags: Математика;Линейная алгебра и аналитическая геометрия;Линейная алгебра;Матрицы и определители;
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 10
1.1 History and Summary......Page 14
1.2 Notation and Terminology......Page 18
2.1 Introduction......Page 22
2.2 The Antosik-Mikusinski Matrix Theorem......Page 23
2.3 The Nikodym Convergence Theorem......Page 29
3.2 K-Convergence :......Page 34
3.3 C-Boundedness......Page 42
3.4 A-Spaces......Page 46
3.5 An Abstract Hellinger-Toeplitz Theorem......Page 48
3.6 Variants of K Convergence......Page 50
4.2 The Uniform Boundedness Principle......Page 54
4.4 Equicontinuity......Page 57
4.5 Ptak's Generalization of the UBP......Page 59
4.6 Mate's UBP......Page 61
4.7 The Nikodym Boundedness Theorem......Page 64
4.8 An Abstract Uniform Boundedness Result......Page 67
5.2 The General Banach-Steinhaus Theorem......Page 72
5.3 Equicontinuity......Page 73
5.4 The Hahn-Schur Summability Theorem......Page 76
5.5 Phillips' Lemma......Page 79
6.2 Single Bilinear Mapping......Page 82
6.3 Families of Bilinear Maps......Page 86
7.2 The Adjoint Theorem......Page 92
7.3 Closed Graph Theorem......Page 93
8.1 Subseries Convergent Series......Page 96
8.2 Bounded Multiplier Convergent Series......Page 102
8.3 Other Types of Convergence for Series......Page 108
8.4 Phillips' Lemma......Page 109
8.5 The Antosik Interchange Theorem......Page 111
9.2 Definitions and Examples......Page 114
9.3 The Abstract Hahn-Schur Theorem......Page 117
9.4 Special Cases......Page 125
9.5 Other Abstract Hahn-Schur Results......Page 130
10.1 Introduction......Page 132
10.2 ?stormed Linear Spaces......Page 134
10.3 Locally Convex Spaces......Page 135
10.4 Spaces with a Schauder Basis......Page 138
10.5 Linear Operators......Page 142
10.6 Abstract Orlicz-Pettis Theorems......Page 145
10.7 C-Convergence and the Orlicz-Pettis Theorem......Page 149
11.2 Imbedding co......Page 152
11.3 Imbedding t?......Page 158
12.1 Introduction......Page 162
12.2 Weak Sequential Completeness of a-Duals......Page 163
12.3 Weak Sequential Completeness of a-duals......Page 165
12.4 Weak Sequential Completeness of /3-duals......Page 168
12.5 Banach-Steinhaus Results......Page 173
12.6 Automatic Continuity for Matrix Mappings......Page 180
12.7 The Transpose of a Summability Matrix......Page 187
12.8 The Kernel Theorem for Kothe Sequence Spaces......Page 199
Index......Page 206
References......Page 210
