Bornologies and Functional Analysis

Bornologies and Functional Analysis......Page 4
Copyright Page......Page 5
Introduction......Page 6
CONTENTS......Page 10
0.A Vector Spaces......Page 14
0.B Preliminaries of General Topology and Normed Spaces......Page 24
0.C Topological Vector Spaces......Page 25
1:1 Definitions......Page 31
1:2 Bounded Linear Maps......Page 32
1:3 Fundamental Examples of Bornologies......Page 33
1:4 Bornological Convergence......Page 38
2:1 Initial Bornologies......Page 42
2:2 Product Bornologies......Page 43
2:4 Bornologies Generated by a Family of Subsets......Page 44
2:6 Final Bornologies......Page 45
2:8 Bornological Inductive Limits......Page 46
2:9 Bornological Direct Sums: Finite-Dimensional Bornologies......Page 47
2:10 Stability of the Separation Property......Page 49
2:11 Bornologically Closed Sets: Separation of Bornological Quotients......Page 50
2:12 The Associated Separated Bornological Vector Space......Page 51
2:13 The Structure of a Convex Bornological Space: Comparison with the Structure of a Locally Convex Space......Page 52
3:1 Completant Bounded Disks......Page 53
3:2 Complete Convex Bornological Spaces......Page 55
3:3 Separated Bornological Vector Spaces of Finite Dimension......Page 57
3:4 The Complete Bornology Associated with a Separated Vector Bornology......Page 58
3:5 Bornologically Complete Topological Vector Spaces......Page 59
4:1 Compatible Topologies and Bornologies]......Page 60
4:2 Characterisation of Bornological Topologies......Page 64
4:3 Completely Bornological Spaces......Page 66
4:4 The Closed Graph Theorem......Page 68
CHAPTER V. "TOPOLOGY –BORNOLOGY": EXTERNAL DUALITY I: THE FUNDAMENTAL PRINCIPLES OF DUALITY......Page 75
5:0 Preliminaries: The Hahn-Banach Theorem and its Consequences......Page 76
5:1 The External Duality Between Topology and Bornology......Page 81
5:2 Duality Between Equicontinuous and Equibounded Sets in a Dual Space......Page 84
5:3 Completeness of the Equicontinuous Bornology: Completely Bornological Topology on A Dual Space......Page 88
5:5 External Duality Between Bounded and Continuous Linear Maps : Dual Maps......Page 90
CHAPTER VI. "TOPOLOGY – BORNOLOGY": EXTERNAL DUALITY II: WEAKLY COMPACT BORNOLOGIES AND REFLEXIVITY......Page 94
6:1 Weak Compactness of Equicontinuous Sets......Page 95
6:2 The Bornology of Weakly Compact Disks and the Mackey-Arens Theorem......Page 96
6:3 Weakly Compact Bornologies: Reflexivity......Page 99
6:4 Completely Reflexive Locally Convex Spaces......Page 102
CHAPTER VII. COMPACT BORNOLOGIES......Page 104
7:1 Hypo–Montel Spaces......Page 105
7:2 Schwartz spaces......Page 106
7:3 Silva Spaces......Page 111
8:0 Multi-Dimensional Notation......Page 116
8:1 The Bornological Spaces E(o) and D(o)......Page 117
8:2 Distributions as Bounded Linear Functionals......Page 118
8:3 Differential Operators and Partial Differential Equations......Page 119
8:4 The Silva Space E'(o)......Page 121
8:5 The Spaces E'(K) and the Bornological Structure of E'(o)......Page 122
8:6 The General Existence Theorem for Infinitely Differentiable Solutions......Page 123
8:7 Proof of the Existence Theorem: Sufficiency......Page 124
8:8 Proof of the Existence Theorem: Necessity......Page 125
8:9 Existence Theorem for Partial Differential Equations with Constant Coefficients......Page 126
Appendix: Existence of a Fundamental Solution......Page 127
Exercises on Chapter I......Page 131
Exercises on Chapter II......Page 136
Exercises on Chapter III......Page 139
Exercises on Chapter IV......Page 142
Exercises on Chapter V......Page 145
Exercises on Chapter VI......Page 148
Exercises on Chapter VII......Page 150
Index......Page 152
Bibliography......Page 156
References for Advanced Studies......Page 157