Bornologies and Functional Analysis (eBook)

Introductory course on the theory of duality topology-bornology and its use in functional analysis

eBook Download: PDF

1977 | 1. Auflage
143 Seiten
Elsevier Science (Verlag)
978-0-08-087137-0 (ISBN)

Infos

Bornologies and functional analysis

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 1.1.1977
• Mitarbeit: Herausgeber (Serie): Henri Hogbe-Nlend
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087137-2 / 0080871372
• ISBN-13: 978-0-08-087137-0 / 9780080871370