Cobzas S. Functional analysis in asymmetric normed spaces (Basel, 2013). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаCobzas S. Functional analysis in asymmetric normed spaces. - Basel: Birkhäuser/Springer, 2013. - x, 219 p. - (Frontiers in mathematics). - Bibliogr.: p.201-214. - Ind.: p.215-219. - ISBN 978-3-0348-0477-6
 

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Оглавление / Contents
 
Introduction .................................................. vii
1    Quasi-metric and Quasi-uniform Spaces
1.1  Topological properties of quasi-metric and quasi-uniform 
     spaces ..................................................... 2
     1.1.1  Quasi-metric spaces and asymmetric normed spaces .... 2
     1.1.2  The topology of a quasi-semimetric space ............ 4
     1.1.3  More on bitopological spaces ....................... 14
     1.1.4  Compactness in bitopological spaces ................ 21
     1.1.5  Topological properties of asymmetric seminormed 
            spaces ............................................. 25
     1.1.6  Quasi-uniform spaces ............................... 30
     1.1.7  Asymmetric locally convex spaces ................... 36
1.2  Completeness and compactness in quasi-metric and quasi-
     uniform spaces ............................................ 45
     1.2.1  Various notions of completeness for quasi-metric 
            spaces ............................................. 45
     1.2.2  Compactness, total boundedness and precompactness .. 60
     1.2.3  Baire category ..................................... 71
     1.2.4  Baire category in bitopological spaces ............. 73
     1.2.5  Completeness and compactness in quasi-uniform 
            spaces ............................................. 77
     1.2.6  Completions of quasi-metric and quasi-uniform 
            spaces ............................................. 92
2    Asymmetric Functional Analysis
2.1  Continuous linear operators between asymmetric normed 
     spaces .................................................... 99
     2.1.1  The asymmetric norm of a continuous linear 
            operator .......................................... 100
     2.1.2  Continuous linear functionals on an asymmetric
            seminormed space .................................. 103
     2.1.3  Continuous linear mappings between asymmetric 
            locally convex spaces ............................. 106
     2.1.4  Completeness properties of the normed cone of 
            continuous linear operators ....................... 110
     2.1.5  The bicompletion of an asymmetric normed space .... 112
     2.1.6  Asymmetric topologies on normed lattices .......... 114
2.2  Hahn-Banach type theorems and the separation of convex 
     sets ..................................................... 124
     2.2.1  Hahn-Banach type theorems ......................... 124
     2.2.2  The Minkowski gauge functional - definition
            and properties .................................... 128
     2.2.3  The separation of convex sets ..................... 129
     2.2.4  Extreme points and the Krein-Milman theorem ....... 131
2.3  The fundamental principles ............................... 134
     2.3.1  The Open Mapping and the Closed Graph Theorems .... 134
     2.3.2  The Banach-Steinhaus principle .................... 137
     2.3.3  Normed cones ...................................... 139
2.4  Weak topologies .......................................... 142
     2.4.1  The wb-topology of the dual space Xbp ............. 142
     2.4.2  Compact subsets of asymmetric normed spaces ....... 144
     2.4.3  Compact sets in LCS ............................... 145
     2.4.4  The conjugate operator, precompact operators
            and a Schauder type theorem ....................... 151
     2.4.5  The bidual space, reflexivity and Goldstine 
            theorem ........................................... 155
     2.4.6  Weak topologies on asymmetric LCS ................. 161
     2.4.7  Asymmetric moduh of rotundity and smoothness ...... 165
2.5  Applications to best approximation ....................... 170
     2.5.1  Characterizations of nearest points in convex
            sets and duality .................................. 171
     2.5.2  The distance to a hyperplane ...................... 177
     2.5.3  Best approximation by elements of sets with 
            convex complement ................................. 179
     2.5.4  Optimal points .................................... 181
     2.5.5  Sign-sensitive approximation in spaces of 
            continuous or integrable functions ................ 181
2.6  Spaces of semi-Lipschitz functions ....................... 183
     2.6.1  Semi-Lipschitz functions - definition and the 
            extension property ................................ 183
     2.6.2  Properties of the cone of semi-Lipschitz
            functions - linearity ............................. 187
     2.6.3  Completeness properties of the spaces of
            semi-Lipschitz functions .......................... 190
     2.6.4  Applications to best approximation in quasi-
            metric spaces ..................................... 199
Bibliography .................................................. 201
Index ......................................................... 215


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