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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