Functional Analysis in Asymmetric Normed Spaces

An asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn–Banach type theorems and separation results for convex sets, Krein–Milman type theorems, analogs of the fundamental principles – open mapping, closed graph and uniform boundedness theorems – an analog of the Schauder’s theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis – completeness, compactness and Baire category – which drastically differ from those in metric or uniform spaces. The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text. Table of Contents Cover Functional Analysis in Asymmetric Normed Spaces ISBN 9783034804776 e-ISBN 9783034804783 Contents Introduction Chapter 1 Quasi-metric and Quasi-uniform Spaces 1.1 Topological properties of quasi-metric and quasi-uniform spaces 1.1.1 Quasi-metric spaces and asymmetric normed spaces 1.1.2 The topology of a quasi-semimetric space 1.1.3 More on bitopological spaces 1.1.4 Compactness in bitopological spaces 1.1.5 Topological properties of asymmetric seminormed spaces 1.1.6 Quasi-uniform spaces 1.1.7 Asymmetric locally convex spaces 1.2 Completeness and compactness in quasi-metric and quasi-uniform spaces 1.2.1 Various notions of completeness for quasi-metric spaces 1.2.2 Compactness, total boundedness and precompactness 1.2.3 Baire category 1.2.4 Baire category in bitopological spaces 1.2.5 Completeness and compactness in quasi-uniform spaces 1.2.6 Completions of quasi-metric and quasi-uniform spaces Chapter 2 Asymmetric Functional Analysis 2.1 Continuous linear operators between asymmetric normed spaces 2.1.1 The asymmetric norm of a continuous linear operator 2.1.2 Continuous linear functionals on an asymmetric seminormed space 2.1.3 Continuous linear mappings between asymmetric locally convex spaces 2.1.4 Completeness properties of the normed cone of continuous linear operators 2.1.5 The bicompletion of an asymmetric normed space 2.1.6 Asymmetric topologies on normed lattices 2.2 Hahn-Banach type theorems and the separation of convex sets 2.2.1 Hahn-Banach type theorems 2.2.2 The Minkowski gauge functional - definition and properties 2.2.3 The separation of convex sets 2.2.4 Extreme points and the Krein-Milman theorem 2.3 The fundamental principles 2.3.1 The Open Mapping and the Closed Graph Theorems 2.3.2 The Banach-Steinhaus principle 2.3.3 Normed cones 2.4 Weak topologies 2.4.1 The wb-topology of the dual space Xbp 2.4.2 Compact subsets of asymmetric normed spaces 2.4.3 Compact sets in LCS 2.4.4 The conjugate operator, precompact operators and a Schauder type theorem 2.4.5 The bidual space, reflexivity and Goldstine theorem 2.4.6 Weak topologies on asymmetric LCS 2.4.7 Asymmetric moduli of rotundity and smoothness 2.5 Applications to best approximation 2.5.1 Characterizations of nearest points in convex sets and duality 2.5.2 The distance to a hyperplane 2.5.3 Best approximation by elements of sets with convex complement 2.5.4 Optimal points 2.5.5 Sign-sensitive approximation in spaces of continuous or integrable functions 2.6 Spaces of semi-Lipschitz functions 2.6.1 Semi-Lipschitz functions - definition and the extension property 2.6.2 Properties of the cone of semi-Lipschitz functions - linearity 2.6.3 Completeness properties of the spaces of semi-Lipschitz functions 2.6.4 Applications to best approximation in quasi-metric spaces Bibliography Index