Functional Analysis in Asymmetric Normed Spaces
by Stefan Cobzas

Persian Title: تجزیه و تحلیل عملکرد در فضاهای Normed نامتقارن

Summary and Info
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 nonnegative entries in the first argument. The asymmetric dual of X, meaning the set of all realvalued upper semicontinuous 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 semiLipschitz functions on quasimetric spaces. Since the basic topological tools come from quasimetric spaces and quasiuniform 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 selfcontained, 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 ContentsCoverFunctional Analysis in Asymmetric Normed SpacesISBN 9783034804776 eISBN 9783034804783ContentsIntroductionChapter 1 Quasimetric and Quasiuniform Spaces 1.1 Topological properties of quasimetric and quasiuniform spaces 1.1.1 Quasimetric spaces and asymmetric normed spaces 1.1.2 The topology of a quasisemimetric 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 Quasiuniform spaces 1.1.7 Asymmetric locally convex spaces 1.2 Completeness and compactness in quasimetric and quasiuniform spaces 1.2.1 Various notions of completeness for quasimetric 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 quasiuniform spaces 1.2.6 Completions of quasimetric and quasiuniform spacesChapter 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 HahnBanach type theorems and the separation of convex sets 2.2.1 HahnBanach 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 KreinMilman theorem 2.3 The fundamental principles 2.3.1 The Open Mapping and the Closed Graph Theorems 2.3.2 The BanachSteinhaus principle 2.3.3 Normed cones 2.4 Weak topologies 2.4.1 The wbtopology 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 Signsensitive approximation in spaces of continuous or integrable functions 2.6 Spaces of semiLipschitz functions 2.6.1 SemiLipschitz functions  definition and the extension property 2.6.2 Properties of the cone of semiLipschitz functions  linearity 2.6.3 Completeness properties of the spaces of semiLipschitz functions 2.6.4 Applications to best approximation in quasimetric spacesBibliographyIndex
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