Functional Analysis in Asymmetric Normed Spaces

Abstract

The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space X is not a linear space, but merely a convex cone in the space of all linear functionals on X. Due to this fact, a careful treatment of the duality problems (e.g. reflexivity) and of other results as, for instance, the extension of fundamental principles of functional analysis -the open mapping theorem and the closed graph theorem -to this setting, is needed. Contents 1. Introduction: 1.1 Quasi-metric spaces and asymmetric normed spaces; 1.2 The topology of a quasisemimetric space; 1.3 Quasi-uniform spaces.2. Completeness and compactness in quasi-metric and in quasi-uniform spaces: 2.1 Various notions of completeness for quasi-metric spaces; 2.2 Compactness, total bondedness and precompactness; 2.3 Completeness in quasi-uniform spaces; 2.4 Baire category.3. Continuous linear operators between asymmetric normed spaces: 3.1 The asymmetric norm of a continuous linear operator; 3.2 The normed cone of continuous linear operators -completeness; 3.3 The bicompletion of an asymmetric normed space; 3.4 Open mapping and closed graph theorems for asymmetric normed spaces; 3.5 Normed cones; 3.6 The w ♭ topology of the dual space X ♭ p ; 3.7 Compact subsets of an asymmetric normed space; 3.8 The conjugate operator, precompact operators between asymmetric normed spaces and a Schauder type theorem; 3.9 Asymmetric moduli of smoothness and rotundity; 3.10 Asymmetric topologies on normed lattices. 4. Linear functionals on an asymmetric normed space: 4.1 Some properties of continuous linear functionals; 4.2 Hahn-Banach type theorems; 4.3 The bidual space, reflexivity and Goldstine theorem. 5. The Minkowski functional and the separation of convex sets: 5.1 The Minkowski gauge functional -definition and properties; 5.2 The separation of convex sets; 5.3 Extremal points and Krein-Milman theorem.6. Applications to best approximation: 6.1 Characterizations of nearest points in convex sets and duality; 6.2 The distance to a hyperplane; 6.3 Best approximation by elements of sets with convex complement; 6.4 Optimal points; 6.5 Sign-sensitive approximation in spaces of continuous or integrable functions.7. Spaces of semi-Lipschitz functions: 7.1 Semi-Lipschitz functions -definition, the extension property, applications to best approximation in quasi-metric spaces; 7.2 Properties of the cone of semi-Lipschitz functions -linearity, completeness. ReferencesThis research was supported by Grant CNCSIS 2261, ID 543.Remark 1.1. Since the terms "quasi-norm", "quasi-normed space" and "quasi-Banach space" are already "registered trademarks" (see, for instance, th...

Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

“…It is T 1 if and only if ρ(x, y) > 0 for any pair of distinct elements x, y ∈ X. A characterization of asymmetric norms inducing a Hausdorff topology was given in [8], see also [5].…”

“…As the proof given in [5] is not correct (see the comments in the next section) we give here a different one. Since X is right p-K-complete, it is of secondp-category (see Theorem 1.9).…”

“…In [5] one asserts that the proof given to the Uniform Boundedness Principle ([5, Theorem 2.3.7]) yields the principle of the condensation of singularities. In the proof one supposes that the family T is q-pointwise bounded but notq-pointwise bounded, that is T satisfies (5.1) but not (5.2), that is obviously false since, as we have remarked, these two conditions are equivalent.…”

Quasi-uniform convergence topologies on function spaces- Revisited

“…In [39], Pervin greatly simplified Csaszar's proof by giving a direct method of constructing a compatible quasi-uniformity for an arbitrary topological space. For more information, see ([29], p. 14-16; [7], p. 34). Definition 2.5.…”

“…In particular, a net {y α : α ∈ D} in a quasi-uniform or locally uniform space (Y, U) is said to be T (U)-convergent to y ∈ Y if, for each U ∈ U, there exists an α 0 ∈ D such that y α ∈ U [y] for all α ≥ α 0 . Definition 2.6 ( [41,26,24,7]). Let (Y, U) be a quasi-uniform space.…”