Functional analysis in asymmetric normed spaces

TL;DR

This paper surveys recent developments in the theory of asymmetric normed spaces, highlighting their unique duality properties and adaptations of classical functional analysis principles.

Contribution

It provides a comprehensive overview of the structure, duality, and operator theory in asymmetric normed spaces, emphasizing differences from classical normed space theory.

Findings

Dual space of an asymmetric normed space is a convex cone, not a linear space.

Extension of the open mapping and closed graph theorems to asymmetric normed spaces.

Analysis of reflexivity and duality 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.