Index of symmetry and topological classification of asymmetric normed spaces

Abstract

Let X,Y be asymmetric normed spaces and Lc(X,Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X,Y) is not a vector space. The aim of this note is to give, using the Baire category theorem, a complete characterization on X and a finite dimensional Y so that Lc(X,Y) is a vector space. For this, we introduce an index of symmetry of the space X denoted c(X)∈[0,1] and we give the link between the index c(X) and the fact that Lc(X,Y) is in turn an asymmetric normed space for every asymmetric normed space Y. Our study leads to a topological classification of asymmetric normed spaces.