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/ 2 T0; 1U 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.