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 cracterization 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 [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.