Index of symmetry and topological classification of asymmetric normed spaces

Abstract

Let $X,Y$ be asymmetric normed spaces and $L_c\left(X,Y\right)$ the convex cone of all linear continuous operators from $X$ to $Y$. It is known that in general, $L_c\left(X,Y\right)$ 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 $L_c\left(X,Y\right)$ is a vector space. For this, we introduce an index of symmetry of the space $X$ denoted $c\left(X\right)\in \left[0,1\right]$ and we give the link between the index $c\left(X\right)$ and the fact that $L_c\left(X,Y\right)$ is in turn an asymmetric normed space for every asymmetric normed space $Y$. Our study leads to a topological classification of asymmetric normed spaces.