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