  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 prove, using the Baire
category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed
space Y , then X is isomorphic to its associated normed space (the converse is
true for every asymmetric normed space Y and is easy to establish). 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 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.
