Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms

We obtain characterizations of those paratopological vector spaces that are quasi-metrizable, locally bounded, quasi-metrizable locally convex, and quasi-normable, respectively, in terms of appropriate kinds of fuzzy quasi-norms. Characterizations of distinguished kinds of metrizable topological vector spaces, some of them known already, are deduced as particular cases of our approach.