Metrizability vs. Frechet-Urysohn property

In metrizable spaces, points in the closure of a subset A are limits of sequences in A; i.e., metrizable spaces are Frechet-Urysohn spaces. The aim of this paper is to prove that metrizability and the Frechet-Urysohn property are actually equivalent for a large class of locally convex spaces that includes (LF)- and (DF)-spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Frechet-Urysohn property. We provide applications of our results to, for instance, the space of distributions D'(Omega).The space D'(Omega) is not Frechet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.