The κ-Fréchet--Urysohn property for locally convex spaces

A topological space X is κ-Fréchet--Urysohn if for every open subset U of X and every x∈U there exists a sequence in U converging to x. We prove that every κ-Fréchet--Urysohn Tychonoff space X is Ascoli. We apply this statement and some of known results to characterize the κ-Fréchet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that Cp(X) is Ascoli iff X has the property (κ).