ON CLOSEDNESS IN THE $\mathcal{L}$-TOPOLOGY OF T.V.S.

Abstract

Let $X$ and $Y$ be Hausdorff and locally convex topological vector spaces. In this paper, we prove that a convex subset of $X$ is closed if and only if it is closed in the topology on $X$ induced by the set of continuous linear mappings from $X$ into $Y$ . As applications, some existence results for vector equilibrium problems and vector variational inequalities associated with discontinuous mappings are given.