Separability and Submetrizability in Locally Convex Spaces

Abstract: We introduce the property of countable separation for a locally convex Hausdorff space $X$ and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of $X$ is equivalent to the existence of a locally convex topology on the dual $X'$ that is metrizable and coarser than the weak topology $\sigma(X', X)$. This result generalizes known conditions for separability and provides a precise duality between separability and metrizability. We also show how to derive new and known conditions for the separability of $X$ from this characterization.