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 $$X'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> that is metrizable and coarser than the weak topology $$\sigma (X', X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.