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<p>We introduce the property of countable separation for a locally convex Hausdorff space <i>X</i> and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of <i>X</i> is equivalent to the existence of a locally convex topology on the dual <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> that is metrizable and coarser than the weak topology <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma (X', X)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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 <i>X</i> from this characterization.</p>

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Separability and submetrizability in locally convex spaces

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  • 发表日期：

    2026年6月29日

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  • Thomas Ruf

摘要

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 \(X'\) X ′ that is metrizable and coarser than the weak topology \(\sigma (X', X)\) σ ( 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.