= Continuous-dual separation theorem for Hausdorff locally convex spaces
{title2=$X^*\text{ separates points of }X$}
= Continuous-dual point separation
{synonym}
In a Hausdorff <locally convex space>, a separating family of continuous <seminorms> has $p(x)>0$ for any chosen nonzero $x$ and some $p$. Define a <linear functional> on its one-dimensional span attaining $p(x)$ and extend it under that <seminorm> by the real <Hahn-Banach theorem>. The extension is continuous because it is seminorm-bounded, and distinguishes $x$ from zero. The Hausdorff condition is essential.
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