= Separation of a point from a closed linear subspace
{title2=$f(x_0)=1,\quad f|_Y=0$}
If $Y$ is a <closed linear subspace> of a Hausdorff <locally convex space> and $x_0\notin Y$, choose a continuous <seminorm> $q$ with $q(x_0-y)\geq1$ for all $y\in Y$, using a basic neighborhood disjoint from $Y$. The <linear functional> $g(y+tx_0)=t$ is bounded by $q$. The <Hahn-Banach theorem> extends it to a continuous <linear functional> with the displayed values. This is the linear-subspace form of separation, sharper than merely distinguishing the two sets.
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