= Linear functional with closed kernel
{title2=$l\text{ continuous}\iff\ker l\text{ closed}$}
A <linear functional> on a <normed vector space> is continuous exactly when its kernel is closed. The forward direction follows because zero is closed in the scalar field. Conversely, if $l\ne0$, choose $v$ with $l(v)=1$ and write $M=\ker l$. Closedness gives $d=\operatorname{dist}(v,M)>0$. Since $x/l(x)-v\in M$ when $l(x)\ne0$, $|l(x)|\leq\|x\|/d$. The zero functional is continuous separately. Thus the criterion does not require completeness or the <closed graph theorem>.
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