Continuous linear functional

A continuous linear functional is a continuous linear map from a topological vector space to its scalar field. The collection of all such functionals is the continuous dual space.

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 $l\ne 0$, choose $v$ with $l(v)=1$ and write $M=\ker l$. Closedness gives $d=\mathrm{dist}(v,M)>0$. Since $x/l(x)-v\in M$ when $l(x)\ne 0$, $|l(x)|\le \Vert x\Vert /d$. The zero functional is continuous separately. Thus the criterion does not require completeness or the closed graph theorem.

For a function space on a set $X$ and a point $x\in X$, the point evaluation functional is ${\delta }_x(f)=f(x)$. It is bounded on $C(X)$ with the supremum norm.