Continuity of Linear Functionals via Closed Level Sets

A linear functional on a normed space is continuous iff one of its level sets is closed.

Continuity of Linear Functionals via Closed Level Sets

Let $X$ be a real normed space , let $f:X\to\mathbb{R}$ be a nonzero linear functional, and let $\alpha\in\mathbb{R}$ .

Theorem: The functional $f$ is continuous if and only if the level set

A:=\{x\in X\mid f(x)=\alpha\}

is a closed subset of $X$ .

Context: This links geometric closedness of a hyperplane level set to analytic boundedness of $f$ (compare bounded linear functionals ). It is used to upgrade algebraic separation in vector spaces to separation by closed hyperplanes in normed spaces; see closed hyperplane separation .