Continuity of Linear Functionals via Closed Level Sets

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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 $X$ .

Remarks

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.