Strict Separation of Compact and Closed Convex Sets

Disjoint compact convex and closed convex sets in a normed space admit strict separation by a continuous functional.

Strict Separation of Compact and Closed Convex Sets

Let $X$ be a real normed space . Let $K,F\subset X$ be nonempty convex sets and assume:

$K$ is compact,
$F$ is closed , and
$K\cap F=\emptyset$ .

Theorem: The sets $K$ and $F$ can be strictly separated by a closed hyperplane . Equivalently, there exists $x^\ast \in X^\ast$ (see dual space ) such that

\sup_{x\in K}\langle x^\ast ,x\rangle < \inf_{y\in F}\langle x^\ast ,y\rangle.

Context: This is a strong separation result in normed spaces. The proof in the notes applies closed hyperplane separation to a ball around $0$ and a translated difference set $F-K$ , using compactness/closedness to guarantee closedness and a positive gap.