Proposition 4

— Let $A$ be a closed non-empty convex set in a locally convex space $E$ and let $K$ be a compact non-empty convex set in $E$, that does not meet $A$. Then there exists a closed hyperplane $H$ that strictly separates $A$ from $K$.

For there exists an open convex neighbourhood $V$ of 0 in $E$ such that $A + V$ and $K + V$ do not meet (GT, II, § 4.3, prop. 4). As $A + V$ and $K + V$ are convex and open in $E$, prop. 1 of II, p. 37 shows that there exists a closed hyperplane $H$ that strictly separates $A + V$ from $K + V$, and $a$ fortiori $A$ from $K$.