Disjoint Compact Set and Closed Set in Topological Vector Space separated by Open Neighborhood

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\Bbb F$.

Let $K$ be a compact subspace of $X$.

Let $C \subseteq X$ be convex such that:


 * $K \cap C = \O$

Then there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:


 * $\paren {K + V} \cap \paren {C + V} = \O$