Hahn-Banach Theorem/Real Vector Space/Lemma 1

Lemma
Let $X$ be a vector space over $\R$.

Let $p : X \to \R$ be a convex function. Let $G$ be a linear subspace of $X$.

Let $g_0 : G \to \R$ be a linear functional such that:


 * $\map {g_0} x \le \map p x$ for each $x \in G$.

Then there exists a linear subspace $G'$ of $X$ and a linear functional $g : G' \to \R$ such that:


 * $(1): \quad$ if $G \ne X$, then $G'$ is a proper superset of $G$
 * $(2): \quad$ $g$ extends $g_0$ with $\map g x \le \map p x$ for each $x \in G'$.