Hahn-Banach Theorem/Real Vector Space

Theorem
Let $E$ be a vector space over $\R$ and let $p: E \to \R$ be a Minkowski functional.

Let $G \subset E$ be a linear subspace of $E$ and let $f : G \to \R$ be a linear functional such that:
 * $\forall x \in G: f \left({x}\right) \le p \left({x}\right)$

Then there exists a linear functional $\tilde f$ defined on the whole space $E$ which extends $f$, that is:
 * $\forall x \in G: \tilde f \left({x}\right) = f \left({x}\right)$

such that:
 * $\forall x \in E: \tilde f \left({x}\right) \le p\left({x}\right)$