Hahn-Banach Theorem/Real Vector Space

Theorem
Let $E$ be a vector space over $\R$.

Let $p: E \to \R$ be a Minkowski functional.

Let $G \subseteq E$ be a linear subspace of $E$.

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

Proof
Let a linear functional $g$ be called admissible
 * $\forall x \in \operatorname{Dom} \left({g}\right): g \left({x}\right) \le p \left({x}\right)$

A linear functional $h_1$ extends a linear functional $h_2$ :
 * $\operatorname{Dom} \left({h_2}\right) \subseteq \operatorname{Dom} \left({h_1}\right)$

and:
 * $\forall x \in \operatorname{Dom} \left({h_2}\right): h_2 \left({x}\right) = h_1 \left({x}\right)$

The proof consists of two steps:

First, the set of admissible linear functionals that extend $f$ is inductive.

Using Zorn's Lemma the existence of a maximal element is derived.

Second, it is proved by contradiction that this functional is defined on the whole space $E$.