Hahn-Banach Theorem

Theorem

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

Let $G \subseteq E$ 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:

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

Proof

We call a linear functional $g$ admissible if $g(x) \le p(x)$ for all $x$ in the domain $D(g)$ of $g$.

We say that a linear functional $h_1$ extends a linear functional $h_2$ if $D(h_2) \subset D(h_1)$ and $h_2(x) = h_1(x)$ for all $x$ in $D(h_2)$.

The proof consists of two steps: first, we show that the set of admissible linear functionals that extend $f$ is inductive. Using Zorn's Lemma we can derive the existence of a maximal element. Second, we show by contradiction that this functional is defined on the whole space $E$.

Source of Name

This entry was named for Hans Hahn and Stefan Banach.