Hahn-Banach Theorem

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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 if and only if

$\forall x \in \operatorname{Dom} \left({g}\right): g \left({x}\right) \le p \left({x}\right)$

A linear functional $h_1$ extends a linear functional $h_2$ if and only if:

$\operatorname{Dom} \left({h_2}\right) \subseteq \operatorname{Dom} \left({h_1}\right)$

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 $E$.



Source of Name

This entry was named for Hans Hahn and Stefan Banach.