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:

$\forall x \in G: \map f x \le \map p x$


Then there exists a linear functional $\tilde f$ defined on the whole space $E$ which extends $f$.


That is:

$\forall x \in G: \map {\tilde f} x = \map f x$

such that:

$\forall x \in E: \map {\tilde f} x \le \map p x$


Proof

Let a linear functional $g$ be called admissible if and only if

$\forall x \in \Dom g: \map g x \le \map p x$

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

$\Dom {h_2} \subseteq \Dom {h_1}$

and:

$\forall x \in \Dom {h_2}: \map {h_2} x = \map {h_1} x$

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.


Historical Note

The Hahn-Banach Theorem was first proved by Eduard Helly in around $1912$, some $15$ years before Stefan Banach and Hans Hahn developed it independently.


Sources