# Hahn-Banach Theorem

## Theorem

### Real Vector Space

Let $X$ be a vector space over $\R$.

Let $p : X \to \R$ be a sublinear functional.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \R$ be a linear functional such that:

$\map {f_0} x \le \map p x$ for each $x \in X_0$.

Then there exists a linear functional $f$ defined on the whole space $X$ which extends $f_0$ and satisfies:

$\map f x \le \map p x$ for each $x \in X$.

That is, there exists a linear functional $f : X \to \R$ such that:

$\map f x \le \map p x$ for each $x \in X$

and:

$\map f x = \map {f_0} x$ for each $x \in X_0$.

### Complex Vector Space

Let $X$ be a vector space over $\C$.

Let $p : X \to \R$ be a seminorm on $X$.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \C$ be a linear functional such that:

$\cmod {\map {f_0} x} \le \map p x$ for each $x \in X_0$.

Then there exists a linear functional $f$ defined on the whole space $X$ which extends $f_0$ and satisfies:

$\cmod {\map f x} \le \map p x$ for each $x \in X$.

That is, there exists a linear functional $f : X \to \C$ such that:

$\cmod {\map f x} \le \map p x$ for each $x \in X$

and:

$\map f x = \map {f_0} x$ for each $x \in X_0$.

## 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.