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.

Sources

There are no source works cited for this page. In particular: The Hahn-Banach Theorem is arguably one of the most important theorems in functional analysis to this date, so citation is highly desirable. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.