# Hahn-Banach theorem

The Hahn-Banach theorem is a foundational result in functional analysis. Roughly speaking, it asserts the existence of a great variety of bounded (and hence continuous) linear functionals on an normed vector space, even if that space happens to be infinite-dimensional. We first consider an abstract version of this theorem, and then give the more classical result as a corollary.

Let $V$ be a real, or a complex vector space, with $K$ denoting the corresponding field of scalars, and let

 $\operatorname{p}:V\rightarrow\mathbb{R}^{+}$

be a seminorm on $V$.

###### Theorem 1

Let $f:U\to K$ be a linear functional defined on a subspace $U\subset V$. If the restricted functional satisfies

 $|f(\mathbf{u})|\leq\operatorname{p}(\mathbf{u}),\quad\mathbf{u}\in U,$

then it can be extended to all of $V$ without violating the above property. To be more precise, there exists a linear functional $F:V\to K$ such that

 $\displaystyle F(\mathbf{u})$ $\displaystyle=f(\mathbf{u}),\quad\mathbf{u}\in U$ $\displaystyle|F(\mathbf{u})|$ $\displaystyle\leq\operatorname{p}(\mathbf{u}),\quad\mathbf{u}\in V.$
###### Definition 2

We say that a linear functional $f:V\to K$ is bounded if there exists a bound $B\in\mathbb{R}^{+}$ such that

 $|f(\mathbf{u})|\leq B\operatorname{p}(\mathbf{u}),\quad\mathbf{u}\in V.$ (1)

If $f$ is a bounded linear functional, we define $\|f\|$, the norm of $f$, according to

 $\|f\|=\sup\{|f(\mathbf{u})|:\operatorname{p}(\mathbf{u})=1\}.$

One can show that $\|f\|$ is the infimum of all the possible $B$ that satisfy (1)

###### Theorem 3 (Hahn-Banach)

Let $f:U\to K$ be a bounded linear functional defined on a subspace $U\subset V$. Let $\|f\|_{U}$ denote the norm of $f$ relative to the restricted seminorm on $U$. Then there exists a bounded extension $F:V\to K$ with the same norm, i.e.

 $\|F\|_{V}=\|f\|_{U}.$
Title Hahn-Banach theorem HahnBanachTheorem 2013-03-22 12:54:09 2013-03-22 12:54:09 rmilson (146) rmilson (146) 10 rmilson (146) Theorem msc 46B20 bound bounded