Hahn-Banach theorem
HahnBanachTheorem
2002-08-01 12:24:14
2003-04-13 17:20:38
Theorem
bound
bounded
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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
$$\pnorm:V\rightarrow\reals^+$$
be a seminorm on $V$.
\begin{theorem}
Let $f:U\to K$ be a linear functional defined on a subspace
$U\subset V$. If the restricted functional satisfies
$$\vert f(\bu)\vert\leq \snorm{\bu},\quad \bu\in U,$$
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
\begin{align*}
F(\bu) &= f(\bu),\quad \bu\in U\\
\vert F(\bu) \vert &\leq \snorm{\bu},\quad \bu\in V.
\end{align*}
\end{theorem}
\begin{definition}
We say that a linear functional $f:V\to K$ is \emph{bounded} if
there exists a bound $B\in\reals^+$ such that
\begin{equation}
\label{eq:bdef}
\vert f(\bu)\vert \leq B \snorm{\bu},\quad \bu\in V.
\end{equation}
If $f$ is a bounded linear functional, we define $\Vert f\Vert$, the
norm of $f$, according to
$$\Vert f \Vert = \sup \{ \vert f(\bu)\vert : \snorm{\bu} = 1 \}.$$
One can show that $\Vert f\Vert$ is the infimum of all the possible
$B$ that satisfy \eqref{eq:bdef}
\end{definition}
\begin{theorem}[Hahn-Banach]
Let $f:U\to K$ be a bounded linear functional defined on a subspace
$U\subset V$. Let $\Vert f \Vert_U$ 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.
$$\Vert F\Vert_V = \Vert f\Vert_U.$$
\end{theorem}