Editing Functional analysis (section)

===Hahn–Banach theorem===
{{main|Hahn–Banach theorem}}
The [[Hahn–Banach theorem]] is a central tool in functional analysis. It allows the extension of [[Bounded operator|bounded linear functionals]] defined on a subspace of some [[vector space]] to the whole space, and it also shows that there are "enough" [[continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] to make the study of the [[dual space]] "interesting".

{{math theorem | name = Hahn–Banach theorem:<ref name="rudin">{{Cite book | last=Rudin | first=Walter | url={{google books |plainurl=y |id=Sh_vAAAAMAAJ}} | title=Functional Analysis | date=1991 | publisher=McGraw-Hill | isbn=978-0-07-054236-5 | language=en}}</ref>
| math_statement = If <math>p:V\to\mathbb{R}</math> is a [[sublinear function]], and <math>\varphi:U\to\mathbb{R}</math> is a [[linear functional]] on a [[linear subspace]] <math>U\subseteq V</math> which is dominated by <math>p</math> on <math>U</math>; that is,
<math display="block">\varphi(x) \leq p(x)\qquad\forall x \in U</math>
then there exists a linear extension <math>\psi:V\to\mathbb{R}</math> of <math>\varphi</math> to the whole space <math>V</math> which is dominated by <math>p</math> on <math>V</math>; that is, there exists a linear functional <math>\psi</math> such that
<math display="block">\begin{align}
\psi(x) &= \varphi(x) &\forall x\in U, \\
\psi(x) &\le p(x) &\forall x\in V.
\end{align}</math>}}