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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".
<blockquote>'''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> 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 [[dominate (mathematics)|dominated]] by <math>p</math> on <math>U</math>; that is,
:<math>\varphi(x) \leq p(x)\qquad\forall x \in U</math>
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