Functional analysis: Difference between revisions

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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|''φ'' >\varphi: ''U'' → '''\to\mathbb{R'''}} is a [[linear functional]] on a [[linear subspace]] {{<math|''>U''\subseteq ⊆ ''V''}}</math> which is [[dominate (mathematics)|dominated]] by {{mvar|<math>p}}</math> on {{mvar|<math>U}}</math>; that is,

:<math>\varphi(x) \leq p(x)\qquad\forall x \in U</math>

then there exists a linear extension {{<math|''ψ'' >\psi: ''V'' → '''\to\mathbb{R'''}}</math> of {{mvar|φ}}<math>\varphi</math> to the whole space {{mvar|<math>V}}</math> which is [[dominate (mathematics)|dominated]] by {{mvar|<math>p}}</math> on {{mvar|<math>V}}</math>; that is, there exists a linear functional {{mvar|ψ}}<math>\psi</math> such that

:<math>\psi(x)=\varphi(x)\qquad\forall x\in U,</math>

:<math>\psi(x) \le p(x)\qquad\forall x\in V.</math></blockquote>

===Open mapping theorem===