Editing Seminorm (section)

===Hahn–Banach theorem for seminorms===

Seminorms offer a particularly clean formulation of the [[Hahn–Banach theorem]]: 
:If <math>M</math> is a vector subspace of a seminormed space <math>(X, p)</math> and if <math>f</math> is a continuous linear functional on <math>M,</math> then <math>f</math> may be extended to a continuous linear functional <math>F</math> on <math>X</math> that has the same norm as <math>f.</math>{{sfn|Wilansky|2013|pp=21-26}}

A similar extension property also holds for seminorms:

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=150}}{{sfn|Wilansky|2013|pp=18-21}}|note=Extending seminorms|math_statement=
If <math>M</math> is a vector subspace of <math>X,</math> <math>p</math> is a seminorm on <math>M,</math> and <math>q</math> is a seminorm on <math>X</math> such that <math>p \leq q\big\vert_M,</math> then there exists a seminorm <math>P</math> on <math>X</math> such that <math>P\big\vert_M = p</math> and <math>P \leq q.</math> 
}}

:'''Proof''': Let <math>S</math> be the [[convex hull]] of <math>\{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}.</math> Then <math>S</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X</math> and so the [[Minkowski functional]] <math>P</math> of <math>S</math> is a seminorm on <math>X.</math> This seminorm satisfies <math>p = P</math> on <math>M</math> and <math>P \leq q</math> on <math>X.</math> <math>\blacksquare</math>