Editing Hahn–Banach theorem (section)

===Invariant Hahn–Banach===
{{See also|Vector-valued Hahn–Banach theorems}}

A set <math>\Gamma</math> of maps <math>X \to X</math> is {{em|{{visible anchor|commutative set of maps|text=commutative}}}} (with respect to [[function composition]] <math>\,\circ\,</math>) if <math>F \circ G = G \circ F</math> for all <math>F, G \in \Gamma.</math> 
Say that a function <math>f</math> defined on a subset <math>M</math> of <math>X</math> is {{em|{{visible anchor|invariant map|text=<math>\Gamma</math>-invariant}}}} if <math>L(M) \subseteq M</math> and <math>f \circ L = f</math> on <math>M</math> for every <math>L \in \Gamma.</math>

{{Math theorem
| name = {{visible anchor|An invariant Hahn–Banach theorem}}{{sfn|Rudin|1991|p=141}}
| math_statement = 
Suppose <math>\Gamma</math> is a [[#commutative set of maps|commutative set]] of continuous linear maps from a [[normed space]] <math>X</math> into itself and let <math>f</math> be a continuous linear functional defined some vector subspace <math>M</math> of <math>X</math> that is [[#invariant map|<math>\Gamma</math>-invariant]], which means that <math>L(M) \subseteq M</math> and <math>f \circ L = f</math> on <math>M</math> for every <math>L \in \Gamma.</math> 
Then <math>f</math> has a continuous linear extension <math>F</math> to all of <math>X</math> that has the same [[operator norm]] <math>\|f\| = \|F\|</math> and is also <math>\Gamma</math>-invariant, meaning that <math>F \circ L = F</math> on <math>X</math> for every <math>L \in \Gamma.</math>
}}

This theorem may be summarized:

:Every [[#invariant map|<math>\Gamma</math>-invariant]] continuous linear functional defined on a vector subspace of a normed space <math>X</math> has a <math>\Gamma</math>-invariant Hahn–Banach extension to all of <math>X.</math>{{sfn|Rudin|1991|p=141}}