Editing Hahn–Banach theorem (section)

===Maximal dominated linear extension===

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}
| note = Andenaes, 1970
| math_statement = Let <math>p : X \to \R</math> be a sublinear function on a real vector space <math>X,</math> let <math>f : M \to \R</math> be a linear functional on a vector subspace <math>M</math> of <math>X</math> such that <math>f \leq p</math> on <math>M,</math> and let <math>S \subseteq X</math> be any subset of <math>X.</math> 
Then there exists a linear functional <math>F : X \to \R</math> on <math>X</math> that extends <math>f,</math> satisfies <math>F \leq p</math> on <math>X,</math> and is (pointwise) maximal on <math>S</math> in the following sense: if <math>\widehat{F} : X \to \R</math> is a linear functional on <math>X</math> that extends <math>f</math> and satisfies <math>\widehat{F} \leq p</math> on <math>X,</math> then <math>F \leq \widehat{F}</math> on <math>S</math> implies <math>F = \widehat{F}</math> on <math>S.</math>
}}

If <math>S = \{s\}</math> is a singleton set (where <math>s \in X</math> is some vector) and if <math>F : X \to \R</math> is such a maximal dominated linear extension of <math>f : M \to \R,</math> then <math>F(s) = \inf_{m \in M} [f(s) + p(s - m)].</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}