Editing Linear form (section)

=== Hahn–Banach theorem ===
{{Main|Hahn–Banach theorem}}

Any (algebraic) linear functional on a [[vector subspace]] can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of <math>\R.</math>  However, this extension cannot always be done while keeping the linear functional continuous. The Hahn–Banach family of theorems gives conditions under which this extension can be done.  For example,

{{math theorem|name=Hahn–Banach dominated extension theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{harv|Rudin|1991|loc=Th. 3.2}}|math_statement=
If <math>p : X \to \R</math> is a [[sublinear function]], and <math>f : M \to \R</math> is a [[linear functional]] on a [[linear subspace]] <math>M \subseteq X</math> which is dominated by {{mvar|p}} on {{mvar|M}}, then there exists a linear extension <math>F : X \to \R</math> of {{mvar|f}} to the whole space {{mvar|X}} that is dominated by {{mvar|p}}, i.e., there exists a linear functional {{mvar|F}} such that
<math display="block">F(m) = f(m)</math>
for all <math>m \in M,</math> and
<math display="block">|F(x)| \leq p(x)</math>
for all <math>x \in X.</math>
}}