Editing Hahn–Banach theorem (section)

===Non-locally convex spaces===

The [[#Hahn–Banach continuous extension theorem|continuous extension theorem]] might fail if the [[topological vector space]] (TVS) <math>X</math> is not [[Locally convex topological vector space|locally convex]]. For example, for <math>0 < p < 1,</math> the [[Lp space|Lebesgue space]] <math>L^p([0, 1])</math> is a [[Complete metric space|complete]] [[Metrizable topological vector space|metrizable TVS]] (an [[F-space]]) that is {{em|not}} locally convex (in fact, its only convex open subsets are itself <math>L^p([0, 1])</math> and the empty set) and the only continuous linear functional on <math>L^p([0, 1])</math> is the constant <math>0</math> function {{harv|Rudin|1991|loc=§1.47}}. Since <math>L^p([0, 1])</math> is Hausdorff, every finite-dimensional vector subspace <math>M \subseteq L^p([0, 1])</math> is [[TVS-isomorphism|linearly homeomorphic]] to [[Euclidean space]] <math>\Reals^{\dim M}</math> or <math>\Complex^{\dim M}</math> (by [[F. Riesz's theorem]]) and so every non-zero linear functional <math>f</math> on <math>M</math> is continuous but none has a continuous linear extension to all of <math>L^p([0, 1]).</math> 
However, it is possible for a TVS <math>X</math> to not be locally convex but nevertheless have enough continuous linear functionals that its [[continuous dual space]] <math>X^*</math> [[Separating set|separates points]]; for such a TVS, a continuous linear functional defined on a vector subspace {{em|might}} have a continuous linear extension to the whole space. 

If the [[topological vector space|TVS]] <math>X</math> is not [[Locally convex topological vector space|locally convex]] then there might not exist any continuous seminorm <math>p : X \to \R</math> {{em|defined on <math>X</math>}} (not just on <math>M</math>) that dominates <math>f,</math> in which case the Hahn–Banach theorem can not be applied as it was in [[#Proof of the continuous extension theorem for locally convex topological vector spaces|the above proof]] of the continuous extension theorem. 
However, the proof's argument can be generalized to give a characterization of when a continuous linear functional has a continuous linear extension: If <math>X</math> is any TVS (not necessarily locally convex), then a continuous linear functional <math>f</math> defined on a vector subspace <math>M</math> has a continuous linear extension <math>F</math> to all of <math>X</math> if and only if there exists some continuous seminorm <math>p</math> on <math>X</math> that [[#dominated complex functional|dominates]] <math>f.</math>  Specifically, if given a continuous linear extension <math>F</math> then <math>p := |F|</math> is a continuous seminorm on <math>X</math> that dominates <math>f;</math> and conversely, if given a continuous seminorm <math>p : X \to \Reals</math> on <math>X</math> that dominates <math>f</math> then any dominated linear extension of <math>f</math> to <math>X</math> (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension.