Hahn-Banach theorem



The Hahn–Banach theorem explains why the concept of locally convex spaces is of interest in the analysis of topological vector spaces: It ensures that such a space will have enough continuous linear functionals such that the topological dual space is interesting.



Let VV be a vector space over a local field KK (usually \mathbb{R} or \mathbb{C}), equipped with a seminorm p:V[0,)p: V \to [0, \infty). Let WVW \subseteq V be a subspace, and f:WKf: W \to K a linear functional such that |f(x)|p(x){|f(x)|} \leq p(x) for all xWx \in W, then there exists an extension of ff to a linear functional g:VKg: V \to K such that |g(x)|p(x){|g(x)|} \leq p(x) for all xVx \in V.

Foundational issues

The full Hahn–Banach theorem may be seen as a weak form of the axiom of choice; this is the perspective taken in, for example, HAF. It fails in dream mathematics and is generally not accepted in constructive mathematics. (Does it actually imply excluded middle?)

The Hahn-Banach theorem can be proven in set theory with the axiom of choice, or more weakly in set theory assuming the ultrafilter theorem, itself a weak form of choice. (To be continued…)

However, the Hahn–Banach theorem for separable spaces is much weaker. It may be proved constructively using only dependent choice. There is also a version of the theorem for locales (or so I heard).



Revised on April 13, 2015 02:06:42 by Todd Trimble (