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 be a vector space over a local field (usually or ), equipped with a seminorm . Let be a linear subspace, and a linear functional such that for all , then there exists an extension of to a linear functional such that for all .
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? No, but the ultrafilter principle frequently used to prove the Hahn-Banach theorem implies De Morgan's law which is usually not accepted in constructive mathematics.)
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 proven in Pelletier 1991. This constructs a locale of functionals (whose points are the unit ball of the dual space) and proves that it is compact and completely regular.
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See also:
Discussion in the generality of locales:
Last revised on September 21, 2021 at 04:49:15. See the history of this page for a list of all contributions to it.