nLab
Hahn-Banach theorem

Contents

Idea

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.

Foundational issues

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

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).

References

Revised on August 22, 2012 21:56:28 by Toby Bartels (98.19.40.130)