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 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?)