nLab
Hahn-Banach theorem

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.

Statement

Theorem

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

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).

Consequences

References

• Wikipedia on the Hahn-Banach theorem