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 linear 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 $x \in V$.

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.

Joan Wick Pelletier, Locales in Functional Analysis, Journal of Pure and Applied Algebra Volume 70, Issues 1–2, 15 March 1991, Pages 133-145 (doi:10.1016/0022-4049(91)90013-R)

Last revised on September 21, 2021 at 00:49:15.
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