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 $p \colon V \to [0, \infty)$. Let $W \subseteq V$ be a linear subspace, and $f \colon W \to K$ 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. (It does not actually imply excluded middle, 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, under the additional assumption that the bound $p$ in the statement is continuous, the Hahn–Banach theorem for a restricted class of spaces, including e.g. separable spaces is much weaker. It may be proved constructively using only dependent choice. This is false if the continuity assumption is removed. See Dodu & Morillon 2010.

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.

Consequences

(…)

References

See also:

• Wikipedia, Hahn-Banach theorem

Discussion in constructive mathematics:

• Juliette Dodu, Marianne Morillon: The Hahn-Banach Property and the Axiom of Choice, Mathematical Logic Quarterly 45 3 (2010) 299-314 [doi:10.1002/malq.19990450303]

Discussion in the generality of locales:

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