The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.
[2] The first Hahn–Banach theorem was proved by Eduard Helly in 1912 who showed that certain linear functionals defined on a subspace of a certain type of normed space (
Helly did this through the technique of first proving that a one-dimensional extension exists (where the linear functional has its domain extended by one dimension) and then using induction.
[3] The Hahn–Banach theorem arose from attempts to solve infinite systems of linear equations.
It wasn't until 1932 that Banach, in one of the first important applications of the Hahn–Banach theorem, solved the general functional problem.
is a sublinear function (such as a norm or seminorm for example) defined on a real vector space
The dominated extension theorem for real linear functionals implies the following alternative statement of the Hahn–Banach theorem that can be applied to linear functionals on real or complex vector spaces.
With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly: Proof The following observations allow the Hahn–Banach theorem for real vector spaces to be applied to (complex-valued) linear functionals on complex vector spaces.
The Hahn–Banach theorem for real vector spaces ultimately follows from Helly's initial result for the special case where the linear functional is extended from
The standard proof of the general case uses Zorn's lemma although the strictly weaker ultrafilter lemma[11] (which is equivalent to the compactness theorem and to the Boolean prime ideal theorem) may be used instead.
Hahn–Banach can also be proved using Tychonoff's theorem for compact Hausdorff spaces[12] (which is also equivalent to the ultrafilter lemma) The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.
is a locally convex space then this statement remains true when the linear functional
The continuous extension theorem might fail if the topological vector space (TVS)
The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets:
This sort of argument appears widely in convex geometry,[18] optimization theory, and economics.
[19][20] They are generalizations of the hyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space
When the convex sets have additional properties, such as being open or compact for example, then the conclusion can be substantially strengthened: Theorem[3][21] — Let
be convex non-empty disjoint subsets of a real topological vector space
Mazur's theorem clarifies that vector subspaces (even those that are not closed) can be characterized by linear functionals.
Since points are trivially convex, geometric Hahn–Banach implies that functionals can detect the boundary of a set.
For example, many results in functional analysis assume that a space is Hausdorff or locally convex.
However, suppose X is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set M. Then geometric Hahn–Banach implies that there is a hyperplane separating M from any other point.
then X becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space separates points.
Theorem[26] — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
In 1992, Kakol proved that any infinite dimensional vector space X, there exist TVS-topologies on X that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on X.
[31] The proof of the Hahn–Banach theorem for real vector spaces (HB) commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory (ZF) is equivalent to the axiom of choice (AC).
It was discovered by Łoś and Ryll-Nardzewski[12] and independently by Luxemburg[11] that HB can be proved using the ultrafilter lemma (UL), which is equivalent (under ZF) to the Boolean prime ideal theorem (BPI).
In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.
[38] For separable Banach spaces, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic that takes a form of Kőnig's lemma restricted to binary trees as an axiom.
In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of reverse mathematics.