In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry.

The most general formulation of the theorem needs some preparation. Given a real vector space V, a function f : V → R is called sublinear if

Every seminorm on V (in particular, every norm on V) is sublinear. Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

Hahn–Banach theorem (Rudin 1991, Th. 3.2). If p : V → R is a sublinear function, and φ : U → R is a linear functional on a linear subspace U ⊆ V which is dominated by p on U, i.e.

then there exists a linear extension ψ : V → R of φ to the whole space V, i.e., there exists a linear functional ψ such that

Hahn–Banach theorem (alternative version). Set K = R or C and let V be a K-vector space with a seminorm p : V → R. If φ : U → K is a K-linear functional on a K-linear subspace U of V which is dominated by p on U in absolute value,

then there exists a linear extension ψ : V → K of φ to the whole space V, i.e., there exists a K-linear functional ψ such that

In the complex case of the alternate version, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, we have ix ∈ U and φ(ix) = iφ(x).

The extension ψ is in general not uniquely specified by φ and the proof gives no explicit method as to how to find ψ. The usual proof for the case of an infinite dimensional space V uses Zorn's lemma or, equivalently, the axiom of choice. It is now known (see below) that the ultrafilter lemma, which is slightly weaker than the axiom of choice, is actually strong enough.

It is possible to relax slightly the subadditivity condition on p, requiring only that (Reed and Simon, 1980):

It is further possible to relax the positive homogeneity and the subadditivity conditions on p, requiring only that p is convex (Schechter, 1996).

This reveals the intimate connection between the Hahn–Banach theorem and convexity.

The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.^[3]

The theorem has several important consequences, some of which are also sometimes called "Hahn–Banach theorem":

Another version of Hahn–Banach theorem is known as the Hahn–Banach separation theorem.^[4] It has numerous uses in convex geometry,^[5] optimization theory, and economics. The separation theorem is derived from the original form of the theorem.

Theorem. Set K = R or C and let V be a topological vector space over K. If A, B are convex, non-empty disjoint subsets of V, then:

One form of Hahn–Banach theorem is known as the Geometric Hahn–Banach theorem, or Mazur's theorem.^[6]

This can be generalized to an arbitrary topological vector space, which need not be locally convex or even Hausdorff, as:^[7]

Theorem. Let M be a vector subspace of the topological vector space X. Suppose K is a non-empty convex open subset of X with K ∩ M = ∅. Then there is a closed hyperplane N in X containing M with K ∩ N = ∅.

As mentioned earlier, the axiom of choice implies the Hahn–Banach theorem. The converse is not true. One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case.

The Hahn–Banach theorem is equivalent to the following:^[8]

(The Boolean prime ideal theorem is easily seen to be equivalent to the statement that there are always probability charges which take only the values 0 and 1.)

In ZF, one can show that the Hahn–Banach theorem is enough to derive the existence of a non-Lebesgue measurable set.^[9] Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.^[10]

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.^[11][12]

If X is a topological vector space, not necessarily Hausdorff or locally convex, then there exists a non-zero continuous linear form if and only if X contains a nonempty, proper, convex, open set U.^[13] So if the continuous dual space of X, X*, is non-trivial then by considering X with the weak topology induced by X*, X becomes a locally convex topological vector space with a non-trivial topology that is weaker than original topology on X. If in addition, X* separates points on X (which means that for each x ∈ X there is a linear functional in X* that's non-zero on x) then X with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

We have the following consequence of the Hahn–Banach theorem.

Proposition. Let −∞ < a < b < ∞. Then, F ∈ C[a, b]* if and only if there exists a (complex) measure ρ : [a, b] → R of bounded variation such that

for all u ∈ C[a, b]. In addition, |F| = V(ρ), where V(ρ) denotes the total variation of ρ.