Hahn–Banach theorem

Formulation

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

Positive homogeneity: f(γx) = γ f(x) for all γ ∈ R₊, x ∈ V,

Subadditivity: f(x + y) ≤ f(x) + f(y) for all x, y ∈ V.

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.

φ ( x ) ≤ p ( x ) ∀ x ∈ U

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

ψ ( x ) = φ ( x ) ∀ x ∈ U , ψ ( x ) ≤ p ( x ) ∀ x ∈ V .

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,

| φ ( x ) | ≤ p ( x ) ∀ x ∈ U

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

ψ ( x ) = φ ( x ) ∀ x ∈ U , | ψ ( x ) | ≤ p ( x ) ∀ x ∈ V .

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 section 4.0) 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):

p ( a x + b y ) ≤ | a | p ( x ) + | b | p ( y ) , x , y ∈ V , | a | + | b | ≤ 1.

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.

Important consequences

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

If V is a normed vector space with linear subspace U (not necessarily closed) and if φ : U → K is continuous and linear, then there exists an extension ψ : V → K of φ which is also continuous and linear and which has the same norm as φ (see Banach space for a discussion of the norm of a linear map). In other words, in the category of normed vector spaces, the space K is an injective object.

If V is a normed vector space with linear subspace U (not necessarily closed) and if z is an element of V not in the closure of U, then there exists a continuous linear map ψ : V → K with ψ(x) = 0 for all x in U, ψ(z) = 1, and ||ψ|| = dist(z, U)⁻¹.

In particular, if V is a normed vector space and if z is any element of V, then there exists a continuous linear map ψ : V → K with ψ(z) = ||z|| and ||ψ|| ≤ 1. This implies that the natural injection J from a reflexive normed space V into its double dual V′′ is isometric.

Hahn–Banach separation theorem

Another version of Hahn–Banach theorem is known as the Hahn–Banach separation theorem. It has numerous uses in convex geometry, 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:

If A is open, then there exists a continuous linear map λ : V → K and t ∈ R such that Re(λ(a)) < t ≤ Re(λ(b)) for all a ∈ A, b ∈ B.

If V is locally convex, A is compact, and B closed, then there exists a continuous linear map λ : V → K and s, t ∈ R such that Re(λ(a)) < t < s < Re(λ(b)) for all a ∈ A, b ∈ B.

Geometric Hahn–Banach theorem

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

Theorem. Let K be a convex set having a nonempty interior in a real normed linear vector space X. Suppose V is a linear variety in X containing no interior points of K. Then there is a closed hyperplane in X containing V but containing no interior points of K; i.e., there is an element x* ∈ X* and a constant c such that ⟨ v , x ∗ ⟩ = c for all v ∈ V and ⟨ v , x ∗ ⟩ < c for all k ∈ int(K).

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

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 = ∅.

Relation to axiom of choice

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:

(∗): On every Boolean algebra B there exists a "probability charge", that is: a nonconstant finitely additive map from B into [0, 1].

(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. Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.

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 König'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.

Topological vector spaces

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. 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.

The dual space C[a, b]*

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

F ( u ) = ∫ a b u ( x ) d ρ ( x ) ,

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