Hahn–Banach theorem

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 Hahn–Banach theorem is known as Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry. It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier (in 1912) by Eduard Helly, and a general extension theorem from which the Hahn–Banach theorem can be derived was proved in 1923 by Marcel Riesz.

Formulation

The most general formulation of the theorem needs some preparation. Given a real vector space Template:Mvar, 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 Template:Mvar (in particular, every norm on Template:Mvar) is sublinear. Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

Hahn–Banach Theorem Template:Harv. 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 Template:Mvar on Template:Mvar, i.e.

$\varphi (x)\leq p(x)\qquad \forall x\in U$

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

$\psi (x)=\varphi (x)\qquad \forall x\in U,$

$\psi (x)\leq p(x)\qquad \forall x\in V.$

Hahn–Banach Theorem (Alternate Version). Set K = R or C and let Template:Mvar 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 Template:Mvar of Template:Mvar which is dominated by Template:Mvar on Template:Mvar in absolute value,

$|\varphi (x)|\leq p(x)\qquad \forall x\in U$

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

$\psi (x)=\varphi (x)\qquad \forall x\in U,$

$|\psi (x)|\leq p(x)\qquad \forall x\in V.$

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 Template:Mvar is in general not uniquely specified by Template:Mvar and the proof gives no explicit method as to how to find Template:Mvar. The usual proof for the case of an infinite dimensional space Template:Mvar 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 Template:Mvar, requiring only that (Reed and Simon, 1980):

$p(ax+by)\leq |a|\,p(x)+|b|\,p(y),\qquad x,y\in V,\quad |a|+|b|\leq 1.$

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 Template:Mvar is a normed vector space with linear subspace Template:Mvar (not necessarily closed) and if φ : U → K is continuous and linear, then there exists an extension ψ : V → K of Template:Mvar which is also continuous and linear and which has the same norm as Template:Mvar (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 Template:Mvar is a normed vector space with linear subspace Template:Mvar (not necessarily closed) and if Template:Mvar is an element of Template:Mvar not in the closure of Template:Mvar, then there exists a continuous linear map ψ : V → K with ψ(x) = 0 for all Template:Mvar in Template:Mvar, ψ(z) = 1, and Template:!!ψTemplate:!! = dist(z, U)⁻¹.
• In particular, if Template:Mvar is a normed vector space and if Template:Mvar is any element of Template:Mvar, then there exists a continuous linear map ψ : V → K with ψ(z) = Template:!!zTemplate:!! and Template:!!ψTemplate:!! ≤ 1. This implies that the natural injection Template:Mvar from a normed space Template:Mvar 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 Template:Mvar be a topological vector space over K. If A, B are convex, non-empty disjoint subsets of Template:Mvar, then:

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

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

Theorem. Let Template:Mvar be a vector subspace of the topological vector space Template:Mvar. Suppose Template:Mvar is a non-empty convex open subset of Template:Mvar with K ∩ M = ∅. Then there is a closed hyperplane Template:Mvar in Template:Mvar containing Template:Mvar 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 Template:Mvar there exists a "probability charge", that is: a nonconstant finitely additive map from Template:Mvar 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.

Consequences

Topological Vector Spaces

If Template:Mvar is a topological vector space, not necessarily Hausdorff or locally convex, then there exists a non-zero continuous linear form if and only if Template:Mvar contains a nonempty, proper, convex, open set Template:Mvar. So if the continuous dual space of X, X*, is non-trivial then by considering Template:Mvar 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 Template:Mvar. If in addition, X* separates points on Template:Mvar (which means that for each x ∈ X there is a linear functional in X* that's non-zero on Template:Mvar) then Template:Mvar 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

$F(u)=\int _{a}^{b}u(x)d\rho (x),$

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

See also

• M. Riesz extension theorem
• Separating axis theorem
• Farkas lemma

References

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• Lawrence Narici and Edward Beckenstein, "The Hahn–Banach Theorem: The Life and Times", Topology and its Applications, Volume 77 (1997), 193–211.
• Lothar M Schmitt, An Equivariant Version of the Hahn-Banach Theorem, Houston J. of Math. 18 (1992), 429-447
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
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• Terence Tao, The Hahn–Banach theorem, Menger’s theorem, and Helly’s theorem
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• Gerd Wittstock, Ein operatorwertiger Hahn-Banach Satz, J. of Functional Analysis 40 (1981), 127–150
• Eberhard Zeidler, Applied Functional Analysis: main principles and their applications, Springer, 1995.