Hahn–Banach theorem


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 hyperplane separation theorem, and has numerous uses in convex geometry.

History

The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.
The special case of the theorem for the space of continuous functions on an interval was proved earlier by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz.
The first Hahn-Banach theorem was proved by Eduard Helly in 1921 who showed that certain linear functionals defined on a subspace of a certain type of normed space had an extension of the same norm.
Helly did this through the technique of first proving that a one-dimensional extension exists and then using induction.
In 1927, Hahn defined general Banach spaces and used Helly's technique to prove a norm preserving version of Hahn-Banach theorem for Banach spaces.
In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses sublinear functions.
Whereas Helly's proof used mathematical induction, Hahn and Banach both used transfinite induction.
The Hahn-Banach theorem arose from attempts to solve infinite systems of linear equations.
This is needed to solve problems such as the moment problem, whereby given all the potential moments of a function one must determine if a function having these moments exists and if so then find it.
Another such problem is the Fourier cosine series problem, whereby given all the potential Fourier cosine coefficients one must determine if a function having those coefficients exists and if so then find it.
Riesz and Helly solved the problem for certain classes of spaces and C) where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals.
In effect, they needed to solve the following problem:
To solve this, if is reflexive then it suffices to solve the following dual problem:
Riesz went on to define Lp space| in 1910 and the spaces in 1913.
While investing these spaces he proved a special case of the Hahn-Banach theorem.
Helly also proved a special case of the Hahn-Banach theorem in 1912.
In 1910, Riesz solved the functional problem for some specific spaces and in 1912, Helly solved it for a more general class of spaces.
It wasn't until 1932 that Banach, in one of the first important applications of the Hahn-Banach theorem, solved the general functional problem.
The following theorem states the general functional problem and characterizes its solution.
One can use the above theorem to deduce the Hahn-Banach theorem.
If is reflexive, then this theorem solves the vector problem.

Prerequisites and definitions

The most general formulation of the theorem needs some preparation. Given a real or complex vector space, a function is called sublinear if the following conditions hold:
  1. Positive homogeneity: for all,,
  2. Subadditivity: for all.
A sublinear function is called a seminorm if it also satisfies:

  1. Absolute homogeneity: for all and all scalars.
A seminorm is called a norm if in addition it satisfies the following condition:

  1. Positive definiteness/Separates points: If is such that, then.
Every seminorm on and every linear functional on is sublinear.
Note that unlike a seminorm, a sublinear function is allowed to take on negative values.
Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.
If is a linear functional on a topological vector space and if is a continuous sublinear function on then implies that is continuous.
If is a linear functional on a real or complex vector space and if is a seminorm on, then on if and only if on .
If is a complex-valued linear functional on a complex vector space then for all, so that I is completely determined by 's real part R, which we will also denote by, and this in turn implies that is entirely determined by.
In particular, it follows that is bounded if and only if is bounded.
If is a subset of and if is a function then we say that dominates on or that is bounded above by on if for all.
We call a function and extension of to if for all ;
if in addition is a linear map then we call a linear extension of.

Formulation

The general template for the various versions of the Hahn-Banach theorem presented in this article is as follows:
Sometimes is assumed to have additional structure, such as a topology or a norm, but many of the statements are purely algebraic.
We may apply a purely algebraic version of this theorem to the case where is a topological vector space as follows:
by choosing to be continuous, the inequality allows us to conclude that is necessarily continuous.
.
Note that because it is always possible to extend a continuous linear functional on to a unique continuous linear functional on the closure of in, some of these Hahn-Banach theorems assume that is a closed vector subspace of.
Unless indicated otherwise, it is to be assumed that is a vector space over the real or the complex numbers.
Some of the statements are given only for real vector spaces or only real-valued linear functionals while others are given for real or complex vector spaces.
One may apply a result that applies only to real-valued linear functionals to the complex case by recalling that a complex-valued linear functional is continuous if and only if its real part, R, is continuous and that furthermore, the real part R completely determines the imaginary part I and thus completely determines c.
The sublinear function is always real-valued.
If is assumed to be a real vector space then all linear functionals are assumed to be real-valued, unless indicated otherwise.
If the linear functional is real-valued then you'll often see the condition whereas if is complex-valued then you're more likely to see or.
Sometimes a functional is assumed to be bounded and other times it is assumed to be continuous.
If a is pseudometrizable TVS then a linear map from into any other TVS is continuous if and only if it is a bounded map.
In particular, a linear functional on a normed or Banach space is continuous if and only if it is bounded.

Hahn-Banach theorem

The following lemma is fundamental to proving the general Hahn-Banach theorem and its basic prove first appeared in a 1912 paper by Helly where it was proved for the space C.
The extension is in general not uniquely specified by and the proof gives no explicit method as to how to find.
The usual proof for the case of an infinite dimensional space uses Zorn's lemma or, equivalently, the axiom of choice.
It is now known 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, requiring only that for all and all scalars and satisfying,
It is further possible to relax the positive homogeneity and the subadditivity conditions on, requiring only that is convex.
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":
For normed spaces we have the following results:
Let be a TVS.
Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on.
Say that has the Hahn-Banach extension property if every vector subspace of has the extension property.
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
For complete metrizable TVSs there is a converse:
If a vector space has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
A vector subspace of a TVS has the separation property if for every element of such that, there exists a continuous linear functional on such that and for all.
Clearly, the continuous dual space of a TVS separates points on if and only if has the separation property.
In 1992, Kakol proved that any infinite dimensional vector space, there exist TVS-topologies on that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on.
However, if is a TVS then every vector subspace of has the extension property if and only if every vector subspace of has the separation property.

Hahn–Banach separation theorems

Hahn–Banach separation theorems are the geometrical versions of the Hahn–Banach Theorem.
They have numerous uses in convex geometry, optimization theory, and economics.
The separation theorem is derived from the original form of the theorem.
Let be a real vector space, and non-empty subsets of, a real linear functional on, a scalar, and let.
We also define the lower half space to be .
We define the strict lower half space to be .
We say that separates and if sup or equivalently, if for all and.
The separation is:
Note that and are separated if and only if the same is true of and.
If and are convex then they are strongly separated by a hyperplane if and only if there exists an absorbing convex such that.
We say that and are united if they cannot be properly separated.
If and separates and then is called a supporting hyperplane of at, is called a support point of, and is called a support functional.
If is convex and, then we call a smooth point of if there exists a unique hyperplane such that.
We call a normed space smooth if at each point in its unit ball there exists a unique closed hyperplane to the unit ball at.
Köthe showed in 1983 that a normed space is smooth at a point if and only if the norm is Gateaux differentiable at that point.

Characterizations of when a separating functional exists

Many of the separation theorems below may be generalized to the following theorem:

Subsets of a half space

Separation of sets

The following theorem may be used if the sets are not necessarily disjoint.

Separation of a point and set

The following is the Hahn–Banach separation theorem for a point and a set.

Separation of a closed and compact set

Separation of points and disked neighborhoods of 0

Reflexive Banach spaces

Hahn-Banach sandwich theorem

Geometric Hahn–Banach theorem

One form of Hahn–Banach theorem is known as the Geometric Hahn–Banach theorem or Mazur's theorem.
This can be generalized to an arbitrary topological vector space, which need not be locally convex or even Hausdorff, as:

Mazur-Orlicz theorem

The following theorem of Mazur-Orlicz is equivalent to the Hahn-Banach theorem.

Generalizations

Due to its importance, the Hahn-Banach theorem has been generalized many times.

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, 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:
In Zermelo–Fraenkel set theory, 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 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.

Consequences

Topological vector spaces

If is a topological vector space, not necessarily Hausdorff or locally convex, then there exists a non-zero continuous linear form if and only if contains a nonempty, proper, convex, open set.
So if the continuous dual space of, is non-trivial then by considering with the weak topology induced by becomes a locally convex topological vector space with a non-trivial topology that is weaker than original topology on.
If in addition, separates points on then 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

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

Partial differential equations

The Hahn–Banach theorem is often useful when one wishes to apply the method of a priori estimates.
Suppose that we wish to solve the linear differential equation for, with given in some Banach space.
If we have control on the size of in terms of and we can think of as a bounded linear functional on some suitable space of test functions, then we can view as a linear functional by adjunction:.
At first, this functional is only defined on the image of, but using the Hahn–Banach theorem, we can try to extend it to the entire codomain.
We can then reasonably view this functional as a weak solution to the equation.