Hahn–Banach theorem


In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. 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 1912 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, find it in terms of those moments. 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, again, find it if so.
Riesz and Helly solved the problem for certain classes of spaces 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:
If happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem:
Riesz went on to define space in 1910 and the spaces in 1913. While investigating 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.
The Hahn–Banach theorem can be deduced from [|the above theorem]. If is reflexive then this theorem solves the vector problem.

Hahn–Banach theorem

A real-valued function defined on a subset of is said to be a function if for every
For this reason, the following version of the Hahn–Banach theorem is called.
The theorem remains true if the requirements on are relaxed to require only that be a convex function:
A function is convex and satisfies if and only if for all vectors and all non-negative real such that Every sublinear function is a convex function.
On the other hand, if is convex with then the function defined by is positively homogeneous
, hence, being convex, it is sublinear. It is also bounded above by and satisfies for every linear functional So the extension of the Hahn–Banach theorem to convex functionals does not have a much larger content than the classical one stated for sublinear functionals.
If is linear then if and only if
which is the conclusion that some authors write instead of
It follows that if is also, meaning that holds for all then if and only
Every norm is a seminorm and both are symmetric balanced sublinear functions. A sublinear function is a seminorm if and only if it is a balanced function. On a real vector space, a sublinear function is a seminorm if and only if it is symmetric. The identity function on is an example of a sublinear function that is not a seminorm.

For complex or real vector spaces

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.
The theorem remains true if the requirements on are relaxed to require only that for all and all scalars and satisfying
This condition holds if and only if is a convex and balanced function satisfying or equivalently, if and only if it is convex, satisfies and for all and all unit length scalars
A complex-valued functional is said to be if for all in the domain of
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 linear functionals on complex vector spaces.
Every linear functional on a complex vector space is completely determined by its real part through the formula
and moreover, if is a norm on then their dual norms are equal:
In particular, a linear functional on extends another one defined on if and only if their real parts are equal on .
The real part of a linear functional on is always a and if is a real-linear functional on a complex vector space then defines the unique linear functional on whose real part is
If is a linear functional on a vector space and if is a seminorm then
Stated in simpler language, a linear functional is dominated by a seminorm if and only if its real part is dominated above by
The proof above shows that when is a seminorm then there is a one-to-one correspondence between dominated linear extensions of and dominated real-linear extensions of the proof even gives a formula for explicitly constructing a linear extension of from any given real-linear extension of its real part.
Continuity
A linear functional on a topological vector space is continuous if and only if this is true of its real part if the domain is a normed space then .
Assume is a topological vector space and is sublinear function.
If is a continuous sublinear function that [|dominates] a linear functional then is necessarily continuous. Moreover, a linear functional is continuous if and only if its absolute value is continuous. In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.

Proof

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 to a larger vector space in which has codimension
This lemma remains true if is merely a convex function instead of a sublinear function.
Assume that is convex, which means that for all and Let and be as in the lemma's statement. Given any and any positive real the positive real numbers and sum to so that the convexity of on guarantees
and hence
thus proving that which after multiplying both sides by becomes
This implies that the values defined by
are real numbers that satisfy As in [|the above proof] of the one–dimensional dominated extension theorem above, for any real define by
It can be verified that if then where follows from when .
The lemma above is the key step in deducing the dominated extension theorem from Zorn's lemma.
When has countable codimension, then using induction and the lemma completes the proof of the Hahn–Banach theorem. The standard proof of the general case uses Zorn's lemma although the strictly weaker ultrafilter lemma may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact Hausdorff spaces
The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.

Continuous extension theorem

The Hahn–Banach theorem can be used to guarantee the existence of continuous linear extensions of continuous linear functionals.
In category-theoretic terms, the underlying field of the vector space is an injective object in the category of locally convex vector spaces.
On a normed space, a linear extension of a bounded linear functional is said to be if it has the same dual norm as the original functional:
Because of this terminology, the second part of the above theorem is sometimes referred to as the "norm-preserving" version of the Hahn–Banach theorem. Explicitly:

Proof of the [|continuous extension theorem]

The following observations allow the continuous extension theorem to be deduced from the Hahn–Banach theorem.
The absolute value of a linear functional is always a seminorm. A linear functional on a topological vector space is continuous if and only if its absolute value is continuous, which happens if and only if there exists a continuous seminorm on such that on the domain of
If is a locally convex space then this statement remains true when the linear functional is defined on a vector subspace of
Proof for normed spaces
A linear functional on a normed space is continuous if and only if it is bounded, which means that its dual norm
is finite, in which case holds for every point in its domain.
Moreover, if is such that for all in the functional's domain, then necessarily
If is a linear extension of a linear functional then their dual norms always satisfy
so that equality is equivalent to which holds if and only if for every point in the extension's domain.
This can be restated in terms of the function defined by which is always a seminorm:
Applying the Hahn–Banach theorem to with this seminorm thus produces a dominated linear extension whose norm is equal to that of which proves the theorem:

Non-locally convex spaces

The continuous extension theorem might fail if the topological vector space is not locally convex. For example, for the Lebesgue space is a complete metrizable TVS that is locally convex and the only continuous linear functional on is the constant function. Since is Hausdorff, every finite-dimensional vector subspace is linearly homeomorphic to Euclidean space or and so every non-zero linear functional on is continuous but none has a continuous linear extension to all of
However, it is possible for a TVS to not be locally convex but nevertheless have enough continuous linear functionals that its continuous dual space separates points; for such a TVS, a continuous linear functional defined on a vector subspace have a continuous linear extension to the whole space.
If the TVS is not locally convex then there might not exist any continuous seminorm that dominates in which case the Hahn–Banach theorem can not be applied as it was in the above proof of the continuous extension theorem.
However, the proof's argument can be generalized to give a characterization of when a continuous linear functional has a continuous linear extension: If is any TVS, then a continuous linear functional defined on a vector subspace has a continuous linear extension to all of if and only if there exists some continuous seminorm on that dominates Specifically, if given a continuous linear extension then is a continuous seminorm on that dominates and conversely, if given a continuous seminorm on that dominates then any dominated linear extension of to will be a continuous linear extension.

Geometric Hahn–Banach (the Hahn–Banach separation theorems)

The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: and This sort of argument appears widely in convex geometry, optimization theory, and economics. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.
They are generalizations of the hyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space can be separated by some, which is a fiber of the form where is a non-zero linear functional and is a scalar.
When the convex sets have additional properties, such as being open or compact for example, then the conclusion can be substantially strengthened:
Then following important corollary is known as the Geometric Hahn–Banach theorem or Mazur's theorem. It follows from the first bullet above and the convexity of
Mazur's theorem clarifies that vector subspaces can be characterized by linear functionals.

Supporting hyperplanes

Since points are trivially convex, geometric Hahn–Banach implies that functionals can detect the boundary of a set. In particular, let be a real topological vector space and be convex with If then there is a functional that is vanishing at but supported on the interior of
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.

Balanced or disked neighborhoods

Let be a convex balanced neighborhood of the origin in a locally convex topological vector space and suppose is not an element of Then there exists a continuous linear functional on such that

Applications

The Hahn–Banach theorem is the first sign of an important philosophy in functional analysis: to understand a space, one should understand its continuous functionals.
For example, linear subspaces are characterized by functionals: if is a normed vector space with linear subspace and if is an element of not in the closure of, then there exists a continuous linear map with for all and Moreover, if is an element of, then there exists a continuous linear map such that and This implies that the natural injection from a normed space into its double dual is isometric.
That last result also suggests that the Hahn–Banach theorem can often be used to locate a "nicer" topology in which to work. For example, many results in functional analysis assume that a space is Hausdorff or locally convex. However, suppose is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set. Then geometric Hahn–Banach implies that there is a hyperplane separating from any other point. In particular, there must exist a nonzero functional on — that is, the continuous dual space is non-trivial. Considering with the weak topology induced by then becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space separates points.
Thus 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.

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. The resulting functional is often defined to be a weak solution to the equation.

Example from Fredholm theory

To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem.
The above result may be used to show that every closed vector subspace of is complemented because any such space is either finite dimensional or else TVS–isomorphic to

Generalizations

General template
There are now many other versions of the Hahn–Banach theorem. The general template for the various versions of the Hahn–Banach theorem presented in this article is as follows:

For seminorms

So for example, suppose that is a bounded linear functional defined on a vector subspace of a normed space so its the operator norm is a non-negative real number.
Then the linear functional's absolute value is a seminorm on and the map defined by is a seminorm on that satisfies on
The Hahn–Banach theorem for seminorms guarantees the existence of a seminorm that is equal to on and is bounded above by everywhere on .

Maximal dominated linear extension

If is a singleton set and if is such a maximal dominated linear extension of then

Invariant Hahn–Banach

A set of maps is if for all
Say that a function defined on a subset of is if and on for every
This theorem may be summarized:

For nonlinear functions

The following theorem of Mazur–Orlicz is equivalent to the Hahn–Banach theorem.
The following theorem characterizes when scalar function on has a continuous linear extension to all of

Converse

Let be a topological vector space. A vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on, and we 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 topological vector spaces there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex. On the other hand, a vector space of uncountable dimension, endowed with the finest vector topology, then this is a topological vector spaces with the Hahn–Banach extension property that is neither locally convex nor 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 vector subspace of has the extension property if and only if vector subspace of has the separation property.

Relation to axiom of choice and other theorems

The proof of the Hahn–Banach theorem for real vector spaces commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory is equivalent to the axiom of choice. It was discovered by Łoś and Ryll-Nardzewski and independently by Luxemburg that HB can be proved using the ultrafilter lemma, which is equivalent to the Boolean prime ideal theorem. BPI is strictly weaker than the axiom of choice and it was later shown that HB is strictly weaker than BPI.
The ultrafilter lemma is equivalent to the Banach–Alaoglu theorem, which is another foundational theorem in functional analysis. Although the Banach–Alaoglu theorem implies HB, it is not equivalent to it.
However, HB is equivalent to a certain weakened version of the Banach–Alaoglu theorem for normed spaces.
The Hahn–Banach theorem is also equivalent to the following statement:
In ZF, the Hahn–Banach theorem suffices 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.