Hahn–Banach theorem explained

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a vector 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 (in 1912) 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.^[1]

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 (where the linear functional has its domain extended by one dimension) 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 (where a bounded linear functional on a subspace has a bounded linear extension of the same norm to the whole space). 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 (such as

and

C([a,b])

) 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:

() Given a collection

of bounded linear functionals on a normed space and a collection of scalars determine if there is an such that for all

If

happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem:

(The functional problem) Given a collection

of vectors in a normed space and a collection of scalars determine if there is a bounded linear functional on such that for all

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 Hence the reason why 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 }\; \inf_ \frac is positively homogeneous (because for all and one has ), 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 $$-p(-x)\; \backslash leq\; F(x)\; \backslash leq\; p(x)\; \backslash quad\; \backslash text\; x\; \backslash in\; X,$$ which is the (equivalent) 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 (although not on a complex 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 $$p(a\; x\; +\; b\; y)\; \backslash leq\; |a|\; p(x)\; +\; |b|\; p(y).$$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:

Hahn–Banach dominated extension theorem: If

is a seminorm defined on a real or complex vector space then every dominated linear functional defined on a vector subspace of has a dominated linear extension to all of In the case where is a real vector space and is merely a convex or sublinear function, this conclusion will remain true if both instances of "dominated" (meaning ) are weakened to instead mean "dominated " (meaning ).

Proof

The following observations allow the Hahn–Banach theorem for real vector spaces to be applied to (complex-valued) linear functionals on complex vector spaces.

Every linear functional

on a complex vector space is completely determined by its real part through the formula^[2] $$F(x)\; \backslash ;=\backslash ;\; \backslash operatorname\; F(x)\; -\; i\; \backslash operatorname\; F(i\; x)\; \backslash qquad\; \backslash text\; x\; \backslash in\; X$$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 (in other words, a linear functional extends if and only if extends ). The real part of a linear functional on is always a (meaning that it is linear when is considered as a real vector space) 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 (complex or real) vector space and if is a seminorm then^[3] $$|F|\; \backslash ,\backslash leq\backslash ,\; p\; \backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname\; F\; \backslash ,\backslash leq\backslash ,\; p.$$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 (where one side is infinite if and only if the other side is infinite). 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 (which is a seminorm that dominates ) 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 guaranteesand hencethus proving that which after multiplying both sides by becomes $$\backslash tfrac\; [-\; p(m\; -\; r\; x)\; +\; f(m)]\; ~\backslash leq~\; \backslash tfrac\; [p(n\; +\; s\; x)\; -\; f(n)].$$ This implies that the values defined by $$a\; =\; \backslash sup\_\; \backslash tfrac\; [-\; p(m\; -\; r\; x)\; +\; f(m)]\; \backslash qquad\; \backslash text\; \backslash qquad\; c\; =\; \backslash inf\_\; \backslash tfrac\; [p(n\; +\; s\; x)\; -\; f(n)]$$ 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 (respectively, 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 (which is equivalent to the compactness theorem and to the Boolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact Hausdorff spaces (which is also equivalent to the ultrafilter lemma)

The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.^[4]

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 (or seminormed) 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 $$\backslash |f\backslash |\; =\; \backslash sup\; \backslash$$

\

m\

\leq 1, m \in \operatorname f\

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:^[5] is norm-preserving if and only if the extension is dominated by the seminorm

Applying the Hahn–Banach theorem to

with this seminorm thus produces a dominated linear extension whose norm is (necessarily) equal to that of which proves the theorem:

Non-locally convex spaces

The continuous extension theorem might fail if the topological vector space (TVS)

is not locally convex. For example, for the Lebesgue space is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself and the empty set) 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 (by F. Riesz's theorem) 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. is not locally convex then there might not exist any continuous seminorm (not just on ) 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 (not necessarily locally convex), 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 (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension.

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

See also: Hyperplane separation theorem.

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,^[6] optimization theory, and economics. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.^{[7] [8]} 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 (level set) 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 (also known as Ascoli–Mazur theorem^[9]). It follows from the first bullet above and the convexity of

Mazur's theorem clarifies that vector subspaces (even those that are not closed) 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 (not necessarily closed) and if

is an element of not in the closure of, then there exists a continuous linear map with for all and (To see this, note that is a sublinear function.) 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. 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:

is a sublinear function (possibly a seminorm) on a vector space is a vector subspace of (possibly closed), and is a linear functional on satisfying on (and possibly some other conditions). One then concludes that there exists a linear extension of to such that on (possibly with additional properties).

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 (since ) and is bounded above by everywhere on (since ).

Maximal dominated linear extension

If

is a singleton set (where is some vector) and if is such a maximal dominated linear extension of then

Vector valued Hahn–Banach

See also: Vector-valued Hahn–Banach theorems.

Invariant Hahn–Banach

See also: Vector-valued Hahn–Banach theorems.

A set

of maps is (with respect to function composition ) if for all Say that a function defined on a subset of is if and on for every

This theorem may be summarized:

Every

-invariant continuous linear functional defined on a vector subspace of a normed space

has a -invariant Hahn–Banach extension to all of

For nonlinear functions

The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.

The following theorem characterizes when scalar function on

(not necessarily linear) 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 (HBEP) 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 (HB) commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory (ZF) is equivalent to the axiom of choice (AC). It was discovered by Łoś and Ryll-Nardzewski and independently by Luxemburg that HB can be proved using the ultrafilter lemma (UL), which is equivalent (under ZF) to the Boolean prime ideal theorem (BPI). 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 (under ZF) to the Banach–Alaoglu theorem, which is another foundational theorem in functional analysis. Although the Banach–Alaoglu theorem implies HB,^[10] it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than HB). However, HB is equivalent to a certain weakened version of the Banach–Alaoglu theorem for normed spaces.^[11] The Hahn–Banach theorem is also equivalent to the following statement:^[12]

(∗): On every Boolean algebra there exists a "probability charge", that is: a non-constant finitely additive map from

into

(BPI is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.)

In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.^[13] Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.^[14]

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.^{[15] [16]}

Notes

Proofs

Bibliography

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Notes and References

1. See M. Riesz extension theorem. According to 0256837. Gårding. L.. Lars Gårding. Marcel Riesz in memoriam. Acta Math.. 124. 1970. 1. I–XI . 10.1007/bf02394565. free., the argument was known to Riesz already in 1918.
2. If

   z=a+ib\in\Complex

   has real part

   \operatorname{Re}z=a

   then

   -\operatorname{Re}(iz)=b,

   which proves that

   z=\operatorname{Re}z-i\operatorname{Re}(iz).

   Substituting

   F(x)

   in for

   z

   and using

   iF(x)=F(ix)

   gives

   F(x)=\operatorname{Re}F(x)-i\operatorname{Re}F(ix).

   \blacksquare

3. Let

   F

   be any homogeneous scalar-valued map on

   X

   (such as a linear functional) and let

   p:X\to\R

   be any map that satisfies

   p(ux)=p(x)

   for all

   x

   and unit length scalars

   u

   (such as a seminorm). If

   |F|\leqp

   then

   \operatorname{Re}F\leq|\operatorname{Re}F|\leq|F|\leqp.

   For the converse, assume

   \operatorname{Re}F\leqp

   and fix

   x\inX.

   Let

   r=|F(x)|

   and pick any

   \theta\in\R

   such that

   F(x)=reⁱ;

   it remains to show

   r\leqp(x).

   Homogeneity of

   F

   implies

   F\left(e^-ix\right)=r

   is real so that

   \operatorname{Re}F\left(e^-ix\right)=F\left(e^-ix\right).

   By assumption,

   \operatorname{Re}F\leqp

   and

   p\left(e^-ix\right)=p(x),

   so that

   r=\operatorname{Re}F\left(e^-ix\right)\leqp\left(e^-ix\right)=p(x),

   as desired.

   \blacksquare

4. http://mizar.uwb.edu.pl/JFM/Vol5/hahnban.html HAHNBAN file
5. Like every non-negative scalar multiple of a norm, this seminorm

   \|f\|\| ⋅ \|

   (the product of the non-negative real number

   \|f\|

   with the norm

   \| ⋅ \|

   ) is a norm when

   \|f\|

   is positive, although this fact is not needed for the proof.
6. Harvey. R.. Lawson. H. B.. 1983. An intrinsic characterisation of Kähler manifolds. Invent. Math.. 74. 2. 169–198. 10.1007/BF01394312. 1983InMat..74..169H. 124399104.
7. Book: Zălinescu, C.. Convex analysis in general vector spaces. World Scientific Publishing Co., Inc. River Edge, NJ . 2002. 5–7. 981-238-067-1. 1921556.
8. Gabriel Nagy, Real Analysis lecture notes
9. Book: Semen. Kutateladze. 1996. 40. Fundamentals of Functional Analysis. Kluwer Texts in the Mathematical Sciences . 12. 978-90-481-4661-1. 10.1007/978-94-015-8755-6.
10. Book: Muger, Michael. Topology for the Working Mathematician. 2020.
11. Bell. J.. Fremlin. David. A Geometric Form of the Axiom of Choice. Fundamenta Mathematicae. 1972. 77. 2. 167–170. 10.4064/fm-77-2-167-170. 26 Dec 2021.
12. Book: Schechter, Eric. Handbook of Analysis and its Foundations. 620. Eric Schechter.
13. Foreman. M.. Wehrung. F.. 1991. The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set. Fundamenta Mathematicae. 138. 13–19. 10.4064/fm-138-1-13-19. free.
14. Pawlikowski. Janusz. 1991. The Hahn–Banach theorem implies the Banach–Tarski paradox. Fundamenta Mathematicae. 138. 21–22. 10.4064/fm-138-1-21-22. free.
15. Brown. D. K.. Simpson. S. G.. 1986. Which set existence axioms are needed to prove the separable Hahn–Banach theorem?. Annals of Pure and Applied Logic. 31. 123–144. 10.1016/0168-0072(86)90066-7 . Source of citation.
16. Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press,,