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 normed vector space to the whole space. In a different guise the 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.See M. Riesz extension theorem. According to , the argument was known to Riesz already in 1918.
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 ) 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 implies the above statement. In order to see it, simply observe that the existence of the continuous linear functional , where is either or , implies the thesis, simply taking as the (finite) norm of , which is a non-negative real number. For the converse, let be the vector subspace of generated by the family . Then, the inequality implies that there is a well-defined linear functional , such that , for all . Moreover, the inequality, as stated, also implies that this linear functional is also continuous and its norm is bounded by as continuous linear operator. Anyway, taking , defined as , we immediately conclude that is a semi-norm and that , for all . Hence, Hahn-Banach Theorem (cf. the "real or complex" version below) implies the existence of extending , having norm, as linear operator, bounded above by and, hence, continuous. As extends g, then satisfying the first claim. In other words, Hahn-Banach implies this theorem as stated. In the case is reflexive, both statements are equivalent.
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 (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 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
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 formulaIf has real part then which proves that Substituting in for and using gives
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 thenLet be any homogeneous scalar-valued map on (such as a linear functional) and let be any map that satisfies for all and unit length scalars (such as a seminorm). If then For the converse, assume and fix Let and pick any such that it remains to show Homogeneity of implies is real so that By assumption, and so that as desired.
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 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 (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.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 (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.