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 $C[a,b]$ 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.

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 ($\mathbb {C} ^{\mathbb {N} }$) 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 $L^{p}([0,1])$ 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:

(The vector problem) Given a collection $\left(f_{i}\right)_{i\in I}$ of bounded linear functionals on a normed space $X$ and a collection of scalars $\left(c_{i}\right)_{i\in I},$ determine if there is an $x\in X$ such that $f_{i}(x)=c_{i}$ for all $i\in I.$

If $X$ 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 $\left(x_{i}\right)_{i\in I}$ of vectors in a normed space $X$ and a collection of scalars $\left(c_{i}\right)_{i\in I},$ determine if there is a bounded linear functional $f$ on $X$ such that $f\left(x_{i}\right)=c_{i}$ for all $i\in I.$

Riesz went on to define $L^{p}([0,1])$ space ($1<p<\infty$) in 1910 and the $\ell ^{p}$ 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.

Theorem (The functional problem) — Let $\left(x_{i}\right)_{i\in I}$ be vectors in a real or complex normed space $X$ and let $\left(c_{i}\right)_{i\in I}$ be scalars also indexed by $I\neq \varnothing .$

There exists a continuous linear functional $f$ on $X$ such that $f\left(x_{i}\right)=c_{i}$ for all $i\in I$ if and only if there exists a $K>0$ such that for any choice of scalars $\left(s_{i}\right)_{i\in I}$ where all but finitely many $s_{i}$ are $0,$ the following holds:

$$\left|\sum _{i\in I}s_{i}c_{i}\right|\leq K\left\|\sum _{i\in I}s_{i}x_{i}\right\|.$$

The Hahn–Banach theorem can be deduced from the above theorem. If $X$ is reflexive then this theorem solves the vector problem.

Hahn–Banach theorem

A real-valued function $f:M\to \mathbb {R}$ defined on a subset $M$ of $X$ is said to be dominated (above) by a function $p:X\to \mathbb {R}$ if $f(m)\leq p(m)$ for every