Hahn–Banach theorem

# Hahn–Banach theorem

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hahn–Banach theorem is a central tool in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. It allows the extension of bounded linear functionals
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

defined on a subspace of some vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

to make the study of the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

"interesting." Another version of Hahn–Banach theorem is known as Hahn-Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry
Convex geometry
Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...

. It is named for Hans Hahn
Hans Hahn
Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...

and Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....

who proved this theorem independently in the late 1920s, although a special case was proved earlier (in 1912) by Eduard Helly
Eduard Helly
Eduard Helly was a mathematician and the eponym of Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem. In 1912, Helly published a proof of Hahn–Banach theorem, 15 years before Hahn and Banach discovered it independently...

, and a general extension theorem from which the Hahn–Banach theorem can be derived was proved in 1923 by Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...

.

## Formulation

The most general formulation of the theorem needs some preparation. Given a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

V over the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

R of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

is called sublinear if
for any and any x ∈ V (positive homogeneity),
for any xy ∈ V (subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive...

).

Every seminorm on V (in particular, every norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

on V) is sublinear. Other sublinear functions can be useful as well, especially Minkowski functional
Minkowski functional
In functional analysis, given a linear space X, a Minkowski functional is a device that uses the linear structure to introduce a topology on X.-Example 1:...

s of convex sets.

The Hahn–Banach theorem states that if is a sublinear function, and is a linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

on a linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

UV which is dominated by on U,

then there exists a linear extension of to the whole space V, i.e., there exists a linear functional ψ such that

and

Another version of Hahn–Banach theorem states that if V is a vector space over the scalar field K (either the real numbers R or the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s C), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

then there exists a linear extension of φ to the whole space V, i.e., there exists a K-linear functional ψ such that

and

In the complex case of this theorem, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, the vector be also in U and .

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...

, one formulation of the axiom of choice.

It is possible to relax slightly the sublinearity condition on , requiring only that

according to (Reed and Simon, 1980). This reveals the intimate connection between the Hahn–Banach theorem and convexity
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

.

The Mizar project
Mizar system
The Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs, a computer program which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referenced and used in new articles. Mizar has goals...

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":
• If V is a normed vector space with linear subspace U (not necessarily closed) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ (see Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

for a discussion of the norm of a linear map). In other words, in the category of normed vector spaces, the space K is an injective object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

.
• If V is a normed vector space with linear subspace U (not necessarily closed) and if z is an element of V not in the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

of U, then there exists a continuous linear map with ψ(x) = 0 for all x in U, ψ(z) = 1, and ||ψ|| = 1 / dist(z, U).
• In particular, if V is a normed vector space and if z is any element of V, then there exists a continuous linear map with ψ(z) = ||z|| and ||ψ|| ≤ 1. This implies that the natural injection J from a normed space V into its double dual  is isometric.

## Hahn-Banach separation theorem

Another version of Hahn–Banach theorem is known as the Hahn-Banach separation theorem. It has numerous uses in convex geometry
Convex geometry
Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...

, optimization theory
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

, and economics. The separation theorem is derived from the original form of the theorem.

Theorem: Let V be a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

over  = ℝ or ℂ,
and A, B convex, non-empty subsets of V.
Assume that A ∩ B = ∅.
Then

(i) If A is open,
then there exists a continuous linear map
and
such that
for all ,

(ii) If V is locally convex,
A is compact,
and B closed,
then there exists a continuous linear map
and
such that
for all , .

## Relation to the 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 can in fact be proved using even weaker hypotheses than the ultrafilter lemma. For separable Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s, Brown and Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic
Second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their...

that takes Konig's Lemma
König's lemma
König's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig . It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic,...

as an axiom.