Dual pair
Encyclopedia
In functional analysis
and related areas of mathematics
a dual pair or dual system is a pair of vector spaces with an associated bilinear form.
A common method in functional analysis, when studying normed vector space
s, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed by a bilinear form. Using the bilinear form, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.
consisting of two vector space
s
and 
over the same (real
or complex) field
and a bilinear form
with
and
We say
puts 
and 
in duality.
We call two elements
and 
orthogonal if
We call two sets
and 
orthogonal if any two elements of 
and 
are orthogonal.
together with its algebraic dual 
and the bilinear form defined as
forms a dual pair.
A locally convex topological vector space
space
together with its topological dual 
and the bilinear form defined as
forms a dual pair. (to show this, the Hahn–Banach theorem
is needed)
For each dual pair
we can define a new dual pair 
with
A sequence space
and its beta dual 
with the bilinear form defined as
form a dual pair.
is an injective
linear map from
to 
given by
There is an analogous injective map from
to 
.
In particular, if either of
or 
is finite dimensional, these maps are isomorphisms.
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...
and related areas of 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...
a dual pair or dual system is a pair of vector spaces with an associated bilinear form.
A common method in functional analysis, when studying 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....
s, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed by a bilinear form. Using the bilinear form, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.
Definition
A dual pair is a 3-tuple consisting of two vector spaceVector 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...
s
and over the same (realReal 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...
or complex) 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...
and a bilinear formwith
and
We say
puts and in duality.We call two elements
and orthogonal ifWe call two sets
and orthogonal if any two elements of and are orthogonal.Example
A vector space together with its algebraic dual and the bilinear form defined asforms a dual pair.
A locally convex topological vector space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
space
together with its topological dual and the bilinear form defined asforms a dual pair. (to show this, the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, 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...
is needed)
For each dual pair
we can define a new dual pair withA sequence space
Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers...
and its beta dual with the bilinear form defined asform a dual pair.
Comment
Associated with a dual pair is an injectiveInjective function
In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
linear map from
to given byThere is an analogous injective map from
to .In particular, if either of
or is finite dimensional, these maps are isomorphisms.See also
- dual topologyDual topologyIn functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space....
- polar set
- polar topology
- reductive dual pairReductive dual pairIn the mathematical field of representation theory, a reductive dual pair is a pair of subgroups of the isometry group Sp of a symplectic vector space W, such that G is the centralizer of G ′ in Sp and vice versa, and these groups act reductively on W...