Dual pair
From Wikipedia, the free 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 spaces, 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.
[edit] Definition
A dual pair is a 3-tuple 
consisting of two vector spaces X and Y over the same (real or complex) field 
and a bilinear form
with
and
We say 
puts X and Y in duality.
We call two elements 
and 
orthogonal if
We call two sets 
and 
orthogonal if any two elements of X and Y are orthogonal.
[edit] Example
A vector space V together with its algebraic dual V * and the bilinear form defined as
forms a dual pair.
For each dual pair we can define a new dual pair 
with
A sequence space E and its beta dual Eβ with the bilinear form defined as
form a dual pair.
[edit] See also
- dual topology
- polar set
- polar topology
- reductive dual pair
