In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear map to the base field.
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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 as a bilinear map. Using the bilinear map, 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
with
and
If the vector spaces are finite dimensional this means that the bilinear form is non-degenerate.
We call
When the two spaces are a vector space
We call two elements
We call two sets
Example
A vector space
forms a dual pair.
A locally convex topological vector space
forms a dual pair. (To show this, the Hahn–Banach theorem is needed.)
For each dual pair
A sequence space
form a dual pair.
Comment
Associated with a dual pair
There is an analogous injective map from
In particular, if either of