Dual pair
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.
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.
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Definition
A dual pair^{[1]} is a 3-tuple consisting of two vector spaces and over the same field and a bilinear map
with
and
If the vector spaces are finite dimensional this means that the bilinear form is non-degenerate.
We call the duality pairing, and say that it puts and in duality.
When the two spaces are a vector space (or a module over a ring in general) and its dual , we call the canonical duality pairing the natural pairing.
We call two elements and orthogonal if
We call two sets and orthogonal if each pair of elements from and are orthogonal.
Example
A vector space together with its algebraic dual and the bilinear map defined as
forms a dual pair.
A locally convex topological vector space together with its topological dual and the bilinear map 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 map defined as
form a dual pair.
Comment
Associated with 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.
See also
References
- ^ Jarchow, Hans (1981). Locally convex spaces. Stuttgart. pp. 145–146. ISBN 9783519022244.