# Dual pair

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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.

## Definition

A **dual pair**^{[1]} is a 3-tuple consisting of two vector spaces and over the same (real or complex) field and a bilinear form

with

and

We call two elements and **orthogonal** if

We 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 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.

## 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

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