# 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.

## 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.