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 form.

A common method in functional analysis, when studying a vector space, is to analyze the relationship of the space to its dual space. The dual of a vector space is the set of all possible linear functions on the original space, endowed with a vector space structure. A dual pair generalizes this concept by considering arbitrary vector spaces, with the duality being expressed by a bilinear form.

Definition

A dual pair is a 3-tuple $(X,Y,\langle ,\rangle )$ consisting of two vector space $X$ and $Y$ over the same (real or complex) field $\mathbb {F}$ and a bilinear form

$\langle ,\rangle :X\times Y\mapsto \mathbb {F}$

with

$\forall x\in X\setminus \{0\}\quad \exists y\in Y:\langle x,y\rangle \neq 0$

and

$\forall y\in Y\setminus \{0\}\quad \exists x\in X:\langle x,y\rangle \neq 0$

Example

A vector space $V$ together with its algebraic dual $V^{*}$ and the bilinear form defined as

$\langle x,f\rangle :=f(x)\qquad x\in V{\mbox{ , }}f\in V^{*}$

forms a dual pair.

Weak topology

Given a dual pair $(X,Y,\langle ,\rangle )$ for every $y$ in $Y$

$p_{y}:X\to \mathbb {R}$

with

$p_{y}(x):=\vert \langle x,y\rangle \vert \qquad x\in X$

defines a semi norm on $X$. $X$ together with this family of semi norms $p_{y}$ is a locally convex space. The locally convex topology is called weak topology and denoted $\sigma (X,Y)$.

See also

• polar set
• polar topology
• reductive dual pair.