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 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 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

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

We say $\langle ,\rangle$ puts $X$ and $Y$ in duality.

We call two elements $x\in X$ and $y\in Y$ orthogonal if

$\langle x,y\rangle =0.$

We call two sets $M\subseteq X$ and $N\subseteq Y$ orthogonal if any two elements of $M$ and $N$ are orthogonal.

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.

For each dual pair $(X,Y,\langle ,\rangle )$ we can define a new dual pair $(Y,X,\langle ,\rangle ')$ with

$\langle ,\rangle ':(y,x)\to \langle x,y\rangle$

A sequence space $E$ and its beta dual $E^{\beta }$ with the bilinear form defined as

$\langle x,y\rangle :=\sum _{i=1}^{\infty }x_{i}y_{i}\quad x\in E,y\in E^{\beta }$

forms a dual pair.

See also

• dual topology
• polar set
• polar topology
• reductive dual pair