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Dual pair

In functional analysis and related areas of mathematics a dual pair is a 3-tuple (X,Y,\langle , \rangle) consisting of two vector space X and Y over some 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.

See also

Last updated: 05-26-2005 20:57:29
Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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