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.

A dual pair is a 3-tuple ${\displaystyle (X,Y,\langle ,\rangle )}$ consisting of two vector spaces ${\displaystyle X}$ and ${\displaystyle Y}$ over the same field ${\displaystyle F}$ and a bilinear map

...
Wikipedia