Pairing

A mapping defined on the Cartesian product of two sets. The requirements of bilinearity, continuity and others may be imposed upon this mapping, according to the context. A pairing $X\times Y\to Z$ defines a mapping from $X$ into the set of functions acting from $Y$ into $Z$ (or into some subset of the latter, for example consisting of homomorphisms, continuous mappings, etc.), cf. Exponential law for sets. Statements about the properties of the mapping thus obtained form the essence of various duality theorems in algebra, topology and functional analysis.