Dual system

In mathematics, a dual system, dual pair, or a duality over a field ${\displaystyle \mathbb {K} }$ is a triple ${\displaystyle (X,Y,b)}$consisting of two vector spaces ${\displaystyle X}$ and ${\displaystyle Y}$ over ${\displaystyle \mathbb {K} }$ and a non-degenerate bilinear map.${\displaystyle b:X\times Y\to \mathbb {K} }$.

Mathematical duality theory, the study of dual systems, has an important place in functional analysis and has extensive applications to quantum mechanics via the theory of Hilbert spaces.