Dual system

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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 ${\displaystyle b:X\times Y\to \mathbb {K} }$.

In mathematics, duality is the study of dual systems and is important in functional analysis. Duality plays crucial roles in quantum mechanics because it has extensive applications to the theory of Hilbert spaces.