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

Mathematical duality is the study of dual systems and is important in functional analysis. It has extensive applications to quantum mechanics arising in the theory of Hilbert spaces. In addition, it can be very helpful when working with quantum mechanics and quantum physics.