Vector space/Perfect pairing/Definition

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ denote vector spaces over ${\displaystyle {}K}$, and let

${\displaystyle \Psi \colon V\times W\longrightarrow K}$

denote a bilinear form. We say that ${\displaystyle {}\Psi }$ defines a perfect pairing, if the mapping

${\displaystyle V\longrightarrow {W}^{*},v\longmapsto {\left(w\mapsto \Psi (v,w)\right)},}$

is bijective.