Vector space/Perfect pairing/Definition

Perfect pairing

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

$\Psi \colon V\times W\longrightarrow K$

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

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

is bijective.