Linear mapping/Dual mapping/Definition

Dual mapping

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

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping. Then the mapping

$\varphi ^{*}\colon \operatorname {Hom} _{K}{\left(W,K\right)}={W}^{*}\longrightarrow \operatorname {Hom} _{K}{\left(V,K\right)}={V}^{*},f\longmapsto f\circ \varphi ,$

is called the dual mapping of ${}\varphi$.