Linear mapping/Dual mapping/Definition

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

${\displaystyle \varphi \colon V\longrightarrow W}$

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

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

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