Vector spaces over C/Mapping/Antilinear/Definition

Let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be vector spaces over the complex numbers ${\displaystyle {}\mathbb {C} }$. A mapping

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

is called antilinear, if

${\displaystyle {}\varphi (u+v)=\varphi (u)+\varphi (v)\,}$

holds for all ${\displaystyle {}u,v\in V}$, and if

${\displaystyle {}\varphi (\lambda v)={\overline {\lambda }}\varphi (v)\,}$

holds for all ${\displaystyle {}v\in V}$ and ${\displaystyle {}\lambda \in \mathbb {C} }$.