Vector spaces over C/Mapping/Antilinear/Definition

Antilinear mapping

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

$\varphi \colon V\longrightarrow W$

is called antilinear, if

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

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

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

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