Vector space over C/Sesquilinear form/Definition

Sesquilinear form

Let ${}V$ be a ${}\mathbb {C}$-vector space. A mapping

$V\times V\longrightarrow \mathbb {C} ,(v,w)\longmapsto \left\langle v,w\right\rangle ,$

is called a sesquilinear form, if, for all ${}v\in V$, the induced mappinga

$V\longrightarrow \mathbb {C} ,w\longmapsto \left\langle v,w\right\rangle ,$

are ${}\mathbb {C}$-antilinear, and, for all ${}w\in V$, the induced mappings

$V\longrightarrow \mathbb {C} ,v\longmapsto \left\langle v,w\right\rangle ,$

are ${}\mathbb {C}$-linear.