Vector space/Bilinear form/Definition

Bilinear form

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. A mapping

$V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,$

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

$V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,$

and for all ${}w\in V$, the induced mappings

$V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,$

are ${}K$-linear.