Vector space/Bilinear form/Definition

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

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

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

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

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

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

are ${\displaystyle {}K}$-linear.