Vector space/Bilinear form/Definition

Bilinear form

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

$V\times V⟶K,(v,w)⟼⟨v,w⟩,$

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

$V⟶K,w⟼⟨v,w⟩,$

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

$V⟶K,v⟼⟨v,w⟩,$

are $K$-linear.