Let \(F\) be a field and \(V\) be a vector space over \(F\). A function. \[B:\cases{V\times V\longrightarrow F,\cr(v,w)\longmapsto \left\langle v,w\right\rangle,}\]
(i.e. a function mapping all pairs of vectors \((v,w)\) to their respective dot products) \(\langle v,w\rangle\)) is called a bilinear form, if it is linear over \(F\) in each argument separately, i.e.:
\(\langle u + v, w\rangle = \langle u, w) + \langle v, w\rangle\) \(\langle su, v\rangle = s\langle u, v\rangle\) \(\langle u, v + w\rangle = \langle u, v\rangle + \langle u, w\rangle\) \(\langle u, sv\rangle = s\langle u, v\rangle\)
for all \(u,v,w\in V\) and \(s\in F\).
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