# Definition
$$(\cdot | \cdot) : V \times V \to \mathbb{C}$$
> [!info]
> $$V \times V \ni (u, v) \mapsto (u | v) \in \mathbb{C}$$
> Such that $(au + bv | w) = a(u | w) + b(v|w)$
> and $\overline{(u | v)} = (v | u)$
> >[!example]-
> > $$(w | au + bv) = \overline{(au + bv | w)} = \overline{a(u | w) + b (v | w)} = \bar{a} \overline{(u | w)} + \bar{b} \overline{( v | w)} = \dots$$
>
> and $(v | v) \geq 0$ and $(u | u) = 0 \implies u = 0$