Inner Product

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 $\overline{(u|v)}=(v|u)$

$(w|au+bv)=\overline{(au+bv|w)}=\overline{a(u|w)+b(v|w)}=\bar{a}\overline{(u|w)}+\bar{b}\overline{(v|w)}=\dots$

and $(v|v)\ge 0$ and $(u|u)=0⟹u=0$