Cauchy-Schwarz Inequality

Definition

$|(u|v)|\le \Vert u{\Vert }_2\times \Vert v{\Vert }_2$
$\Vert u+v{\Vert }_2\le \Vert u{\Vert }_2+\Vert v{\Vert }_2$

Proof of Cauchy-Schwartz

Insert $a\equiv -\frac{\overline{(u|v)}}{\Vert u{\Vert }_2^2}$ for $u\ne 0$ into
$f(a)\equiv |a{|}^2\times \Vert u{\Vert }_2^2+\text{Re}(a\times (u|v))+\Vert v{\Vert }_2^2=\Vert au+v{\Vert }_2^2\ge 0$
QED