# Definition
$$| (u|v) | \leq \| \, u \, \|_{2} \times \| \, v \, \|_{2}$$
$$\| \, u + v \, \|_{2} \leq \| \, u \, \|_{2} + \| \, v \, \|_{2}$$
> [!example] Proof of Cauchy-Schwartz
> Insert $a \equiv - \frac{\overline{(u | v)}}{\| \, u \, \|_{2}^2}$ for $u \neq 0$ into
> $$f(a) \equiv |a|^2 \times \| \, u \, \|_{2}^2 + \text{Re}(a \times (u | v)) + \| \, v \, \|_{2}^2 = \| \, au + v \, \|_{2}^2 \geq 0$$
> [[QED]]