ACIT4330-Page/content/Definitions/Cauchy-Schwarz Inequality.md

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