A.3 Inequalities

Theorem A..6 (Cauchy-Schwarz-Buniakovsky Inequality)   For all vectors $a,b\in\mathbb{R}^n$ the inequality

$$\vert\langle a,b \rangle\vert \le \Vert a\Vert _2 \Vert b\Vert _2... ...\sum a_k b_k \right)^2 \le \left( \sum a_k^2 \right) \left( \sum b_k^2 \right)$$

holds.

Corollary A..7   For all $a,b\in\mathbb{R}^n$ and all $c\in\mathbb{R}_+^n$ the inequality

$$\left( \sum a_k b_k c_k \right)^2 \le \left( \sum a_k^2 c_k \right) \left( \sum b_k^2 c_k \right)$$

holds.