A.3 Inequalities

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

$\displaystyle \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

$\displaystyle \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.

Clemens Heitzinger 2003-05-08