MathItems

For all real numbers $a_i$, $b_i$, $i = 1, 2, \ldots, n$,

$$(a_1^2 + a_2^2 + \ldots + a_n^2) (b_1^2 + b_2^2 + \ldots + b_n^2) \geq (a_1 b_1 + a_2 b_2 + \ldots + a_n b_n)^2 \; ,$$

with equality if and only if $a_i$ and $b_i$ are proportional for all $i$.