Cauchy–Schwarz inequality

Cauchy–Schwarz inequality: In an inner product space $(V,\langle\cdot,\cdot\rangle)$, for all $u,v\in V$, $|\langle u,v\rangle|\le \|u\|\,\|v\|, \qquad \text{where } \|u\|=\sqrt{\langle u,u\rangle}.$ Moreover, equality holds if and only if $u$ and $v$ are linearly dependent (i.e., one is a scalar multiple of the other).

Cauchy–Schwarz is a central inequality in analysis and linear algebra. It implies the triangle inequality for the norm induced by an inner product and controls projections and angles.