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.

Proof sketch: If $v=0$ the statement is trivial. Otherwise consider the nonnegative quadratic function in $t\in\mathbb{R}$: $0\le \|u-tv\|^2=\langle u-tv,u-tv\rangle=\|u\|^2-2t\langle u,v\rangle+t^2\|v\|^2.$ Its discriminant must be nonpositive, giving $\langle u,v\rangle^2\le \|u\|^2\|v\|^2$.