Cauchy–Schwarz inequality

Cauchy–Schwarz inequality: In an inner product space $(V,\langle\cdot,\cdot\rangle)$, for all $x,y\in V$,

where $\|x\|=\sqrt{\langle x,x\rangle}$ is the induced norm . Equality holds if and only if $x$ and $y$ are linearly dependent.

In a Euclidean space , this bounds the dot product by the product of Euclidean lengths. It is central for understanding orthogonality and for many norm inequalities in inner product geometry.