Prob Expert: Cauchy–Schwarz inequality

The statement of the Cauchy–Schwarz inequality is:

For all vectors $x$ and $y$ of a real or complex inner product space $V$ the following inequality holds:

$|\langle x,y\rangle|^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle.$

or, equivalently, by taking the square root of both sides, and referring to the norms of the vectors:

$|\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\,$

Prob Expert