Cauchy-Schwarz Inequality in a Complex Inner Product Space

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $u,v\in V$. Then, with the modulus of a complex number,

$$\bigl|\langle u,v\rangle\bigr|^{2}\le\langle u,u\rangle\,\langle v,v\rangle ,$$

an inequality between real numbers, the two factors on the right being nonnegative real numbers by condition 4 of Complex Inner Product Space.