[Math] Proof of cauchy schwarz inequality in inner product space

Question

$0 \le \lVert x-cy \rVert ^2= \langle x-cy,s-cy \rangle = \langle x,x\rangle -\bar{c}\langle x,y\rangle-c\langle y,x\rangle +c\bar{c}\langle y,y\rangle$.
If we set $c=\frac{\langle x,y\rangle}{\langle y,y\rangle}$ then the inequality becomes
$0 \le \langle x,x\rangle – \frac {\lvert \langle x,y\rangle \rvert^2}{\langle y,y\rangle}$.
I calculate the above equation ans wonder why $\langle x,y\rangle \langle y,x\rangle$ becomes $\lvert\langle x,y\rangle\rvert^2$? In complex field that's not true, isn't it?

Answer 1

Note that for any complex number $\alpha$, we have: $$\alpha \overline{\alpha}=|\alpha|^2$$where $|\alpha|$ is the modulus (or length) of the complex number $\alpha$.