As part of Apostol's proof of the fact that in a Euclidean Space $V$, every inner product satisfies the Cauchy-Schwarts inequality, he states that $(x,y)(y,x)=|(x,y)|^2$. Does this not imply that (x,y) is positive, which doesn't seem like a justified assumption to me.
If necessary, I can post the rest of the proof, it is Theorem 1.8 in Apostol's Calculus Volume 2.
Because
$$(y,x)=\overline{(x,y)}\implies (x,y)(y,x)=(x,y)\overline{(x,y)}=|(x,y)|^2$$
since
$$z\in\Bbb C\implies z\overline z=|z|^2$$