I have the Cauchy Schwarz inequality for inner product is $$|<x,y>| \leq ||x|| ||y||$$ for $x , y \in X$
I have a part of the proof. We first assume that x and y are linearly dependant so we let $x=\alpha y$ and $\alpha \in K$ where K is either real or complex.
So we have $|<x,y>|= |<x,y>|= |\alpha| ||y||^2=\sqrt {\alpha \bar\alpha <y,y>} ||y|| = \sqrt{<\alpha y ,\alpha y>} ||y||=||x|| ||y||$
What does $\alpha \bar \alpha$ mean?
And how do you get from $\sqrt {\alpha \bar\alpha <y,y>} ||y||$ to $\sqrt{<\alpha y ,\alpha y>} ||y||$?