I am reading a proof of the following Cauchy-Schwarz Inequality and I don't understand one part of the proof:
Theorem: Let $A$ be a $C^*$-algebra and let $E$ be a semi-inner-product $A$-module. Then $ \langle x,y \rangle ^* \langle x,y \rangle \leq \| \langle x,x \rangle \| \langle y,y \rangle \text{ for all } x,y\in E $
The proof starts of by saying, without loss of generality we can assume $ \| \langle x,x \rangle \| = 1$ But I don't understand why we can assume this.
The rest of the proof is just some calculations which I understand: for $a\in A$, $x,y\in E$ we have $ 0 \leq \langle xa-y,xa-y \rangle = a^* \langle x,x \rangle a - \langle y,x \rangle a - a^* \langle x,y \rangle + \langle y,y \rangle \leq a^*a - \langle y,x \rangle a - a^* \langle x,y \rangle + \langle y,y \rangle $ and by letting $a=\langle x,y \rangle $ we get the desired result.
I just need to understand why we can assume $\| \langle x,x \rangle \| = 1$. My guess is that if it does not equal 1 then we can replace it with an equivalent norm that equals 1, but I don't know if this is the correct reason.