What does this notation mean? (Introductory complex analysis)

Question

I found these online notes by Harvard due toYum-Tong Siu.

We are trying to prove Cauchy schwarz inequality for complex vectors. I don’t understand why $Re(v,w)=||v,w||$ (what does substitution have to do with anything?) I think $(v,w)$ refers to the standard hermitian product.

Answer 1

When you replace $v$ with $uv$ you choose $u$ so that $\arg u = -\arg v$ and this ensures that $v$ is now a real number. This is because $\| uv\|=\|v\|$ for $|u|=1$.