Does Unitary must have norm equal to 1?

Question

Does Unitary must have norm equal to 1?

I know U is unitary then $UU^*=I$, so $\|Ux\|^2=(Ux,Ux)=(U^*Ux,x)=(x,x)$, so $\|U\|=1$. Is this a proof?

Answer 1

You should note what norm you are using (the matrix-$2$-norm $\|U\|_2$). Practically this is correct, though because it shows $$\| Ux \|_2 = \|x\|_2 \Rightarrow \frac{\|U x\|_2}{\|x\|_2} = 1 \Rightarrow \|U\|_2 = \sup_{x\ne 0} \frac{\|U x\|_2}{\|x\|} = 1$$