So I was browsing through my book and along came a question (pic related) that I do not how to prove. I've tried using the triangle inequality and the Cauchy-Schwartz inequality but it just doesn't unravel itself? Any general tips as to how one should prove these types of inequalities? Thank you in advance.
>>8955455
I'm surprised it's not |x|<=|y| at the end there instead of |x|<1.
>>8955455
Note |Ux|=|x| since U is unitary. Then Cauchy-Schwartz gives
| < Ux,y> | <= |Ux| |y| = |x||y| < |y|, since |x|<1.