1. The problem statement, all variables and given/known data Axler's Linear Algebra done right, propositions 7.6 states: An operator T is normal if and only if ||Tv|| = || T*v||, T* being the adjoint of T. I've been stuck on the last equivalence in the proof stated here: <T*Tv,v>=<TT*v> <=> ||Tv||^2=||T*v||^2, with squaring both sides of the second statement proves the theorem. 2. Relevant equations 3. The attempt at a solution Basically all my attempts have been stopped by the fact that i can't either get rid of T or T* in the second slot when working backwards or trying to get T or T* back when working through it forwards. Any help would be highly appreciated.