1. The problem statement, all variables and given/known data Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real. 2. Relevant equations 3. The attempt at a solution Let c be an eigenvalue. Now since T=T*, we have <TT*v, v>=<v, TT*v> if and only if TT*v=cv on both sides and not -cv (-c is the complex conjugate of c made possible by c being a complex number) on one side and cv on the other side. Therefore c must be real.