Hello, I have attached theorem 7.20 in the link above. Context: In the picture, V denotes a finite-dimensional inner product space over F, and F denotes R (field of real numbers) or C (field of complex numbers). < , > denote an inner product. T is an operator on V. T* is the adjoint of T.
Question: I do not understand how the equivalence in the highlighted step was deduced. How does supposing <T*Tv, v> = <TT*v, v> for all v in V imply <Tv, Tv> = <T*v, T*v> for all v in V?