im pretty rusty on this so double check for mistakes... to show T*T is injective u just need to show if T*Tu=0, then u=0.
0=<T*Tu,u>=<Tu,Tu>, so Tu=0. since T is injective, u=0
T*T is linear and injective and goes from V->V, so range of T*T must have same dimension as V, i.e. range of T*T is V...