T* satisfies the equation <T(v),w> = <v,T*(w)> for all v,w in V?
Good! Now choose a basis for Im(T), $\displaystyle w_1$, ... $\displaystyle w_n$, and let $\displaystyle v_i= T^*(w_i)$. Can you show that $\displaystyle v_1$,... $\displaystyle v_n$ is a basis for Im(T*)?
linear independent property: for all a_{1},....,a_{n} ∈ R if a_{1}v_{1}+.......+a_{n}v_{n }= 0
spanning property: for every x in Im(T*) such that x = a_{1}v_{1}+.....+a_{n}v_{n}