by typhoonss821
 P: 41 To show that there exists such a function, let $$v_1, \ldots , v_n$$ be an orthonormal basis for V, so that $$x = \sum_i \langle x,v_i\rangle v_i$$ for any x in V then we have for all x in V and y in W: $$\langle T(x), y\rangle ' = \langle T (\sum_i \langle x,v_i\rangle v_i ), y\rangle '$$ $$= \sum_i \langle x, v_i\rangle \langle T(v_i), y\rangle '$$ $$= \langle x, \sum_i \overline{ \langle T(v_i),y\rangle '} v_i\rangle$$ which is in the form that we'd like. Which shows that $$T^*(y) = \sum_i \overline{\langle T(v_i),y\rangle '} v_i$$ for all y in W works.