Prove that $R(T^{*})^\perp =N(T)$

Question

Let V be an inner product space, with T being a linear operator on V. How do I prove that $R(T^{*})^\perp =N(T)$? I tried setting $x\in R(T^{*})$ and $Ty\in N(T)$, and set up an inner product = 0 since $Ty\in N(T)$ but just got $T^{*}x\perp y$, and I don't know where to go from there.

Edit: and how could I prove it for a more general case $T:V\rightarrow W$?

Answer 1

For $T: V \rightarrow W$, we have: \begin{align} y\in N(T)&\iff \langle x,Ty\rangle_W=0, \ \forall x\in W\\ &\iff \langle T^*x,y\rangle_V = 0, \ \forall x\in W \\ &\iff y\in R(T^*)^\perp. \end{align}