Closed Graph Theorem Application

Question

I'm having trouble working out the proof for this problem. Let $T:L^2(X)\to L^2(X)$ be a linear map such that there is another linear map $T^*:L^2(X)\to L^2(X)$ with $\langle Tu,v\rangle=\langle u,T^*v\rangle$ for all $u,v\in L^2(X).$

Show that $T$ and $T^*$ are bounded.

So I want to used the closed graph theorem and show that $\Gamma(T)$ and $\Gamma(T^*)$ are both closed. This would imply the desired result.

So suppose that if $(v_n)$ is a sequence in $\Gamma(T)$ which converges to a point in $v\in L^2(X)$ and $(Tv_n)$ converges to a point $w\in L^2(X)$ then I want to show $Tv=w$, but I'm not exactly sure how to go about doing this.

Answer 1

Let $u_n \to u$ and $Tu_n \to w$. Then for all $v\in L^2(X)$,

$$\langle w, v\rangle = \lim \langle Tu_n , v\rangle = \lim \langle u_n , T^* v\rangle = \langle u, T^* v\rangle = \langle Tu, v\rangle$$

Thus $Tu=w$ and so the graph of $T$ is closed. Similar for $T^*$.