Application of closed graph theorem 3

Question

I'm trying to prove this:

Let $X\;,\;Y$ Banach spaces and $ A :X\;\longrightarrow \; Y$ a linear operator. Prove using The Closed Graph Theorem , that if $GA\;\in\;X'\;\forall\;G\;\in\;Y'$, then $A\;\in\;\textit{L}(X,Y)$.

I manage to prove that if $A' :X' \longrightarrow Y'$ is bounded, then $A$ is bounded and $\textit{D(A)}=X$. However, I don't know how to show it. Can someone please help me?

Thank you !

Answer 1

Assume $x_n\to x$ and $Ax_n\to y$ in norm. Then $G(Ax_n)\to G(y)$ and $GA(x_n)\to GA(x).$ Thus $G(Ax)=G(y)$ for all $G\in Y'.$