I found this tricky new proof of the closed graph theorem for a Hilbert space $H$. http://arxiv.org/pdf/1601.02600.pdf
It says in the abstract, that it's possible to extend the proof to Banach space. But I wasn't able to do so. The problem was proving Theorem 3.
Theorem 3. If $A$ is a linear closed operator with $D(A) = H$, then $D(A^∗) = H$.
In the case of Banach spaces I did not find a way to prove that adjoint $A^*$ is densely defined. But even if I did - the next problem would have been to deduce that there is a weekly convergent subsequence of $A^* v_n$.
Is there another way to prove Theorem 3?