Let $A$ be a bounded operator defined on $X$ a Banach space, and $R(A)$ a Banach space. Then $\overline{R(A')}=N(A)^{\perp}$.

Question

I am able to prove one of the contentions, which in fact is quite easy to see, $\overline{R(A')} \subseteq N(A)^{\perp}$. But to prove the other contention, it is probably necessary to use the hypothesis that $R(A)$ is Banach, but I cannot see where or why will this be relevant.