I'm having some trouble finishing this exercise:
Suppose $A : X \rightarrow X$ is a bounded linear operator on a Banach space $X$. Show that, if $A^*(X^*) = X^*$, then $A$ has a bounded inverse. (Bachman, Functional Analysis, Exercise 17.7)
(Here $X^*$ denotes de dual space of $X$, and $A^*$ the adjoint operator associated to $A$)
Now, given the fact that $X$ is Banach, we can use the bounded inverse theorem to show that if $A$ admits an inverse then it must be bounded. So the problem reduces to showing that $A$ is injective and surjective.
I have already shown that $A$ must be injective. But I am having problems showing that it must be surjective. If I can show that $A(X)$ is closed and $A^*$ is injective then I can prove the surjectivity of $A$, but I'm doubting that all of this has to be a consequence of just $A^*$ being surjective.
Does anyone have any hint?
Thanks in advance :)