I'm trying to solve the converse of following excercise
where E and F are Banach Space, $E^*$ and $F^*$ are the dual space, $S(E,F)$ is the set of surjective linear and bounded maps and $I(E,F)$ is the set of the linear bounded operators that has closed image.
First, I tried with orthogonality relations and have this $$ \overline{R(T)} =\, ^\perp\ker(T^*) =\, ^\perp\{0\} = F. $$ But I'm not sure how prove that the image is closed.
Also, I was reading the solution from the book but I'm confused, I suppose that tries to prove that is a open map and futhermore a surjective map.
The book is: Analyse Fonctionnelle et théorie des opérateurs.
Any suggestion or hint would be appreciate