Let $X$ be a Banach space (it is in fact an $L^p$ space) and let $T:X \to X$ be a linear continuous operator (which is not injective and not surjective). I am trying to figure out if the following is true:
For every $g \in Im(T)$ there exist $f \in X$ such that $Tf=g$ and $\lvert \lvert f \lvert \lvert \leq C \lvert \lvert g \lvert \lvert$ for some $C>0$ independent of $g$.
I know that if $T$ was bijective then this would be a consequence of the Open Mapping Theorem (The inverse is continuous). But I cannot see any way to prove my claim from this theorem.