Theorem Let $X$, $Y$ be Banach spaces and $T \in {\cal B}(X,Y)$ a surjective linear map. Then $T$ is an open map (that is, it sends open sets in open sets and the inverse $T^{-1}$ is continuous).
Now, in order $T^{-1}$ to exists in the first place, it's necessary that $T$ is injective. Is every surjective open map also injective?
Secondly, if this is really enough to guarantee the continuity of $T^{-1}$, and hence its boundedness, then what's the difference between this theorem and the so-called bounded inverse theorem?
If $T \in {\cal B}(X,Y)$ is surjective, then, in general, $T$ is not injective.
Example: $X=Y=l^2$ and $T(x_1,x_2,x_3,...)=(x_2,x_3,...).$
If $T \in {\cal B}(X,Y)$ is bijective, then $T^{-1}$ is bounded. That is the bounded inverse theorem.