Suppose we have a surjective bounded linear operator acting between Banach spaces. By the Open Mapping Theorem it maps open sets in the domain to open sets in the codomain. Must the image of a closed linear subspace of the domain be closed then?
[Math] Is an open linear map closed (to some extent)
banach-spacesfunctional-analysis
Best Answer
A construction very similar to that seen in the question Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed? applies here.
If $f:E\to F$ is any bounded linear operator such that $f(E)$ is not closed, then $g:E\oplus F\to F$ defined by $$g(x+y)=f(x)+y$$ is surjective but maps $E\oplus 0$ to the non-closed subspace $f(E)$.