Let $E$ be a (complete) topological vector space, and $u:E\to E$ be continuous. Is it always true that if ${\rm Im}(u)$ is of finite codimension in $E$, then it is closed in $E$ or do we have to assume something on $E$? (It is OK if $E$ is Frechet by the open mapping theorem applied to ${\rm id}\oplus u:F\oplus E\to E$, where $F$ is a supplementary subspace to $E$.)