finite codimension implies closed? 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 $E$.)
 A: No. For $E$ take $X\oplus \ell_2$, where $X$ is the direct sum of continuum many copies of the scalar field under the direct sum topology. This is the largest locally convex topology on $X$ and any linear mapping from $X$ into a locally convex space is continous.  Write $X=X_1 \oplus X_2$ with each $X_i$ isomorphic to $X$ and $\ell_2 = Y_1 \oplus Y_2$ with $Y_1$ and $Y_2$ closed and infinite dimensional.  Let $Y_0$ be a dense codimension one subspace of $Y_1$. Define $T$ by having $T$ map $X_1$ one to one onto $X$, $T$ maps $X_2$ one to one onto $Y_0$, and $T$ maps $\ell_2$ isometrically onto $Y_2$.  Then $T$ is one to one, continuous, and maps $E$ onto a dense codimension one subspace of $E$.
A: Bill: The existence of Hamel basis in (all) vector spaces is equivalent to the axiom of choice (see Blass, Andreas "Existence of bases implies the axiom of choice". Contemporary mathematics 31, 1984). For the existence of a Hamel basis in $l_2$ it is enough to have a well-ordering of the reals. 
Also, it is consistent with ZF that all linear functionals on $l_2$ are continuous (for instance, a model in which every set of reals has the property of Baire).
