Can Open Mapping Theorem be Generalized to Finite Codimensional Subspaces of Banach Spaces? Let $A$ be a bounded linear operator ($A$ is not assumed to be an open mapping!) from a subspace $X_1$ of a Banach space to a subspace $Y_1$ of another Banach space, and that $\mathrm{codim}X_1+\mathrm{codim}Y_1<\infty$. I know that if $A$ maps one open subset $X$ of a Banach space onto one open subset $Y$ of another Banach space, since an open subset of a Banach space is second category, $A$ is surjective and an open map. 
If $A$ maps one open subset $X$ of $X_1$ onto one open subset $Y$ of $Y_1$, is it true that $A$ maps $X_1$ onto $Y_1$, i.e. $A$ is an open map?
 A: Edited in response to clarification:
I believe the following statement is true: If $f : X_1 \to Y_1$ is a linear mapping between normed spaces such that there exists a non-empty open set $U \subset X_1 $ such that $f(U)$ is open in $Y_1$, then $f$ is surjective.
(The fact that $X_1$ and $Y_1$ are finite-codimension subspaces of Banach spaces seems irrelevant.)
To prove this, pick any $x \in U$, and let $\widetilde U = -x + U$. Then $\widetilde U$ is an open set in $X_1$ containing the origin of $X_1$, and $f(\widetilde U) = -f(x) + f(U)$ is an open set in $Y_1$ containing the origin of $Y_1$. Since $f(\widetilde U)$ is open, there exists a $\delta$ such that $B_{Y_1}(\delta) \subset f(\widetilde U)$. But then, the union of the images of $n \widetilde U$ for all $n \in \mathbb N$ contains the union of $B_{Y_1}(n\delta)$ for all $n \in \mathbb N$, hence contains the whole of $Y_1$.
A: Translate $A(\text{open set})$ converting the translated set in a nhood of $0$. Nhoods of $0$ are absorbent.
