TITLE: The image of a Banach space under a continuous, linear, open map is a Banach space.
QUESTION [5 upvotes]: This is an exercise from Royden's Real Analysis. Suppose $X$ is a Banach space, there is a continuous, linear, open map from $X$ onto a normed linear space $Y$. Show that $Y$ is Banach.

REPLY [2 votes]: Hints. Let $T$ be your linear map. Show first that there exists a finite constant $C$ such that, for any $y\in Y$, one can find $x\in X$ such that $Tx=y$ and $\Vert x\Vert\leq C\,\Vert y\Vert$. Use this to show that any series $\sum y_n$ in $Y$ such that $\sum_0^\infty \Vert y_n\Vert<\infty$ is convergent in $Y$.