TITLE: The space is complete iff the subspace and the quotient space is complete.
QUESTION [5 upvotes]: Let $F$ be a closed subspace of a normed space. How to prove that $E$ is complete iff $F$ and $E/F$ are both complete ?

REPLY [5 votes]: Let $\pi: E \to E/F$ be the quotient map.  Let $x_n$ be a Cauchy sequence in $E$.  Then $\pi(x_n)$ is Cauchy in $E/F$, so has a limit which is $\pi(z)$ for some $z \in E$.  Since $\|\pi(x_n - z)\| = \inf \{\|x_n - z - y\|: y \in F\}$, there is $y_n \in F$ with $\|x_n - z - y_n\| < \|\pi(x_n - z) \| + 1/n$, and thus
$\|x_n - z - y_n\| \to 0$.  Show that $y_n$ is Cauchy, and then that $x_n$ converges to $z + \lim_n y_n$.