Natural quotient map

Question

Let $X$ be a Banach space and let $M $ be a closed subspace of $X $.

When is the quotient map $Q :X \to X/M $ closed?

I got somewhere that it happens iff$ M=X$ or $M=(0) $. But I couldn't prove it. Please suggest.

Answer 1

Let $X$ be a Banach space, $M$ a closed subspace such that $\{0\} \ne M \ne X$. Then there is $x_1\not\in M$ and $x_2\in M\setminus \{0\}$.

Define the closed set $$ S:=\{ s x_1 + s^{-1}x_2, \ s>0\}. $$ Then $Q(S)$ is the following set of equivalence classes $$ Q(S)= \{ [sx_1]: \ s>0\}. $$ This is not closed in $X/M$ as $[sx_1]\to0$ for $s\to0$: $$ \|[sx_1]\|_{X/M} = \inf_{x\in [sx_1]}\|x\|_X \le s \|x_1\|_X, $$ but $0\not\in Q(S)$.

This shows that the quotient mapping maps closed sets to closed sets if and only if $M=\{0\}$ or $M=X$.

Answer 2

If you mean by "closed map" that the graph of $Q$ is closed, then $Q$ is closed, since $Q$ is bounded:

$||Qx|| \le ||x||$ for all $x \in X$