Example for $\mathbb{X}/\mathbb{M}$ separable but $\mathbb{X}$ is not.

Question

I have proved that for a Banach space $\mathbb{X}$ and its closed subspace $\mathbb{M}$
a) If $\mathbb{X}/\mathbb{M}$ and $\mathbb{M}$ are separable then $\mathbb{X}$ is separable.

b) If $\mathbb{X}$ separable (obviously then $\mathbb{M}$ is also separable ) then $\mathbb{X}/\mathbb{M}$ is separable.

But now I need a counter example to show that separability of $\mathbb{M}$ is essential for part a). That is to come up with a non separable Banach space with the separable quotient.

Appreciate any help

Answer 1

Consider $\ell_\infty(\mathbb{N})$ with $\|\mathbf{x}\|_\infty=\sup_{n\in\mathbb{N}}|\mathbf{x}(n)|$. This space is known to be non separable. Let $M=\{\mathbf{x}\in\ell_\infty: \mathbf{x}(1)=0\}$. This is a closed subspace of $\ell_\infty$, in fact $M$ and $\ell_\infty$ are isometric: $S:\mathbf{x}\mapsto \mathbf{x}'$ where $\mathbf{x}'(1)=0$ and $\mathbf{x}'(n)=\mathbf{x}(n-1)$ for $n\geq2$ is an isometry with inverse $P:M\rightarrow \ell_\infty(\mathbb{N})$ given by $P:\mathbf{y}\mapsto\mathbf{y}''$ where $\mathbf{y}''(n)=\mathbf{y}(n+1)$.

$\ell_\infty/M$ is isometric to $\mathbb{R}$ (user the usual topolpogy), that later known to be separable.