I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I have found a solution for this but here is my question :

- Is it true that every non-separable normed space $X$ always contains a closed (proper) subspace $M$ such that $X/M$ is linear isometric to a separable normed space whose dimension is infinite ? i.e, are there a map $A$ and a separable normed space $Y$ whose dimension is infinite, st: $A: X/M\to Y$ which is linear, onto, and preserve the distance?

(Edit: I already have an answer for the following question I am thinking a about $l^{\infty}$ : can it contain a closed proper subspace M that $l^{\infty}/M$ is isometric to $l^{1}=l^{1}(\mathbb{N})$?)

Separable quotients of Banach spaces, Rev. Mat. Univ. Complutense Madrid 10 (1997), 299–330. A more recent result, that every Banach space isomorphic to a dual space has a separable quotient, has been shown my Argyros et al inUnconditional families in Banach spaces, Math. Ann. (2008) 341:15–38. – Philip Brooker Jun 15 '11 at 5:30