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 :


1. 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 non-separable Banach spaces?
Your question is the famous "separable quotient problem", as Ady mentioned. From here on, "space" means "infinite dimensional Banach space". A space $X$ has a separable quotient provided $X^\*$ has a reflexive subspace (obvious), a subspace isomorphic to $c\_0$ (Rosenthal and me), or $\ell\_1$ (Hagler and me). A result of PANDELIS DODOS, JORDI LOPEZ-ABAD and STEVO TODORCEVIC is that it is consistent with ZFC that if $X$ has density character at least $\aleph\_\omega$ then $X$ has a separable quotient; see


<http://arxiv.org/pdf/0805.1860.pdf>


Every dual space has a separable quotient (Argyros, Dodos, Kanellopoulos):


<http://users.uoa.gr/~pdodos/Publications/13-Unconditional.pdf>


There are other striking things that I can't locate quickly.


Every non reflexive quotient of a $C(K)$ space contains a subspace isomorphic to $c\_0$ (classical result of Pelczynski), so $\ell\_1$ is not a quotient of $\ell\_\infty$.


