Separable quotients of non-separable Banach spaces? - MathOverflow

Separable quotients of non-separable Banach spaces?

2011-06-14T21:58:39Z

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 $l^{1}=l^{1}(\mathbb{N})$?)

Answer by Bill Johnson for Separable quotients of non-separable Banach spaces?

2011-06-15T05:29:36Z

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 $\ell_\infty$.

Answer by Jun Zhang for Separable quotients of non-separable Banach spaces?

2011-06-16T01:56:23Z

Here seems to be another reference paper from Jorge MUJICA, who transfer this seperable quotient space problem to some other equivalent problems.

http://www.mat.ucm.es/serv/revmat/vol10-2/vol10-2e.pdf