6 got rid of misleading title

# canl-infinitycontainacopySeparablequotients of l1non-separableBanachspaces?

5 added 70 characters in body

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 $l^{1}=l^{1}(\mathbb{N})$?l^{1}=l^{1}(\mathbb{N})$?) 4 added 60 characters in body I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of$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$which is linear, onto, and preserve the distance? I am thinking a about$l^{\infty}$: can it contain a closed proper subspace M that$l^{\infty}/M$is isometric to$l^{1}=l^{1}(\mathbb{N})\$?

3 deleted 4 characters in body
2 removed unnecessary tag
1