If $X/F$ is separable, must $X/F$ isomorphic to $F$?

Question

Suppose $X$ is a Banach space and $F$ is a closed subspace of $X$. Clearly $X/F$ is a Banach space, equipped with quotient norm.

Question: If $X/F$ is separable, must $X/F$ isomorphic to $F$?

The motivation of this question comes from Corollary $3.4$. It seems that I need this fact to conclude that $F$ is linearly complemented in $X$.

Answer 1

Every separable Banach space is isomorphic to a quotient of $\ell_1$. Certainly most of separable Banach spaces are not isomorphic to a subspace of $\ell_1$.

Answer 2

No, in general, this is very false.

A counterexample can already be obtained by taking $X =\Bbb {R}^3$ and $F=\Bbb {R}^2\times \{0\} $.

Then $\dim X/F=1$, but $\dim F=2$, so that they can not be isomorphic.

But since finite dimensional spaces are always separable and complete, your assumptions hold.

Answer 3

In direct sum $X = l^1 \oplus l^2$, take subspace $F = 0 \oplus l^2$, which is reflexive, to get quotient $X / F$ isomorphic to $l^1$, which is separable and not reflexive, and therefore not isomorphic to $F$.