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 $X$.