# Subspaces isomorphic with quotients

Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?

Every separable Banach space is a quotient of $\ell_1$, so in particular every subspace of $\ell_1$ is a quotient of $\ell_1$.
• And $c_0$ has the dual property--every quotient is isomorphic to a subspace. – Bill Johnson May 26 '18 at 17:29
• @Markus, take $\ell_1\oplus C[0,1]$. – Tomek Kania May 26 '18 at 19:50