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 $X$?



Subspaces isomorphic with quotients
Every separable Banach space is a quotient of $\ell\_1$, so in particular every subspace of $\ell\_1$ is a quotient of $\ell\_1$.


