The separable quotient problem for Banach spaces

Theorem 1.1: for $\mathbb F\in\{\mathbb R,\mathbb C\}$, ZFC proves that if $2^{\aleph_0}$ is real-valued measurable then every infinite-dimensional Banach space over $\mathbb F$ has a separable infinite-dimensional quotient, and that under CH there is a counterexample of density $\aleph_1$. So the problem is independent of ZFC if ZFC plus a measurable cardinal is consistent; restricted to density $\aleph_1$ it is independent relative to ZFC alone (Corollary 1.2, with Martin's axiom). It does not give a ZFC answer, does not remove the measurable-cardinal strength for the unrestricted statement, and leaves open which other axioms decide it.

Theorem 1.1: for $\mathbb F\in\{\mathbb R,\mathbb C\}$, ZFC proves that if $2^{\aleph_0}$ is real-valued measurable then every infinite-dimensional Banach space over $\mathbb F$ has a separable infinite-dimensional quotient, and that under CH there is a counterexample of density $\aleph_1$. So the problem is independent of ZFC if ZFC plus a measurable cardinal is consistent; restricted to density $\aleph_1$ it is independent relative to ZFC alone (Corollary 1.2, with Martin's axiom). It does not give a ZFC answer, does not remove the measurable-cardinal strength for the unrestricted statement, and leaves open which other axioms decide it.