The separable quotient problem and the strongly normal sequences

Abstract

We study the notion of a strongly normal sequence in the dual $E^*$ of a Banach space $E$. In particular, we prove that the following three conditions are equivalent: (1) $E$ has a strongly normal sequence, (2) $(E^*;\sigma (E ^*;E))$ has a Schauder basic sequence, (3) $E$ has an infinite-dimensional separable quotient.