Biorthogonal systems and quotient spaces via baire category methods

Abstract

We show that every Banach space X of density smaller that the Baire category number [InlineMediaObject not available: see fulltext.] admits a quotient with a long Schauder basis that can be taken of length ω 1 if X is not separable. So, assuming that the Baire category number [InlineMediaObject not available: see fulltext.] does not take its minimal possible value, a Banach space X is separable if and only if all biorthogonal systems of X are countable.