The uniform bounded deciding property and the separable quotient problem

Abstract

Saxon–Wilansky’s paper The equivalence of some Banach space problems contains six properties equivalent to the existence of an infinite dimensional separable quotient in a Banach space with nice simplified proofs. In the frame of uniform bounded deciding property, we prove that for an infinite dimensional Banach space $$(E,\left\| \cdot \right\| )$$ the following properties are equivalents: 1) The unit sphere $$S_{E}$$ contains a dense and non uniform bounded deciding subset. 2) The unit sphere $$S_{E}$$ contains a dense and non strong norming subset. 3) $$(E,\left\| \cdot \right\| )$$ admits an infinite dimensional separable quotient.