The problem whether every infinite dimensional Banach space has an infiuite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also ifand only if there exists a Banach space Y and a bounded linear operator T ∈ B (Y, X)such that the range of T is nonclosed and dense in X. Besides, the other relevantquestions for such spaces e.g. the question on operator ideals that on H.l. (hereditarilyindecomposable) spaces, that on invariant subspaces of operators, etc. are also discussed.