关键词: Banach space, separable space, quotient space, bounded linear operator

Abstract: The problem whether every infinite dimensional Banach space has all infinite 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 quasi complemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T∈ B(Y, X) such that tile range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc. are also discussed.

Key words: Banach space, separable space, quotient space, bounded linear operator