Lindelof type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family S of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily So with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markucevic-bases, Corson property and related geometric issues are discussed. (c) 2008 Elsevier B.V. All rights reserved.