Lindelof type of generalization of separability in Banach spaces

TL;DR

This paper introduces the countable separation property (CSP) in Banach spaces, exploring its implications for separability, geometric properties, and connections with bases and the Corson property.

Contribution

It defines CSP for Banach spaces, provides examples of non-separable CSP spaces, and discusses its relations with bases and geometric properties.

Findings

Separable Banach spaces all have CSP

Examples of non-separable CSP spaces are identified

Connections between CSP, Markushevich bases, and the Corson property are discussed

Abstract

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \mathcal{F} of X^{\ast}, which separates X, there exists a countable separating subset \mathcal{F}_{0} of \mathcal{F}. 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.