On the property (C) of Corson and other sequential properties of Banach Spaces

2023/03/03 by Martínez-Cervantes, Gonzalo, Poveda, Alejandro
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2303.02122

Abstract

A well-known result of R. Pol states that a Banach space X has property (C) of Corson if and only if every point in the weak*-closure of any convex set C ⊆ B_X^* is actually in the weak*-closure of a countable subset of C. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of B_X^* with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under MA+¬ CH for the class of C(K)-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom (PFA) the following conditions are equivalent for an arbitrary Banach space X: 1) X has property E'; 2) X has weak*-sequential dual ball; 3) X has property (C) of Corson; 4) (B_X^*,w^∗) has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property E' has weak*-convex block compact dual ball.

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