On some properties of Banach spaces of continuous functions. (English) Zbl 0876.46016

Plebanek, Grzegorz

Choquet, G. (ed.) et al., Séminaire d’initiation à l’analyse. 31ème année: 1991/1992. Paris: Université Pierre et Marie Curie, Publ. Math. Univ. Pierre Marie Curie. 107, Exp. No. 20, 9 p. (1994).

Based on a talk given at the Séminaire d’Initiation à l’Analyse in 1992, the author surveys some results concerning the Mazur property and the property of realcompactness for Banach spaces \(C(K)\) of continuous functions on compact spaces \(K\). A Banach space \(E\) is said to have the Mazur property if every \(z\) in \(E^{**}\) that is weak* sequentially continuous on \(E^*\) is, in fact, in \(E\). In a related vein, \(E\) is realcompact provided that every \(z\) in \(E^{**}\) that is weak* continuous on weak* separable subspaces of \(E^*\) is in \(E\). The Mazur property is stronger so that every Banach space having this property is also realcompact. The results surveyed mainly concern conditions on the compact Hausdorff space \(K\) that imply that \(C(K)\) has one or the other of these properties. In the final section dyadic spaces \(K\) are considered. As one expects in a survey article, the results discussed are mainly culled from other papers by the author and others, and the emphasis is on establishing an overall perspective on the subject rather than on providing complete details and proofs.
For the entire collection see [Zbl 0841.00013].

Reviewer: T.Feeman (Villanova)

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
46B20	Geometry and structure of normed linear spaces

Keywords:

Banach spaces of continuous functions; Mazur property; realcompact