On the Gelfand-Phillips property in Banach spaces with PRI. (English) Zbl 0903.46015

Sinha, D. P.; Arora, K. K.

Collect. Math. 48, No. 3, 347-354 (1997).

Summary: It is proved that every Banach space belonging to a certain class called the class \({\mathcal P}\) possesses the Gelfand-Phillips property. Consequently, so does every weakly countably determined Banach space, every Banach space with an \(M\)-basis whose dual unit ball is weak\(^*\) angelic and \(C(K)\) spaces for Valdivia compact \(K\).

MSC:

46B22	Radon-Nikodým, Kreĭn-Milman and related properties
46B10	Duality and reflexivity in normed linear and Banach spaces
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

Gelfand-Phillips property; weakly countably determined Banach space; Banach space with an \(M\)-basis whose dual unit ball is weak\(^*\) angelic; Valdivia compact