On Banach spaces with the Gelfand-Phillips property. (English) Zbl 0629.46020

Drewnowski, Lech

Math. Z. 193, 405-411 (1986).

A bounded subset A of a Banach space E is called limited if for every \(w^*\)-null sequence \((x^*_ n)\) in \(E^*\) we have \(x^*_ n(x)\to 0\) uniformly for \(x\in A\). If all limited subsets of E are relatively compact, then E is said to have the Gelfand-Phillips property, \(E\in (GP)\). The main results are the following:
Theorem 2.2. If \(E^*\) has a norming weak*-conditionally compact subset, then \(E=(GP).\)
Theorem 3.1. If \(E\in (GP)\) and \(F\in (GP)\), then so is their injective tensor product E\({\check \otimes}F.\)
Theorem 4.2. If \(E^*\in (GP)\) and \(F\in (GP)\), then the space of all compact operators K(E,F) belongs to (GP).
Theorem 5.1. If E has a Schauder decomposition \(\sum^{\infty}_{1}E_ n\), where each summand \(E_ n\) belongs to (GP), then \(E\in (GP)\).

Reviewer: M.I.Kadets

MSC:

46B20	Geometry and structure of normed linear spaces
46B25	Classical Banach spaces in the general theory
46B22	Radon-Nikodým, Kreĭn-Milman and related properties
46M05	Tensor products in functional analysis
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46A32	Spaces of linear operators; topological tensor products; approximation properties

Keywords:

limited subsets; Gelfand-Phillips property; norming weak*-conditionally compact subset; injective tensor product; space of all compact operators; Schauder decomposition

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