Web-Compact Spaces and Angelic Theorems
This chapter deals with the class of angelic spaces, introduced by Fremlin, for which several variants of compactness coincide. A remarkable paper of Orihuela introduces a large class of topological spaces X (under the name web-compact) for which the space C p (X) is angelic. Orihuela’s theorem covers many already known partial results providing Eberlein–Šmulian-type results. Following Orihuela, we show that C p (X) is angelic if X is web-compact. This yields, in particular, Talagrand’s result stating that for a compact space X the space C p (X) is K-analytic if and only if C(X) is weakly K-analytic. We present some quantitative versions of Grothendieck’s characterization of the weak compactness for spaces C(X) (for compact Hausdorff spaces X) and quantitative versions of the classical Eberlein–Grothendieck and Krein–Šmulian theorems.
KeywordsBanach Space Topological Space Compact Space Weak Topology Cluster Point
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