×

Every Radon-Nikodým Corson compact space is Eberlein compact. (English) Zbl 0771.46015

By showing that the initial example of M. Talagrand [Ann. of Math., II. Ser. 110, 407-438 (1979; Zbl 0393.46019)]is a Talagrand compact space which is not Radon-Nikodym compact, the authors answer negatively the first two questions raised by I. Namioka in [Mathematika 34, No. 2, 258-281 (1987; Zbl 0654.46017)]. Earlier another example with such properties has been provided by Reznichenko. Motivated by the study of Talagrand’s example, and using the projectional resolution technique of Valdivia, the authors prove the following theorems:
Theorem A. A compact Hausdorff space is Eberlein compact if (and only if) it is Radon-Nikodym and Corson compact.
Theorem B. A Banach space \(E\) is weakly compactly generated if (and only if) its dual unit ball is Corson compact and if it is GSG, i.e. there is a continuous linear map \(T: X\to E\) with dense range defined on an Asplund space \(X\).
From Theorem \(B\) it follows that weakly \(K\)-analytic or more generally, weakly countably determined Banach spaces are not necessarily GSG.

MSC:

46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46A50 Compactness in topological linear spaces; angelic spaces, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML