Measures of weak noncompactness in non-Archimedean Banach spaces (Q2922478)

The classical Krein theorem states that, in a real or complex Banach space \(E\), the convex hull of a weakly relatively compact subset of \(E\) is again weakly relatively compact. A quantitative version of this theorem was proved by \textit{M. Fabian} et al. [Rev. Mat. Iberoam. 21, No. 1, 237--248 (2005; Zbl 1083.46012)]. They asked the following:NEWLINENEWLINELet \(M\) be a bounded set in \(E\) and let \(B_{E''}\) be the closed unit ball in the bidual \(E''\) of \(E\). Assume that \(M\) is \(\epsilon\)-weakly relatively compact, i.e., \(\overline{M}^{\sigma(E'',E')} \subset E + \epsilon B_{E''}\) for some \(\epsilon \geq 0\). Does the same hold for the convex hull of \(M\)?NEWLINENEWLINEThe case \(\epsilon=0\) is the classical Krein theorem. Fabian et al.\ proved in [loc.\,cit.]\ that, whenever \(M\) is \(\epsilon\)-weakly relatively compact for some \(\epsilon >0\), then the convex hull of \(M\) is \(2 \epsilon\)-weakly relatively compact. However, this convex hull does not necessarily have to be \(\epsilon\)-weakly relatively compact, as was proved by \textit{A. S. Granero} et al. [Math. Ann. 328, No. 4, 625--631 (2004; Zbl 1059.46015)].NEWLINENEWLINEIn this paper, the authors consider non-Archimedean Banach spaces \(E\) over a non-Archimedean locally compact non-trivially valued field \(K\). The second and third authors have recently studied weak compactness for those \(E\). Among other things, they proved in [J. Convex Anal. 20, No. 1, 233--242 (2013; Zbl 1277.46043)] that the convex hull of a weakly relatively compact subset of \(E\) is again weakly relatively compact, the non-Archimedean counterpart of the classical Krein theorem. This line of research is continued in the present paper.NEWLINENEWLINEWorking with measures of weak noncompactness, they provide a non-Archimedean version of the result of Fabian et al.\ proved in [loc.\,cit.]\ as well as quantitative versions of Krein's theorem. Among their achievements, we point out the one stating that the convex hull of an \(\epsilon\)-weakly relatively compact subset of a non-Archimedean Banach \(E\) space over \(K\) is again \(\epsilon\)-weakly relatively compact -- a truly non-Archimedean fact, according to the classical results previously shown in this report.NEWLINENEWLINEI encourage the authors to continue working on this so interesting subject, whose results promise to have a non-Archimedean character.