
Journal of Convex Analysis 20 (2013), No. 1, 233242 Copyright Heldermann Verlag 2013 NonArchimedean Quantitative Grothendieck and Krein's Theorems Jerzy Kakol Faculty of Mathematics and Informatics, A. Mickiewicz University, 61614 Poznan, Poland kakol@amu.edu.pl Albert Kubzdela Institute of Civil Engineering, University of Technology, 61138 Poznan, Poland albert.kubzdela@put.poznan.pl [Abstractpdf] We show that the nonarchimedean version of Grothendieck's theorem about weakly compact sets for $C(X,\mathbb{K})$, the space of continuous maps on $% X $ with values in a locally compact nontrivially valued nonarchimedean field $\mathbb{K}$, fails in general. Indeed, we prove that if $X$ is an infinite zerodimensional compact space, then there exists a relatively compact set $H:=\{g_{n}:n\in \mathbb{N}\}\subset C(X,\mathbb{K})$ in the pointwise topology $\tau _{p}$ of $C(X,\mathbb{K})$ which is not $w$% relatively compact, i.e. compact in the weak topology of $C(X,\mathbb{K})$, such that all $\Vert g_{n}\Vert =1$ and $\gamma (H):=\sup \{\lim_{m}\lim_{n}f_{m}(x_{n})\lim_{n}\lim_{m}f_{m}(x_{n}):(f_{m})_{m}% \subset B,(x_{n})_{n}\subset H\}>0$, where $B$ is the closed unit ball in the dual $C(X,\mathbb{K})^{\ast }$ and the involved limits exist. The latter condition $\gamma (H)>0$ shows in fact that a quantitative version of Grothendieck's theorem for real spaces (due to Angosto and Cascales) fails in the nonarchimedean setting. The classical Krein and Grothendieck's theorems ensure that for any compact space $X$ every uniformly bounded set $% H $ in a real (or complex) space $C(X)$ is $\tau _{p}$relatively compact if and only if the absolutely convex hull $acoH$ of $H$ is $\tau _{p}$% relatively compact. In contrast, we show that for an infinite zerodimensional compact space $X$ the absolutely convex hull $acoH$ of a $% \tau _{p}$relatively compact and uniformly bounded set $H$ in $C(X,\mathbb{K% })$ needs not be $\tau _{p}$relatively compact for a locally compact nonarchimedean $\mathbb{K}$. Nevertheless, our main result states that if $% H\subset C(X,\mathbb{K})$ is uniformly bounded, then $acoH$ is $\tau _{p}$% relatively compact if and only if $H$ is $w$relatively compact. Keywords: Grothendieck's theorem, Krein's theorem, locally compact nonarchimedean field, compactness, space of continuous functions. MSC: 46S10, 46A50, 54C35 [ Fulltextpdf (145 KB)] for subscribers only. 