Non-Archimedean Quantitative Grothendieck and Krein's Theorems

We show that the non-archimedean 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 non-trivially valued non-archimedean field $\mathbb{K}$, fails in general. Indeed, we prove that if $X$ is an infinite zero-dimensional 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 non-archimedean 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 $\operatorname{aco}H$ of $H$ is $\tau _{p}$-relatively compact. In contrast, we show that for an infinite zero-dimensional compact space $X$ the absolutely convex hull $\operatorname{aco}H$ 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 non-archimedean $\mathbb{K}$. Nevertheless, our main result states that if $H\subset C(X,\mathbb{K})$ is uniformly bounded, then $\operatorname{aco}H$ is $\tau _{p}-$relatively compact if and only if $H$ is $w$-relatively compact.