Non-archimedean closed graph theorems

Abstract

We consider linear maps $T:X\rightarrow Y$, where $X$ and $Y$ are polar local
convex spaces over a complete non-archimedean field $K$. Recall that $X$ is
called polarly barrelled, if each weakly$^{\ast}$ bounded subset in the dual
$X^{\prime}$ is equicontinuous. If in this definition \textit{weakly}$^{\ast}%
$\textit{ bounded subset} is replaced by \textit{weakly}$^{\ast}$\textit{
bounded sequence} or \textit{sequence weakly}$^{\ast}$\textit{ converging to
zero}, then $X$ is said to be $\ell^{\infty}$-barrelled or $c_{0}$-barrelled,
respectively. For each of these classes of locally convex spaces (as well as
the class of spaces with weakly$^{\ast}$ sequentially complete dual) as domain
class, the maximum class of range spaces for a closed graph theorem to hold is
characterized. As consequences of these results, we obtain non-archimedean
versions of some classical closed graph theorems.

The final section deals with the necessity of the above-named
barrelledness-like properties in closed graph theorems. Among others,
counterparts of the classical theorems of Mahowald and Kalton are proved.