### 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.

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