Non-archimedean closed graph theorems

Abstract

We consider linear maps T: X ? 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* bounded subset in the dual X0 is equicontinuous. If in this definition weakly* bounded subset is replaced by weakly* bounded sequence or sequence weakly* converging to zero, then X is said to be l?-barrelled or c0-barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of spaces with weakly* 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.