On a characterization of spaces satisfying open mapping and equivalent theorems

Abstract

For classes of topological vector spaces, we analyze under which conditions open-mapping, continuous-inverse, and closed-graph properties are equivalent. Here, closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Ptak spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a continuous-inverse theorem, and a closed-graph theorem holds. An analogous, weaker result also holds for the strictly larger class of barreled infra-Ptak spaces.

Update of previous version. It contains a characterization of barreled infra-Ptak spaces and a separation of barreled Ptak and barreled infra-Ptak spaces