# On a characterization of spaces satisfying open mapping and equivalent theorems

## TL;DR

This paper characterizes classes of topological vector spaces where open-mapping, continuous-inverse, and closed-graph theorems are equivalent, identifying the maximal classes with these properties.

## Contribution

It identifies the largest classes of locally-convex topological vector spaces, namely barreled Ptak and infra-Ptak spaces, where key theorems are equivalent and preserved under common operations.

## Key findings

- Barreled Ptak spaces form the largest class with these properties.
- The class is closed under quotients, closed graphs, and continuous images.
- Similar results hold for barreled infra-Ptak spaces.

## 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.