On semi-barrelled spaces

TL;DR

This paper clarifies properties of semi-barrelled spaces, corrects inaccuracies in classical literature, and provides counterexamples to false claims about their completeness and reflexivity.

Contribution

It identifies and corrects six false statements in Bourbaki's classical text regarding semi-barrelled spaces, offering accurate properties and counterexamples.

Findings

Counterexamples disprove previous false claims

Corrected properties of semi-barrelled spaces

Clarification of semi-barrelled space characteristics

Abstract

The aim of this paper is to clarify the properties of semi-barrelled spaces (also called countably quasi-barrelled spaces in the literature). These spaces were studied by several authors, in particular in the classical book of N. Bourbaki "Espaces vectoriels topologiques". However, six incorrect statements can be found in this reference. In particular: a Hausdorff and quasi-complete semi-barrelled space is complete, a semi-barrelled, semi-reflexive space is complete, a locally convex hull of semi-barrelled semi-reflexive spaces is semi-reflexive, a locally convex hull of semi-barrelled reflexive spaces is reflexive. We show through counterexamples that these statements are false. To conclude, we show how these false claims can be corrected and we collect some properties of semi-barrelled spaces.