A characterization of barrelledness of $C_p(X)$

TL;DR

This paper characterizes when the space of continuous functions with the pointwise convergence topology is barrelled, linking this property to being a Mackey group for Tychonoff spaces.

Contribution

It provides a complete characterization of barrelledness in $C_p(X)$ spaces via their Mackey group property for Tychonoff spaces.

Findings

$C_p(X)$ is barrelled if and only if it is a Mackey group.

The characterization applies specifically to Tychonoff spaces.

Establishes a clear equivalence between barrelledness and Mackey group property.

Abstract

We prove that, for a Tychonoff space $X$, the space $C_p(X)$ is barrelled if and only if it is a Mackey group.