On the Mackey problem for free locally convex spaces

TL;DR

This paper establishes that the free locally convex space over a Tychonoff space is a Mackey group or space if and only if the underlying space is discrete, clarifying the conditions for Mackey properties.

Contribution

It provides a complete characterization of when free locally convex spaces are Mackey groups or spaces based on the discreteness of the underlying space.

Findings

Free locally convex space is a Mackey group iff the base space is discrete.

Free locally convex space is a Mackey space iff the base space is discrete.

Discreteness of the base space is necessary and sufficient for Mackey properties.

Abstract

We show that the free locally convex space $L(X)$ over a Tychonoff space $X$ is a Mackey group iff $L(X)$ is a Mackey space iff $X$ is discrete.