The free locally convex space $L(\mathbf{s})$ over a convergent sequence $\mathbf{s}$ is not a Mackey space

TL;DR

This paper proves that the free locally convex space over a convergent sequence is not a Mackey space, providing a negative answer to a previously posed question about its topological properties.

Contribution

It demonstrates that the free locally convex space over a convergent sequence lacks the Mackey space property, which was an open question in the field.

Findings

$L( extbf{s})$ is not a Mackey space

$L( extbf{s})$ is not a Mackey group

Answers a question in the literature negatively

Abstract

We show that the free locally convex space $L(\mathbf{s})$ over a convergent sequence $\mathbf{s}$ is not a Mackey space. Consequently $L(\mathbf{s})$ is not a Mackey group that answers negatively a question posed in [4].