On reflexivity and the Ascoli property for free locally convex spaces

TL;DR

This paper characterizes when the free locally convex space over a Tychonoff space is reflexive or Ascoli, showing these properties hold only under specific conditions related to discreteness and countability.

Contribution

It provides a complete characterization of reflexivity and the Ascoli property for free locally convex spaces over Tychonoff spaces, answering a previously open question.

Findings

$L(X)$ is reflexive iff $X$ is discrete (for Dieudonné complete spaces)

$L(X)$ is an Ascoli space iff $X$ is a countable discrete space

The results apply to metrizable spaces as well

Abstract

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. If $X$ is Dieudonn\'{e} complete (for example, metrizable), then $L(X)$ is a reflexive group if and only if $X$ is discrete. Answering a question posed in [9] we prove also that $L(X)$ is an Ascoli space if and only if $X$ is a countable discrete space.