Free locally convex spaces and the Ascoli property

Saak Gabriyelyan

Published 2018-05-18Version 1

T. Banakh showed in [2] that if $X$ is a Dieudonn\'{e} complete space, then the free locally convex space $L(X)$ on $X$ is an Ascoli space if and only if $X$ is a countable discrete space. We give an independent, short and clearer proof of Banakh's result and remove the condition of being a Dieudonn\'{e} complete space. Thus we have the following result: the free locally convex space $L(X)$ over a Tychonoff space $X$ is an Ascoli space if and only if $X$ is a countable discrete space.