Locally convex properties of free locally convex spaces

TL;DR

This paper investigates the locally convex properties of free locally convex spaces over Tychonoff spaces, establishing equivalences and characterizations related to barrelledness, quasibarrelledness, and other topological properties, depending on the nature of the base space.

Contribution

It provides new characterizations of when free locally convex spaces possess various convexity and barrelledness properties, linking these to the topological features of the underlying space.

Findings

$L(X)$ is $ ext{ell}_ ext{infty}$-barrelled iff $X$ is a $P$-space.

$L(X)$ is a $(DF)$-space iff $X$ is countable and discrete.

$L(X)$ has the Dunford--Pettis property for all Tychonoff spaces.

Abstract

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\ell_\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such that $L(X)$ is a quasi-$(DF)$-space but is not a $c_0$-quasibarrelled space. For each non-metrizable compact space $K$, the space $L(K)$ is a $(df)$-space but is not a quasi-$(DF)$-space. If $X$ is a $\mu$-space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford--Pettis property for every Tychonoff space $X$. If $X$ is a sequential $\mu$-space (for example, metrizable), then $L(X)$ has the sequential Dunford--Pettis property iff $X$ is discrete.