A description of the topology of free topological vector spaces

TL;DR

This paper provides a straightforward description of the topologies of free topological vector spaces and free locally convex spaces over Tychonoff spaces, including the special case of pseudocompact spaces.

Contribution

It offers a simplified characterization of the topologies of free topological vector and locally convex spaces over Tychonoff and pseudocompact spaces.

Findings

Provided a simple description of the topology of $ ext{V}(X)$ and $L(X)$.

Extended the analysis to pseudocompact spaces.

Enhanced understanding of free topological vector space structures.

Abstract

We give a simple description of the topology of free topological vector space $\mathbb{V}(X)$ and the topology of the free locally convex space $L(X)$ over a Tychonoff space $X$. The case when $X$ is a pseudocompact space is also considered.