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A description of the topology of free topological vector spaces

Saak Gabriyelyan

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.

A description of the topology of free topological vector spaces

Abstract

We give a simple description of the topology of free topological vector space and the topology of the free locally convex space over a Tychonoff space . The case when is a pseudocompact space is also considered.

Paper Structure

This paper contains 1 theorem, 11 equations.

Key Result

Theorem 1

Let $X$ be a Tychonoff space. Then the family forms a neighbourhood base at zero of $\mathbb{V}(X)$, and the family where $\mathrm{conv}(W)$ is the convex hull of $W$, is a base at zero of $L(X)$. Moreover, if $X$ is pseudocompact, then all functions $\varphi_n$ can be chosen to be constant.

Theorems & Definitions (2)

  • Theorem 1
  • proof