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A characterization of complete topological vector spaces with applications to spaces of measurable functions

José L. Ansorena, Alejandro Marcos

TL;DR

The paper addresses how to certify and construct complete topological vector spaces of function spaces by giving practical criteria for neighbourhood bases at the origin. It develops a cohesive theory connecting neighborhood-basis criteria, $F$-norms and quasi-$F$-norms, and strong nestedness to obtain complete, first-countable topologies (i.e., $F$-spaces) and workable completeness tests. The authors apply this framework to spaces of measurable functions, including $L_0$-type spaces defined by convergence in measure on directed families and Musielak–Orlicz spaces built from modular gauges, proving completeness and metrizability under standard measure conditions. The work unifies non-locally convex settings and provides constructive tools (Luxemburg-type gauges, modular function norms) to build complete spaces from general growth conditions, with concrete instances such as Orlicz and variable-exponent Lebesgue spaces.

Abstract

The aim of this paper is twofold. Firstly, we give easy-to-handle criteria to determine whether a given family of subsets of a vector space is a neighbourhood basis of the origin for a complete vector topology. Then, we apply these criteria to construct quite general complete topological vector spaces of measurable functions.

A characterization of complete topological vector spaces with applications to spaces of measurable functions

TL;DR

The paper addresses how to certify and construct complete topological vector spaces of function spaces by giving practical criteria for neighbourhood bases at the origin. It develops a cohesive theory connecting neighborhood-basis criteria, -norms and quasi--norms, and strong nestedness to obtain complete, first-countable topologies (i.e., -spaces) and workable completeness tests. The authors apply this framework to spaces of measurable functions, including -type spaces defined by convergence in measure on directed families and Musielak–Orlicz spaces built from modular gauges, proving completeness and metrizability under standard measure conditions. The work unifies non-locally convex settings and provides constructive tools (Luxemburg-type gauges, modular function norms) to build complete spaces from general growth conditions, with concrete instances such as Orlicz and variable-exponent Lebesgue spaces.

Abstract

The aim of this paper is twofold. Firstly, we give easy-to-handle criteria to determine whether a given family of subsets of a vector space is a neighbourhood basis of the origin for a complete vector topology. Then, we apply these criteria to construct quite general complete topological vector spaces of measurable functions.

Paper Structure

This paper contains 3 sections, 5 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Complete metric vector spaces
  3. Applications

Key Result

Theorem 2.1

Let $\mathbb{X}$ be a vector space and $\mathcal{B}$ a family of subsets of $\mathbb{X}$. Then, $\mathcal{B}$ is a local basis at the origin of some vector topology on $\mathbb{X}$ if and only if the following properties hold.

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Example 3.1
  • Lemma 3.2
  • ...and 2 more