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

TL;DR

This paper provides simple criteria to identify neighborhood bases for complete vector topologies and applies these to construct broad classes of complete topological vector spaces of measurable functions.

Contribution

It introduces accessible criteria for neighborhood bases and demonstrates their use in building general complete topological vector spaces of measurable functions.

Findings

Established easy-to-verify criteria for neighborhood bases.

Constructed general classes of complete topological vector spaces.

Applied criteria to spaces of measurable functions.

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.