Net convergence structures with applications to vector lattices
Abstract
Convergence is a fundamental topic in analysis that is most commonly modelled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.
Cite
@article{arxiv.2103.01339,
title = {Net convergence structures with applications to vector lattices},
author = {M. O'Brien and V. G. Troitsky and J. H. van der Walt},
journal= {arXiv preprint arXiv:2103.01339},
year = {2021}
}