Discrete subgroups of normed spaces are free

TL;DR

This paper proves that all discrete subgroups of the additive group of a normed space are free Abelian, removing previous restrictions on their cardinality by employing set-theoretic methods.

Contribution

It extends prior results by showing that every discrete subgroup of a normed space's additive group is free Abelian without size limitations, using elementary submodels and the Singular Compactness Theorem.

Findings

All discrete subgroups are free Abelian.

Removed cardinality constraints from previous theorems.

Applied set-theoretic methods to functional analysis.

Abstract

Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set-theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.