Direct sums and products in topological groups and vector spaces

Abstract

We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\subseteq A$ such that the subgroup generated by $A\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\{z_a:a\in A\}$ of integer numbers, there exists $g\in G$ such that the net $\left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\notin A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $F\subseteq A$ and each set $\{z_a:a\in F\}$ of integers, ${\sum }_{a\in F}{z}_{a}a\in V$ implies that $z_{a}a\in W$ for all $a\in F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $\kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $\kappa$; (2) a topological vector space contains ${\mathbb{R}}^{(\mathbb{N})}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains ${\mathbb{R}}^{\mathbb{N}}$ as its subspace if and only if it has an ${\mathbb{R}}^{(\mathbb{N})}$ multiplier convergent series of non-zero elements. We answer a question of Hušek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.

References in corpus (2)

• Productivity of sequences with respect to a given weight function
• Characterizing Lie groups by controlling their zero-dimensional subgroups

Cited by in corpus (1)

• Topologically independent sets in precompact groups