2015/08/04 by Dikran Dikranjan, Dmitri Shakhmatov, Jan Spěvák · 1 citation
Mathematics · #math.FA #math.GN #math.GR #msc:20K25 #msc:22A05 #msc:46A11 #msc:46A16 #msc:46A30 #msc:46A35
paper · pdf · doi:10.1016/j.jmaa.2016.01.037
published as J. Math. Anal. Appl. 437 (2016) 1257-1282
arxiv created 2015/08/04 · arxiv updated 2016/03/28
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⊆ A such that the subgroup generated by A∖ F is contained in U; (ii) absolutely summable if, for every family \za:a∈ A\ of integer numbers, there exists g∈ G such that the net \∑a∈ F za a: F⊆ A is finite\ converges to g; (iii) topologically independent provided that 0\not ∈ A and for every neighbourhood W of 0 there exists a neighbourhood V of 0 such that, for every finite set F⊆ A and each set \za:a∈ F\ of integers, ∑a∈ Fzaa∈ V implies that zaa∈ W for all a∈ F. We prove that: (1) an abelian topological group contains a direct product (direct sum) of κ-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality κ; (2) a topological vector space contains ℝ(ℕ) as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains ℝℕ as its subspace if and only if it has an ℝ(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.