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⊆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∈Fzaa:F⊆A is finite} converges to g; (iii) topologically independent provided that 0∉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 RN multiplier convergent series of non-zero elements. We answer a question of Hušek and generalize results by Bessaga–Pelczynski–Rolewicz, Dominguez–Tarieladze and Lipecki.