  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 $0\not \in
A$ 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, $\sum_{a\in F}z_aa\in V$ implies that $z_aa\in W$ 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 Hu\v{s}ek and generalize results by
Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.
