  Given a topological group $G$ that can be embedded as a topological subgroup
into some topological vector space (over the field of reals) we say that $G$
has invariant linear span if all linear spans of $G$ under arbitrary embeddings
into topological vector spaces are isomorphic as topological vector spaces.
  For an arbitrary set $A$ let $\mathbb{Z}^{(A)}$ be the direct sum of
$|A|$-many copies of the discrete group of integers endowed with the Tychonoff
product topology. We show that the topological group $\mathbb{Z}^{(A)}$ has
invariant linear span. This answers a question of D. Dikranjan et al. in
positive.
  We prove that given a non-discrete sequential space $X$, the free abelian
topological group $A(X)$ over $X$ is an example of a topological group that
embeds into a topological vector space but does not have invariant linear span.
