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 from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) 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.

给定一个可以作为拓扑子组嵌入到某个拓扑向量空间中（在实数域中）的拓扑组G，我们说，如果任意嵌入到拓扑向量空间中的G的所有线性跨度都是同构的，则G具有不变的线性跨度向量空间。对于任意集A，令\（{{\ mathbb {Z}}} ^ {（A）} \）为|的直接和。一个|赋予了吉洪诺夫积空间离散整数组的-many副本。我们显示拓扑组\（{{\ mathbb {Z}}} ^ {（A）} \）具有不变的线性跨度。这回答了Dikranjan等人的论文中的一个问题。（J Math Anal Appl 437：1257–1282，2016）肯定。我们证明了给定的非离散顺序空间X，自由阿贝尔拓扑群甲（X）超过X是一个拓扑组嵌入到拓扑向量空间，但不具有不变的线性跨度的一个例子。