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Topological groups with invariant linear spans. (English) Zbl 1485.46010

Let \(G\) be a topological group that can be embedded into a topological vector space. The authors 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. As the main result, they prove that the topological group \(\mathbb Z^{(A)}\) has invariant linear span (which is isomorphic to \(\mathbb R^{(A)}\)), where \(A\) is an arbitrary non-empty set and \(\mathbb Z^{(A)}\) is the direct sum of \(|A|\)-many copies of the discrete group of integers endowed with the Tychonoff product topology. They also present an example of a topological group which does not possess the mentioned property.

MSC:

46A99 Topological linear spaces and related structures
22A99 Topological and differentiable algebraic systems
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References:

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