Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space.
Now suppose that A is a topological abelian group (if necessary, we can assume it to be Hausdorff or locally compact). Is there a natural topology on $A \otimes R$ such that it becomes a topological vector space?
If this tensor product turns out to be not unique, I want it to have the property in the last paragraph of this question (a specific example regarding completions).
With natural topology I mean the following: given a bilinear, jointly continuous (if we can achieve only separately continuous here, it would be better) map $\varphi\colon A \times R \to V$, where $V$ is a topological R-vector space (or, if needed, we can assume that V is a locally convex vector space or even normed), then there is a unique continuous map $\Phi\colon A \otimes R \to V$ with the usual universal property relating it to $\varphi$.
To give an example, consider the subgroup A of the infinite product $\prod Z$ of copies of the integers Z consisting of all the bounded integer sequences, equipped with the discrete topology. Then I want the tensor product $A \otimes R$ to have such a topology that the completion of it equals $\ell^\infty$. This should be a special case of the tensor product whose existence I need (and I really need this tensor product not only for the discrete, torsion free case, but in general).