Tensor product of topological abelian groups with the reals

Question

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).

Answer 1

Abelian groups are exactly $Z$-modules ($Z$ - the integers), so it would be better to write $A\otimes_ZR$ for this. Then $A\mapsto A\otimes_Z R$ which the topology you look for is the left adjoint functor from the category of abelian topological groups to the category of topological vector spaces to the forgetful functor which associates to each topological vector space $V$ the topological abelian group $(V,+)$. This suggests a description of the topology: It is the finest one which satisfies your requirement. I guess that some book by Horst Herrlich on category theory contains a more detailed answer.