@Steven Landsburg: Yes, your representation of the elements of $R$ is efficient and absolutely correct.

Dear @YCor: Apologies for my ambiguous formulation. I have made an edit making what I meant more explicit, since I want the fraction you mention to belong to $R$

Thanks a lot for your remarkable work, David. Your extracting an answer to my question so quickly from Nagata's hard to read aricle (and idiosyncratic terminology) is a real *tour de force" .

Thanks again for your clear explanations, LSpice. I have upvoted your answer and your comments.

Dear LSpice, thank you for your explanations. I know nothing about topological vector spaces but I appreciate your point that if I did I would have immediately realized that the question is quite easy, whereas in reality I spent much time coming up with a solution.

OK, I agree, what you write makes sense. However I don't know about weak topology and I thank you for your definition which doesn't make reference to topology.

Yes, it is the same idea, but I posted my answer without seeing yours. The technical details are different and I find your formulation indeed simpler. Also I wrote a comment to your post which you should take into account.

Your symbol $\sum_{b \in \mathcal B} e_b$ does not make sense because you cannot sum infinitely many non-zero vectors in a vector space.

@user131781 Of course we can get by without topology: already two answers show this, just 17 minutes after the question was posted!