Let $S=\mathbb{Z}-\{0\}$. Show the existence or nonexistence of isomorphism between $S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}$ and $\prod_{1}^{\infty}\mathbb{Q}_{i}$ as $\mathbb{Q}$-vector spaces. (Here $\mathbb Z_i=\mathbb Z$ and $\mathbb Q_i=\mathbb Q$ for all $i\ge 1$.)
This is an example used to show that the localization does not commute with infinite products under the natural (canonical) homomorphism. I read this from a lecture note by Ravi Vakil. I think we still need to show these two are really not isomorphic as $\mathbb{Q}$-vector spaces. I tried to follow the hint to consider the element $(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},...)$ but failed.
To user26857: I want a proof of the existence or nonexistence of isomorphism between these two vector spaces. If I get it right, your solution reduced the problem to the existence of basis of these two vector spaces. And in your sense, a basis is a subset $B$ (of the vector space $V$) such that every finite subset of $B$ is a linearly independent set and any vector in $V$ can be expressed as a finite sum of the elements in $B$. If this is what you mean, can you show the existence of such basis? Thank you!
To Ragib Zaman: Your example is a better one. Thanks!