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The problem is to show that in the category of real vector spaces the direct product of countably infinitely many $\mathbb{R}$ is isomorphic to $\mathbb{R}[[t]]:= \sum\limits_{j=1}^\infty a_j t^j, \,\,a_j\in \mathbb{R}$, and the direct sum to polynomials $\mathbb{R}[t]$.

It's not that I don't understand the problem, or that it's not obvious to me; what I am is very uncomfortable with category theory, and pessimistic regarding my attempt to do it in a category-theoretic manner. So here goes: defining the projection maps thus: $\pi_i: \mathbb{R}[[t]] \longrightarrow \mathbb{R}, \quad \sum\limits_{j=1}^\infty a_j t^j \longmapsto a_i,$ $p_i: \prod\limits_{j=1}^\infty \mathbb{R} \longrightarrow \mathbb{R}, \quad (a_1, \cdots,a_j,\cdots)\longmapsto a_i,$ by the universal property of products, $\mathbb{R}[[t]] \cong \prod\limits_{j=1}^\infty \mathbb{R}$. The case of $\mathbb{R}[t] \cong \bigoplus\limits_{j=1}^\infty$ follows from this when all but finitely many of the $a_i = 0$.

Is this wrong? Should I not have done this?

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    @MTurgeon The OP does not seem to answer. Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-13

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To complete the solution in the product case, you have to show "uniqueness up to unique isomorphism".

For $\mathbb R[t]$, what you want to show, is that it is a coproduct. Therefore, you have to give maps $ \iota_j: \mathbb R \to \mathbb R[t], $ and then again you show that $\mathbb R[t]$ and these injections $\iota_j$ gives you a coproduct. This will imply that $\mathbb R[t]\cong\oplus_{i=1}^\infty\mathbb R$.