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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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    First, you may want to use '\prod' instead of '\otimes', since the former denotes the direct product, whereas the second denotes the *tensor* product, which is definitely not the same thing.2012-07-31
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    For your solution, what you want to do is show that $\mathbb{R}[[t]]$ is a product. You have defined the projection maps, and then you want to show that it satisfies the universal property. If it does, it has to be isomorphic to the direct product. For your direct sum, you have to show that $\mathbb{R}[t]$ is a *coproduct*, so you have to define injection maps.2012-07-31
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    Ah, yes, and with that it all clicked into place. Thank you!2012-07-31
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    Glad to hear it! If you want, you can post your (updated) solution as an answer and then MSE users can give feedback.2012-07-31
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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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