Similarly to my previous question about [direct limits][1], I have now basically the same question about inverse limits. It seems in fact, that I only need the result for products.

Question: Is there a natural smooth structure on $\prod \mathbb{R}$ such that $\mathcal{C}^\infty(U,\prod \mathbb{R}) = \prod\mathcal{C}^\infty(U,\mathbb{R})$?


  [1]: https://mathoverflow.net/questions/90974/does-direct-limit-commute-with-functor-of-smooth-sections