Similarly to my previous question about direct limits, 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 $\mathcal{C}^\infty(U,\prod \mathbb{R}) = \prod\mathcal{C}^\infty(U,\mathbb{R})$?