I have two questions, really:
Q1) Suppose $M$ is a noncompact, finite-dimensional smooth manifold. Is it true that $C^{\infty}(M)$, by which I will always mean $C^{\infty}(-; \mathbb{R})$, is a nuclear Frechet space?
Reason for asking: if you quickly google for ``examples of nuclear spaces'', it's usually cited that $C^{\infty}(M)$ is nuclear if $M$ is compact. But compactness seems like rather serious overkill; indeed, Treves, Theorem 51.5 + subsequent Corollaries, suggests $C^{\infty}(U)$ is nuclear for $U \subset \mathbb{R}^n$ open. In this case, I would suspect that if, say, $M$ may be covered by finitely many charts (as is certainly the case if $M$ is compact but is true far more generally), then $C^{\infty}(M)$ is nuclear by embedding $C^{\infty}(M) \hookrightarrow \prod C^{\infty}(U_{\alpha})$ as a closed subspace for $\{U_{\alpha}\}$ a finite open cover of $M$ by charts. This same embedding trick seems to suggest that $C^{\infty}(M)$ is nuclear far more generally still -- in which case, what are the natural conditions on $M$ such that this holds? (e.g., paracompactness, second countability, ...?)
Q2) And then secretly my real reason for asking is to know conditions on manifolds $M, N$ such that $C^{\infty}(M \times N) \simeq C^{\infty}(M) \otimes C^{\infty}(N)$ for some reasonable tensor product. Indeed, if $\otimes$ is taken to be the projective tensor product $\hat{\otimes}_{\pi}$, this statement seems to be commonly asserted - and I think I understand this in the case that at least one space is nuclear so that, e.g., projective and injective tensor products agree. But, right, does this hold for reasonably general noncompact manifolds $M, N$? How about if $M, N$ are in some reasonable class of infinite-dimensional manifolds?
