What is the completed projective tensor product of compactly supported smooth functions on two manifolds? Let $M$ and $N$ be smooth manifolds (not necessarily closed). It is a lovely fact that
$$C^\infty(M \times N) \cong C^\infty(M) \hat{\otimes}_\pi C^\infty(N).$$
See, for the instance, the book ``Topological Vector Spaces, Distributions, and Kernels'' by Francois Tr`eves. He also shows similar statements for distributions and compactly supported distributions.
Is it true or false that
$$C^\infty_c(M \times N) \cong C^\infty_c(M) \hat{\otimes}_\pi C^\infty_c(N)$$
and is there a convenient reference for this fact?
 A: Let $X$ be a smooth compact manifold. You can turn $C_{c}^\infty(X) $ into a Fréchet space by taking as family of semi-norms the suprema of the norms of all partial derivatives. If $X$ is not necessarily compact anymore but admits a countable sequence $K_n$ of compact subsets so that every compact subset of $X$ is contained in at least one $K_n$, then you can  still create, using similar method that what preceeds, a family of semi-norms on $C_{c}^\infty(X) $ turning it into a Fréchet space. Even more : In fact each smooth manifold $X$ of finite dimension can be made into such a increasing union of compact $K_n$. Then endow $X$ with a Riemannian metric $g$ which induces a distance $d(x, y)$, take an $x$ in $X$ (that I assume non empty !) and set $K_n = \{y\in X\;|\;d(x,y)\leq n\}$, and do what preceeds. All of this to say that in most reasonable cases, $C_{c}^\infty(X) $ can be turned into a Fréchet space for a family of semi-norms.
Suppose that $N$ and $M$ are reasonable enough so that $C_{c}^\infty(M) $ and $C_{c}^\infty(N) $ can be turned into Fréchet spaces for respective families $(p_{\alpha})_{\alpha\in I}$ and $(q_{\beta})_{\beta\in J}$ of semi-norms as roughly describe previously. Fix a $(\alpha,\beta)\in I\times J$. Now, for a $\xi\in C_{c}^\infty(M) \otimes C_{c}^\infty(N)$, set $$\pi_{\alpha,\beta} (\xi)= \inf \{ \sum_{i=1}^k p_{\alpha} (m_i) q_{\beta} (n_i) | \xi = \sum_{i=1}^k m_i \otimes n_i \}$$ You can then verify that $(\pi_{\alpha,\beta})_{(\alpha,\beta)\in I\times J}$ is a family of semi-norms on the tensor product $C_{c}^\infty(M) \otimes C_{c}^\infty(N)$. Now $C_{c}^\infty(M) \hat{\otimes}_\pi C_{c}^\infty(N)$ is just the completion of $C_{c}^\infty(M) \otimes C_{c}^\infty(N)$ for the previous family of semi-norms.
Now, thanks to the very construction of $C_{c}^\infty(M) \otimes C_{c}^\infty(N)$, you can show that you that it is sufficient to show the isomorphism $C_{c}^\infty(M \times N) \simeq C_{c}^\infty(M) \otimes C_{c}^\infty(N)$ locally, and then to glue. This allows you to suppose that $M$ and $N$ are open balls of some $\mathbf{R}^n$'s, and then you show the isomorphism explicitely.
