What is the completed projective tensor product of compactly supported smooth functions on two manifolds?

Question

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 Trèves (Thm.51.6 on p.530). 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?

Answer 1

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.