# 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 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?

• Could you please give me the page of Trèves's book where this is stated ? Commented Jan 30, 2015 at 13:39
• See theorem 51.6, on page 530 of the Dover reissue. Note that he states the desired isomorphism for compactly supported functions when the manifolds are compact. Commented Jan 31, 2015 at 11:11
• Thx for the page reference, will have a look at it and will come back to you Commented Jan 31, 2015 at 12:10
• Afair, this is false already for $M=\mathbb R$. This was known by Grothendieck and might be the principal reason for the introduction of inductive tensor products. Commented Mar 17, 2023 at 8:14

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.
• If $M$ is not compact, the topology on $C^\infty_c(M)$ that I mean is not the Frechet topology inherited from all smooth functions. It is the topology arising as a colimit of the compactly supported smooth functions over an exhausting sequence of compact subsets of $M$. This topology is not Frechet; Treves uses the term ''LF topology'' and describes it in chapter 13, pages 131-133. Commented Jan 31, 2015 at 11:16