# projective tensor product of smooth functions

Let $M$ and $N$ be compact smooth manifolds. Is it true, that $C^{\infty}(M) \otimes_{\pi} C^{\infty}(N) \cong C^{\infty}(M \times N)$ as topological vectorspaces if endowed with the familiy of seminorms given by the suprema of differentials of the function over the respective spaces? If yes, is there a reference or easy explanation for this?

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This is indeed true and can be found in many texts,e.g., "Topological vector spaces, distributions and kernels" by F Treves, but also in Köthe's two volume treatise or Pietsch' "Nuclear spaces". If you are interested in the original citation, it is probably "Espaces de fonctions différentiables a valeurs vectorielles" Jour. d' Anal. Math. 4 (1954-6) 88-148 by L. Schwartz. Such topics are also discussed in detail in Grothendieck's thesis (appeared in the series "Memoirs of the American Math. Soc.") Warning: some of these texts might confine themselves to the case of subsets of euclidean space rather than the abstract setting of smooth manifolds---I have no access to the sources to check this at the moment---but the transition from the concrete setting to the abstract one is a standard partition of unity argument.

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