Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.