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?
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 (19546) 88148 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 manifoldsI have no access to the sources to check this at the momentbut the transition from the concrete setting to the abstract one is a standard partition of unity argument. 

