I have two question. Let $M$ and $N$ two compact manifolds.
1) It is true that $C^{\infty}(M\times N)\cong C^{\infty}(M)\otimes C^{\infty}(N)$??.
2) Taking $f$ $\in$ $\Omega^1(M)=\Gamma(T^{\ast}M)$ (differential 1-forms), I want to know if it is true that $$f=\sum^n_i h_idg_i $$ for some $h_i$, $g_i$ $\in$ $C^{\infty}(M)$. Locally, it is clear but I want to know if the compactness property can do it globally.
1) See janacek (https://mathoverflow.net/users/49628/janacek), projective tensor product of smooth functions, URL (version: 2014-04-26): https://mathoverflow.net/q/164443
2) Use a standard partition of unity argument to paste the local representations together. Since the manifold is compact, you can take a finite partition of unity and thus get the desired finite sum.