Question: Any 1-form can be expressed as a finite sum $\displaystyle \sum_{i} f_i dg_i$ for some smooth functions $f_i$ and $g_i$.
My attempt:
Let $M$ be an $n$-dimensional smooth real manifold. Let $\{(U_i,\phi_i)\}_{i \in I}$ be an atlas for $M$. Let $\{\psi_i\}_{i \in I}$ be a partition of unity subordinate to $\{U_i\}_{i \in I}$. Let $\omega \in \Omega^1(M)$ be a $1$-form on $M$. Then, on $U_i$, $\omega$ can be expressed as $$\omega_i=\displaystyle \sum_{j}X^j dx_{i}^j,$$ where $\{X^j\}_{j=1}^n$ are smooth functions and $\{x_{i}^j\}_{j=1}^n$ are local coordinates on $U_i$. Now, globally, we can write $$\omega=\displaystyle \sum_{i} \psi_i \omega_i.$$
Then we have, $$\omega=\displaystyle \sum_{i}\left(\displaystyle \sum_{j}\psi_i X^j\right)dx_{i}^j.$$ Rewrite it as $$\omega=\displaystyle \sum_{i}\left(\displaystyle \sum_{j}\psi_i X^j\right)\left(\displaystyle \sum_{i}\psi_i dx_{i}^j\right).$$
Let $f_i=\displaystyle \sum_{j}\psi_i X^j$. If we have $dg_i=\displaystyle \sum_{i}\psi_i dx_{i}^j$, then we are done. So, we want to find such $g_i$.
I am not able to proceed further. Any help is much appreciated.
Best Answer
Let me assume that $M$ is compact. Take a local coordinate patch $U$ with coordinates $x^i$ taking values in unit ball. Let $V \subset U$ be the set of points with coordinates such that $\sum_i (x^i)^2 < \frac{1}{2}$. We can find functions $y^i$ which are equal to $x^i$ on $V$ and can be extended to whole $M$ (we can for example assume that they vanish for $\sum_i (x^i)^2 > \frac{3}{4}$). Thus $y^i$ are globally smooth and have the property that at every point of $V$ differential forms $dy^1,...,dy^n$ span the cotangent space. Now any point $p$ of $M$ has a neighbourhood $V_p$ which admits functions $y^i_p$ as above. By compactness, we can find $p_1,...,p_m$ such that $V_{p_1},...,V_{p_m}$ cover $M$. Then all $\{ y_{p_1}^1,...,y_{p_1}^n, ..., y_{p_m}^1,...,y_{p_m}^n \}$ are smooth functions defined on whole $M$. We will take them as our $g_i$. Clearly at every point of $M$ we have that $dg_i$ span the cotangent space. From here it should be easy to finish the proof using partitions of unity.
In the non-compact case I doubt you can always find a finite family of $f_i, g_i$ which works, but I'm pretty sure you can find one such that any point has a neighbourhood in which only finitely many $f_i$ and $g_i$ are nonzero. I will leave the exercise to you; I think you can proceed as I did but invoking paracompactness (instead of compactness) to construct the $g_i$. I have not worked out the details though.