Show that $TM$ is bundle isomorphic to $TM^{*}$ for a smooth manifold $M$
It is straight-forward to prove this isomorphism using a Riemannian metric, but suppose we did not have this tool.
As a first attempt at this problem, we could try and define a bundle homomorphism fibre-wise by declaring that given a basis $\displaystyle \left\{\frac{\partial}{\partial x^i}\right\}_{i = 1}^{n}$ for $T_pM$ and a basis $\{dx^i\}_{i = 1}^n$ for $(T_pM)^*$, the basis element $\displaystyle \frac{\partial}{\partial x^i}$ will be sent to $dx^i$. The main problem with this is that this is a coordinate-dependent map, dependent on the coordinates $x^i$, and hence this map may not be well-defined if we used a different set of coordinates.
There is apparently a way to resolve this problem by considering partitions of unity. I was wondering how partitions of unity would be involved and how they would help define a smooth bundle isomorphism between $TM$ and $TM^*$.