Proof

Here is the one-to-one correspondence. Let Form be the family of 1-forms on M (or U) and let Cov be the family of covariant vector fields on M (or U). Define

by

In the homework, we see that CkVk is indeed a scalar by checking the transformation rule:

The linearity property of now follows from the distributive laws of arithmetic. We now define the inverse

by

We need to check that this is a covariant vector field; that is, that it transforms in the correct fashion. But, it x and are two charts, then

That and are in fact inverses is left as an exercise!

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