This question is crossposted from 1.
Carroll writes in his Spacetime and Geometry book on page 68 that
"[...] in fact, however, we could just as well have begun with an intrinsic definition of one-forms and used that to define vectors as the dual space. Roughly speaking, the space of one-forms at $p$ is equivalent to the space of all functions that vanish at $p$ and have the same second partial derivatives. In fact, doing it that way is more fundamental, if anything, since we can provide intrinsic definitions of all $q$-forms (totally antisymmetric tensors with $q$ lower indices), etc."
Could somebody explain this equivalency in a bit more detail?
When I asked this question on 1 I got several replies but the more I thought of the consensus answer, namely that it must be a "typo" the less satisfied I was. So I wrote to Prof. Carroll and asked him directly but his answer was quite laconic and cryptic "Nope, it is correct. If the first derivatives were the same, they'd be the same one-forms."
Now I am totally confused, and I have another question: what does Carroll mean by this?