I was reviewing a definition of a particular integral and was struck for the first time that it actually described a duality pairing between a vector space of integrable functions and a vector space of functions of bounded variation. This is probably old news to people who've taken a differential geometry course, but learning it on my own it was a bit of a revelation.
With that in mind it seemed clear to me that the same sort of pairing would also happen for a vector space of (some suitable class of) integrable functions on a manifold and the space of differential forms on the manifold.
In Terrence Tao's pdf Differential forms and integration he says
We have seen that integration is a duality pairing between manifolds and forms.
It would be easy for me to think that perhaps he is talking about the same thing I was just describing, but then again I also believe Terrence Tao is probably pretty accurate with his words.
Is there actually a duality between manifolds and forms?
Manifolds apparently don't form a vector space, so it doesn't fit my understanding of what a duality pairing is. But perhaps someone can explain that I'm not using the right definition of "pairing" or that Tao already meant what I described before.
Or maybe I even have the wrong end of the stick with what I described before. I'm not sure that one can specify ahead of time a useful space of functions to be paired with any single differential form. Maybe the bilinearity properties of $B(f,g)=\int_M f dg$ only hold on a case-by-case basis, and not on entire vector spaces.