>If I write a sigma to sum some vectors, it doesn't matter if the vectors are coordinates in 3D space, functions defined on an interval, or just numbers.

you're wrong. every "weird" technical hypothesis in a theorem is a leak. even in your super simple example there's a caveat: the sum has to be finite (that mean no you can't perform any sum of any set of vectors).

edit: i got downvoted so here's a better example that i just ran across: manifolds with boundary can have empty boundaries. where does this come up? exact 1-forms on S^1 integrate to zero. why? because by Stokes. how? S^1 has a boundary but it's empty.