A remark in Nelson's "Tensor Analysis" implies there are no non-trivial linear functionals on the module of continuous vector fields on a manifold, when considered as a module over the ring of smooth real-valued functions. This is believable, but I can't see how to get a handle on this. It seems as though this could either be obvious, or deep! :(

A remark in Nelson's "Tensor Analysis" implies there are no non-trivial linear functionals on the module of continuous vector fields on a manifold, when considered as a module over the ring of smooth real-valued functions. This is believable, but I can't see how to get a handle on this. It seems as though this could either be obvious, or deep!