I'm trying to show that there is a bijection between the set of sections of a smooth vector bundle and the set of smooth vector bundle morphisms from the trivial line bundle to .
Since I can't draw a diagram here, the morphisms are given so that (hopefully that makes sense).
Given a section , I get a morphism by .
Given a morphism, I get a section by .
However this does not seem to give a bijection, since sending a morphism to a section and back to a morphism I get for any which doesn't seem right.
Any help would be greatly appreciated, as I have a paper due over Easter.