Is the set of smooth sections of a vector bundle a finite dimensional smooth manifold?

Question

I'm confused about whether or not the there is a manifold structure here. The set of linear bundle homomorphisms between two vector bundles over the same base is clearly a vector bundle. If you take the set of linear bundles morphisms from the projection bundle $M \times R \to M$ into any vector bundle $q: E \to M$, this is a vector bundle for the stronger reason above and it looks like it should be isomorphic to the set of sections of $q$.

Answer 1

The smooth sections of the trivial bundle $\mathbb{R}\times\mathbb{R}$ is just the set of all smooth real functions. It is infinite dimensional. Just as linear maps from $\mathbb{R}$ to a vector space is the vector space itself (each map picks a vector), we can say that bundle maps from the trivial vector bundle $\mathbb{R}$ to any bundle $V$ is isomorphic to the bundle itself. And there is a map of sections. But there isn't a map from the set of sections to the bundle, that's mixing categories.