The space of smooth functions is a vector bundle

Question

I am reading these lecture notes on Calabi-Yau manifolds and, at the beginning of page 4, the author says that space $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a vector bundle. I really do not understand why this is true. I do not even know how to define a projection map $C^\infty(M)\longrightarrow M$. Can you help me?

Answer 1

The author is being imprecise. They're alluding to the Serre-Swan theorem, which gives for a smooth connected manifold $M$ an equivalence of categories between smooth vector bundles on $M$ and finitely generated projective $C^{\infty}(M)$-modules with the equivalence given by taking the space of smooth sections. $C^{\infty}(M)$ itself is the space of smooth sections of the trivial line bundle $\mathbb{R}$; that's the bundle the author means.