I've been reading Swan's paper on vector bundles over compact Hausdorff spaces and projective modules. Since this is my first time learning about the topic, I'd appreciate if anyone can explain to me a few details about his proof. My questions are as follows
In page 265 he notes every section can be written as a combination of sections and scalars which depend on elements of the base space. $s(y)=\sum a_i(y) s_i(y)$. Can someone perhaps elaborate on his argument that a section is continuous if and only if each a_i is continuous?
The second question is how is $\Gamma$ an additive functor. I can't seem to find a proof of this nor be able to come up with one myself.
In corollary 5 in page 268. Why is it clear that $\xi ' \cong \xi$?
In Propositon 1 on page 265. Why is (3) equivalent to (4)?
In corollary 3 on page 267. I'm able to understand the function $s \mapsto s(x)$ is bijective but I can't catch as what structure they are isomorphic as.