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I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here
I have spelled out further elementary detail at topological vector bundle.
In (what is now) the section Transition functions I have added a detailed argument that the thing which is glued from the transition functions of a vector bundle is indeed isomorphic to that vector bundle.
Then in (what is now) the section Basic properties I have spelled out a detailed proof that a homomorphism of topological vector bundles is an isomorphism as soon as it is a fiberwise linear isomorphism.
(I was trying to be really explicit, maybe in contrast to what Hatcher offers. The only thing I should still add for completeness is at general linear group the statement that the inclusion $GL(n,k) \hookrightarrow Maps(k^n, k^n)$ into the mapping space with its compact-open topology is continuous.)
The only thing I should still add for completeness is at general linear group the statement that the inclusion GL(n,k)↪Maps(kn,kn) GL(n,k) \hookrightarrow Maps(k^n, k^n) into the mapping space with its compact-open topology is continuous.)
I wonder if one can see this using the fact $GL(n,k)$ is an open subspace of $End(k^n)$, $End(k^n) \simeq k\otimes k*$, and the resulting linear map $k \to k\otimes Maps(k^n,k^n)$. Here we’d need $Maps(k^n,k^n)$ as a topological vector space. But, hmm, what sort of fields $k$ are you allowing? Just $\mathbb{R}$ and $\mathbb{C}$? (and perhaps $\mathbb{H}$…)
Ah, I see you did this on the other thread!
I have spelled out in some detail the proof that topological vector bundles are classified by the relevant Cech cohomology: here.
I have spelled out more statements and proofs in the section Over closed subspaces
Sorry for not reacting earlier. I’d rather we fix an explanation than just removing it. So I have made it this:
(like every finite dimensional vector space, by the Heine-Borel theorem)
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