I have read that if two smooth vector bundles are isomorphic in the category of continuous vector bundles, then they are actually isomorphic in the category of smooth vector bundles.
Can someone give me an idea of the proof and a reference for this question and related ones? I am trying to think to partition-of-unity- or collar-like-arguments, but I have nothing precise in mind.
Another question, a little stronger: is the subcategory of smooth vector bundles fully embedded into the category of continuous vector bundles? This would imply the latter, but I don't know whether it's true.
Thank you in advance.
This didn't fit into a comment.
Let $BO(n)$ denote the Grassmannian of $n$ planes in $\mathbb{R}^\infty$. This is not itself a manifold as it is infinite dimensional. But it is filtered by finite dimensional manifolds: The Grassmannians of $n$-planes in $\mathbb{R}^N$. Any map from a compact space will land into a finite dimensional Grassmannian. Since smoothness is a local property, this suffices to speak of smooth maps into $BO(n)$ from any manifold.
One way to see that continuous vector bundles are isomorphic to smooth ones is that smooth rank-$n$ vector bundles on $M$ are classified by smooth homotopy classes of smooth maps $M\rightarrow BO(n)$ , while continuous vector bundles are classified by continous maps $M\rightarrow BO(n)$. Now any continuous map is homotopic to a smooth map, which shows that every continuous vector bundle is isomorphic to a smooth one.
I do not have a reference.