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As the title saying,

Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?

Here we are considering vector bundles with finite-dimensional fibers.

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Surely this just boils down to the fact that it's true fibrewise? – Zhen Lin Oct 30 '12 at 18:31
I have no idea.. I'm new to this area and and everything is totally a mess to me now..maybe I will read more and think over it again.. – hxhxhx88 Oct 31 '12 at 4:37
Can you say something about the source of this fact? – Fallen Apart May 2 '15 at 14:09
That is true, since the global section is a finitely generated projective module. – Manan Jul 31 '15 at 6:00

As explained in a comment to this question, we have $\Gamma(E)^\vee\cong\Gamma(E^\vee)$, and hence $\Gamma(E)^{\vee\vee}\cong\Gamma(E^{\vee\vee})\cong\Gamma(E)$.

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