# Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$.

In this question, it is proven that the canonical map $$\Gamma(E)\otimes_{C^\infty(M)}\Gamma(F)\rightarrow\Gamma(E\otimes F)$$ is an isomorphism of $C^\infty(M)$ modules.

There is also the canonical morphism of $C^\infty(M)$ modules given by $$\Gamma(G^*)\rightarrow Hom_{C^\infty (M)}(\Gamma(G),C^\infty(M)),\omega\mapsto[X\mapsto \omega(X)]$$

Is this map an isomorphism, too? If not, is it an iso under certain conditions, for example if $M$ is compact?

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It is. The inverse is given by evaluation; given $\phi\in Hom_{C^\infty (M)}(\Gamma(G),C^\infty(M))$, define $\tilde\phi\in \Gamma(G^*)$, by $\langle\tilde\phi(x), g\rangle=\langle \phi, \tilde g\rangle,$ where $g\in G_x$ and $\tilde g\in \Gamma(G)$ is any section whose value at $x$ is $g$. –  Gil Bor May 25 '14 at 16:38
Yes it is, show locally and glue by partition functions. –  apurv May 25 '14 at 16:58