# How to view $V^*$ as the space of $C^{\infty}(M)$-linear functionals from $V$ to $C^{\infty}(M)$

Suppose $V$ is a vector bundle and $V^*$ is its dual bundle.

I was told that we can view $V^*$ as the space of $C^{\infty}(M)$-linear functionals from $V$ to $C^{\infty}(M)$.

Can anyone show me how?

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Well, the vector bundle corresponding to $C^\infty(M)$ is just the trivial bundle $M \times \mathbb{R}$, and $V^*$ is more-or-less by definition $\textrm{Hom}_M(V, M \times \mathbb{R})$. –  Zhen Lin Nov 13 '12 at 19:22
@ZhenLin, oh, I used to think it is $Hom_M(V,\mathbb{R})$. I should notice that $V$ is a bundle. Thank you. –  hxhxhx88 Nov 14 '12 at 5:04