Let $X$ be a compact Hausdorff space. Let $\mathcat{Vec}(X)$ be the category of complex vector bundles over $X$ . And, let $\mathcat{ProjMod}(C(X))$ be the category of finitely generated projective modules over the $C^*$ -algebra $C(X)$ . There is a functor $\Gamma\colon \mathcat{Vec}(X) \to \mathcat{ProjMod}(C(X))$ which sends each complex vector bundle $E \to X$ to the $C(X)$ -module $\Gamma(X,E)$ of continuous sections.

The functor $\Gamma$ is an equivalence of categories.