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# Serre-Swan theorem

Let $X$ be a compact Hausdorff space. Let $\mathord{\mathbf{Vec}}(X)$ be the category of complex vector bundles over $X$. And, let $\mathord{\mathbf{ProjMod}}(C(X))$ be the category of finitely generated projective modules over the $C^{*}$-algebra $C(X)$. There is a functor $\Gamma\colon\mathord{\mathbf{Vec}}(X)\to\mathord{\mathbf{ProjMod}}(C(X))$ 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.

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Theorem

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## Mathematics Subject Classification

46L85*no label found*

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