Serre-Swan theorem

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

The functor $\mathrm{\Gamma }$ is an equivalence of categories.