Suppose *M* is a compact C^{∞}-manifold, and a smooth vector bundle *V* is given on *M*. The space of smooth sections of *V* is then a module over C^{∞}(*M*) (the commutative algebra of smooth real-valued functions on *M*). Swan's theorem states that this module is finitely generated and projective over C^{∞}(*M*).

Even more: *every* finitely generated projective module over C^{∞}(*M*) arises in this way from some smooth vector bundle on *M*, in essentially only one way. More precisely: the category of smooth vector bundles on *M* is equivalent to the category of finitely generated projective modules over C^{∞}(*M*).

Suppose *X* is a compact Hausdorff space, and C(*X*) is the ring of continuous real-valued functions on *X*. Analogous to the result above, the category of real vector bundles on *X* is equivalent to the category of finitely generated projective modules over C(*X*).