# Serre-Swan Theorem and some K groups

This is an overview on Serre-Swan theorem and some ideas on the construction of K-groups for a Banach category. Serre-Swan theorem establishes equivalences between the categories of topological vector bundles over a compact Hausdorff space $X$, the category of finitely generated projective $C(X)$-modules and the categories of algebraic vector bundles of finite rank over
the affine scheme $\mathrm{Spec}C(X)$. This theorem connects different objects of interest in K-theory.
It also introduces some ideas on the construction of K-groups for a Banach category and
in particular for compact topological spaces and Banach algebras.