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 , the category of finitely generated projective -modules and the categories of algebraic vector bundles of finite rank over
the affine scheme . 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.
This is my course thesis for Algebraic Topology II at UIUC.