Bott periodicity is a theorem about the matrix groups U(n)\mathrm{U}(n) and O(n)\mathrm{O}(n). More specifically, it is about the limiting behaviour as nn \to \infty . For simplicity, we will focus on the case of U(n)\mathrm{U}(n), and describe the corresponding results for O(n)\mathrm{O}(n) at the end.

In these notes, we will formulate the theorem in three different ways — in terms of the groups U(n)\mathrm{U}(n) themselves; in terms of their classifying spaces BU(n)B\mathrm{U}(n); and in terms of topological KK-theory.