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

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