The goal of K-theory is to study and understand a topological space X by associating to it a sequence of abelian groups. The algebraic properties of these groups reflect topological properties of X, and the overarching philosophy of K-theory (and, indeed, of all algebraic topology) is that we can usually distinguish groups more easily than we can distinguish topological spaces. There are many variations on this theme, such as homology and cohomology groups of various sorts. What sets K-theory apart from its algebraic topological brethren is that not only can it be defined directly from X, but also in terms of matrices of continuous complex-valued functions on X. For this reason, we devote a significant part of this chapter to the study of matrices of continuous functions.

Our first step is to look at complex vector spaces equipped with an inner product. The reader is presumably familiar with inner products on real vector spaces, but possibly not the complex case. For this reason, we begin with a brief discussion of complex inner product spaces.

Complex inner product spaces

DefinitionLet V be a finite-dimensional complex vector space and let ℂ denote the complex numbers.