The Weierstrass Theorem

While we now know something about convergence in C(X), there are many more things that we would like to know about C(X). We will find the task unmanageable, however, unless we place some restrictions on the metric space X. If we focus our attention on the case when X is compact, for example, we will be afforded plenty of extra machinery: In this case, C(X) is not only a vector space, an algebra, and a lattice (where algebraic operations are defined pointwise), but also a complete normed space under the supnorm. With all of these tools to work with, we will be able to accomplish quite a bit. And at least a few of our results will apply equally well to the space Cb(X) of bounded continuous functions on a general metric space X. For the remainder of this chapter, then, unless otherwise specified, X will denote a compact metric space.

We will concentrate on two questions in particular, and each of these will lead to some interesting applications:

• Is C(X) separable? More importantly, are there any “useful” dense subspaces, or even dense subalgebras, or sublattices of C(X)?

• What are the compact subsets of C(X)? And are such sets “useful”?

Either question is tough to answer in full generality, but the first one has a very satisfactory and easy to understand answer for C[a, b].