Algebras and Lattices

We continue with our study of B(X), the space of bounded real-valued functions on a set X. As we have seen, B(X) is a Banach space when supplied with the norm ∥f∥∞ = supx∈X |f(x)|. Moreover, convergence in B(X) is the same as uniform convergence. Of course, if X is a metric space, we will also be interested in C(X), the space of continuous real-valued functions on X, and its cousin Cb(X) = C(X) ∩ B(X), the closed subspace of bounded continuous functions in B(X). Finally, if X is a compact metric space, recall that Cb(X) = C(X).

But now we want to add a few more ingredients to the recipe: It's time we made use of the algebraic and lattice structures of B(X). In this chapter we will make formal our earlier informal discussions of algebras and lattices. In particular, we will see how this additional structure leads to a generalization of the Weierstrass approximation theorem in C(X), where X is a compact metric space.

To begin, an algebra is a vector space A on which there is defined a multiplication (f, g) ↦ fg (from A × A into A) satisfying

1. (i) (fg)h = f(gh), for all f, g, h ∈ A;

2. (ii) f(g + h) = fg + fh, (f + g)h = fh + gh, for all f, g, h ∈ A;

3. (iii) α(fg) = (αf)g = f(αg), for all scalars α and all f, g ∈ A.

4. […]