Topologies with the Stone-weierstrass

The two main results in this paper are analogues of the Stone-Weierstrass theorem for real-valued functions, obtained by using different function space topologies. The first (Theorem 2.3) is a Stone-Weierstrass theorem for unbounded functions. The second (Theorem 3.6) is a theorem for bounded functions ; it is stronger than the usual theorem because the topology is larger than the uniform topology. In fact it is the strongest possible theorem in a sense made precise in Theorem 3.5. However, the usefulness of the result is, in both cases, limited by the fact that the topologies used are far from being as well behaved as the uniform topology. Some specific defects of these topologies are discussed in Proposition 2.4, Proposition 3.9, and Example 3.10. (The topologies are defined in §1.) The formulation of the Stone-Weierstrass theorem on which the analogues in the present paper are based is the following: If A' is a completely regular topological space, then X is compact iff every strongly separating subalgebra of C*{X) is «-dense, where u denotes the uniform topology. In [5] Lorch has studied a more general class of function algebras than those of the form C*(X). All of the results for bounded functions in this paper are formulated in this more general setting (see §1).