On the stone-weierstrass theorem

Abstract

Let M be a vector subspace of the Banach space C(Ω) of all real-valued continuous functions on a compact space Ω, and suppose that M contains a subset L and the constant functions. Then in order that L be dense in M it is necessary and sufficient that L, M satisfy a filtering property and that each m in Ω can be approximated on every two points of Ω by functions in L. © 1976, Australian Mathematical Society. All rights reserved.