On von neumann's variation of the weierstrass-stone theorem

Abstract

Let X be a compact HausdorfF space and let D(X) be the set of all continuous real-valued functions f defined on X and such that 0 ≤ f(x) ≤ 1, for all x ∊ X. The set D(X) is equipped with the uniform topology. We characterize the uniform closure of subsets A ⊂ D(X) containing 0 and 1 and ϕψ + (1 − ϕ)η, whenever they contain ϕ, ψ and η