Dominated Convergence and Stone-Weierstrass Theorem

Abstract

Let C(X;R) the algebra of continuous real valued functions defined on a locally compact space X. We consider linear subspaces A ⊂ C(X;R) having the following property: there is a sequence (Φj)j∈N of positive functions in A with limx→∞ Φj(x) = +∞ for every j ∈ N, such that A consists of functions f ∈ C(X;R) bounded above for the absolute value by an homothetic of some Φn (n depends on each f). Dominated convergence of a sequence (gn)n≥1 in A is an estimation of the form |gn(x) − g(x)| ≤ en|h(x)| for all x ∈ X and all n ∈ N where gn, g, h ∈ A and en → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices B ⊂ A and we show that the dominated convergence for sequences is exactly the convergence of sequences when A is endowed with a locally convex (DF)-space topology.