Dominated Convergence and Stone-Weierstrass Theorem

Abstract

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)| ≤ εn|h(x)| for allx ∈ X and all n ∈ N where gn, g, h ∈ A and εn → 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.

Stone-Weierstraß Theorems for Riesz Ideals of Continuous Functions

“…As M a is essentially self-adjoint, [c]is also an element of the domain of the closure of M a , so there exists a sequence[b n ] n∈AE in A G Φ that is • Φ -convergent against [c] and such that [ab n ] n∈AE is a • Φ -Cauchy sequence. Consequently, a corresponding sequence of representatives (b n ) n∈AE in A is • Φ -convergent against c and (ab n ) n∈AE is a • Φ -Cauchy sequence,hence • Φ -convergent against ac by the previous Lemma 6 1…”

“…Then by the previous Lemma 5.1, there also exists a function h ∈ I + such that for all ǫ ∈ ]0, ∞[ there exists a k ∈ AE fulfilling f k ≤ ǫh. This function thus has the property required in the second point.Applying this characterization of strictly I-convergent sequences (hence of I-closed sets) to Theorem 4.3 yields essentially the Stone-Weierstraß-type theorem from[1, Thm. 4.1].…”