Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras

Abstract

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.