Locally convex Riesz spaces and Archimedean quotient spaces

Abstract

A Riesz space with a Hausdorff, locally convex topology determined by Riesz seminorms is called a locally convex Riesz space . A sequence {xn} in a locally convex Riesz space L is said to converge locally to x ϵ L if for some topologically bounded set B and every real r ˃ 0 there exists N (r) and n ≥ N (r) implies x – xn ϵ rb. Local Cauchy sequences are defined analogously, and L is said to be locally complete if every local Cauchy sequence converges locally. Then L is locally complete if and only if every monotone local Cauchy sequence has a least upper bound. This is a somewhat more general form of the completeness criterion for Riesz – normed Riesz spaces given by Luxemburg and Zaanen. Locally complete, bound, locally convex Riesz spaces are barrelled. If the space is metrizable, local completeness and topological completeness are equivalent. Two measures of the non-archimedean character of a non-archimedean Riesz space L are the smallest ideal Ao (L) such that quotient space is Archimedean and the ideal I (L) = { x ϵ L: for some 0 ≤ v ϵ L, n |x| ≤ v for n = 1, 2, …}. In general Ao (L) ᴝ I (L). If L is itself a quotient space, a necessary and sufficient condition that Ao (L) = I (L) is given. There is an example where Ao (L) ≠ I (L). A necessary and sufficient condition that a Riesz space L have every quotient space Archimedean is that for every 0 ≤ u, v ϵ L there exist u1 = sup (inf (n v, u): n = 1, 2, …), and real numbers m1 and m2 such that m1 u1 ≥ v1 and m2 v1 ≥ u1. If, in addition, L is Dedekind σ – complete, then L may be represented as the space of all functions which vanish off finite subsets of some non-empty set.