Some Order and Topological Properties of Locally Solid Linear Topological Riesz Spaces

A theorem of Luxemburg and Zaanen on normed Riesz spaces (Theorem 2.4 below) and one of Nakano (Theorem 2.3 below) have been extended by the author in [1] to metrizable locally solid linear topological Riesz spaces. This note gives an example which shows they cannot be further extended to nonmetrizable Hausdorff locally solid linear topological Riesz spaces. 1. Notation and basic concepts. For notation and terminology concerning Riesz spaces we refer to [5]. Let L be a Riesz space. A vector subspace A of L is called an ideal if |w|^|i-i and v e A implies u e A. An ideal A is called a c-ideal if O^u^u in L and {un}ç:A implies ue A. An ideal A is called a band if O^u^u in L and {uAs^A implies u e A. A subset S of L is called a solid set if |w| ̂ |c| and v e S implies u e S. If r is a linear topology for L (a topology for which both mappings (u, v)\-^u+v, (X, u)t-^>Xu are continuous), with a basis for the neighborhood system of the origin consisting of solid sets, then (L, t) is called a locally solid linear topological Riesz space, or briefly, a locally solid Riesz space. 2. Order and topological continuity. Following Luxemburg and Zaanen (see [4, Notes X, XI]), we introduce the following properties for a locally solid Riesz space (L, t). (A, 0):«„|0 in L and {un} is a T-Cauchy sequence implies u„—>-0< (A, \):un[Q in L implies u^d. (A, ii):uald in L implies u„f+d. (A, \ú):d^un1^u in L, implies that {un} is a T-Cauchy sequence. (A, iv):0^«af ^u in /_, implies that {uA is a r-Cauchy net. Theorem 2.1. Let (L, r) be a locally solid Riesz space. Then: (1) (A, ii) implies (A, i). (2) (L, t) satisfies (A, iii) if and only if' (L, t) satisfies (A, iv). (3) If L is Archimedean, then (A, ii) implies (A, iii).