Order-quasiultrabarrelled vector lattices and spaces

Abstract

1. Introduetion In this paper, we introduce and study a class of topological vector lattices (more generally, ordered topological vector spaces) which we call the class of order-quasiultrabarreUed vector lattices abbreviated to O. Q. U. vector lattices (respectively, O. Q. U. spaces). This class replaces that of order-infrabarrened Riesz spaces (respectively, order-infrabarrelled spaces) [8] in situations where local convexity is not assumed. We obtain an analogue of the Banach--Steinhaus theorem for lattice homom0rphisms on O. Q. U. vector lattices (respectively positive linear maps on O. Q. U. spaces) and the one for O. Q. U. spaces is s used successfully to obtain an analogue of the isomorphism theorem concerning O. Q.U. spaces and similar Schauder bases. Finally, we prove a closed graph theorem for O. Q. U, spaces, analogous to that for ultrabarrelled spa~s [6]. 2. Notations and preliminaries We abbreviate locally convex space, locally convex vector lattice, topelogical vector space, and topological vector lattice to 1.c.s., 1.c.v.l., t.v.s., and t.v.l., respectively. We write (E, C) to denote a vector lattice (or ordered vector space) over the field of reals, with positive cone (or simply, cone) C. A subset B of a vector lattice (E, G) is solid if Ixl ~ ]y], yEB implies xEB; a vector subspace M of (~, C) which is also solid is called a lattice ideal. A linear map of a vector lattice (E, C) into a vector lattice (F, K) is a lattice homomorphism if it preserves lattice operations. A linear mapfof an ordered vector space (E, C) into an ordered vector space (i~, K) is positive (order-bounded) if f(x)~K whenever x~C (f maps order-bounded sets into order-bounded sets). A subset B of an ordered vector space (E, C) is called order-bornivorous if it absorbs all order-bounded subsets of E.