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Order-Quasiultrabarrelled Vector Lattices and Spaces Periodica Mathematica Hungarica YoL 6 (4), (1975), pp. 363--371. ~ ORDER-QUASIULTRABARRELLED VECTOR LATTICES AND SPACES by T. HUSAIN and S. M. KHALEELULLA (Hamilton) 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--Stein- haus theorem for lattice homom0rphisms on O. Q. U. vector lattices (respec- tively 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, tope- logical 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 homo- morphism 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. This research was supported by an N. R. C. grant. AMS (MOS) subject classifications (1970). Primary 46A40. 6* 364 HUSAI~, KHALEELULLA: V~CTOR LATTICES AND SPACES We write (E, C, u) to denote a topological vector lattice (or ordered t.v.s.) with cone C and topology u. An 1.c.v.1. (or ordered 1.c.s.) (E, C, u) is an order-infrabarrelled Riesz space (respectively, order-infrabarrelled space) if every order-bornivorous barrel is a neighbourhood of 0 in E. A sequence {xi} in an ordered t.v.s. (g, C, u) is a basis for E if for each xEE there cor- n responds a unique sequence (~l} C R such that x ~ lim~'~i in the topo- n i-----1 logy u. Each coefficient ~i determines a corresponding linear functional fi on E. In general, the coefficient functionals fl need not be continuous; if they are continuous, we call (~l} a Schauder basis. If (xt} c_ C and each fi is posi- tive (and continuous), we say that {xi} is a positive (Schauder) basis for E. A closed, circled subset B of a t.v.s. (E, u) is an ultrabarrel Ca bomivorous ultrabarrel) if there exists a sequence {Bn}, called defining. sequence for B, of circled absorbing (circled bomivorous) subsets of E such that B 1 -t- B1 c B and Bn§ W Bn+l ~ Bn for all n. A topological vector space (E, u) is called an ultrabarrelled space if any linear topology on E with a base of u-closed neighbourhoods of 0 is necessarily coarser than u iff every ultrabarrel is a neighbourhood of 0. A topological vector space (E, u) is quasiultrabarrelled if each bornivorotts, ultrabarrel is a neighbourhood of 0 in E [2]. 3. O.Q.U. vector lattices DEFINITION 3.1. Let (E,C, u) be an ordered t.v.s. (a t.v.1.). A closed, circled (closed, solid) subset B of E is said to be an order-bornivorous ultra- barrel (solid ultrabarrel)if there exists a sequence (Bn; n~N~ of circled, order-bornivorous (solid, absorbing) subsets of E such that B 1 ~- B 1 c B and Bn+l ~- Bn-1 c B, for all n. {Bn; ~cN} is called a defining sequence for B. DEFim~rm~ 3.2. A t.v.1. (E,C, u) is called an order-quasiultrabarrelled vector lattice (abbreviated to O. Q. U. vector lattice) if everyorder-bornivorous ultrabarrel in E is a neighbourhood of 0 in E. TH]~O~.M 3.3. A t.v.1. (E, C, u) is a~, O. Q. U. vector lattice iff each solid ultrabarrel is a neighbourhood of 0 in E. PROOF. Since a solid, absorbing set is always order-bornivorous and a solid set is circled, the "only if" part follows immediately. To prove the "if" part, let V be an order-bornivorous ultrabarrel with defiming sequence {Vn; hEN}. Consider the solid kernel K(V) of V defined by K(V) = [.Ixl, Ixl] c_ r}. .
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