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Applications of Convergence Spaces to Vector Lattice Theory

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Applications of Convergence Spaces to Vector Lattice Theory Volume 41, 2013 Pages 311{331 http://topology.auburn.edu/tp/ Applications of Convergence Spaces to Vector Lattice Theory by Jan Harm van der Walt Electronically published on October 3, 2012 Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 41 (2013) Pages 311-331 E-Published on October 3, 2012 APPLICATIONS OF CONVERGENCE SPACES TO VECTOR LATTICE THEORY JAN HARM VAN DER WALT Abstract. We introduce the concept of a locally solid convergence vector lattice as a generalization of locally solid Riesz spaces. This notion provides an appropriate context for a number of natural modes of convergence that cannot be described in terms of the usual Hausdorff-Kuratowski-Bourbaki (HKB) notion of topology. We discuss the dual and completion of a locally solid convergence vector lattice. As applications of this new concept we present a Closed Graph Theorem for linear operators on a class of vector lattices, and a duality result for locally convex, locally solid Riesz spaces. 1. Introduction Many of the most important spaces that arise in analysis are vector lattices. Recall [1, page 3] that a subset A of a vector lattice L is called solid whenever 8 f 2 A; g 2 L : (1.1) jgj ≤ jfj ) g 2 A: A vector space topology on L is called locally solid if it has a basis at 0 consisting of solid sets [1, Definition 5.1]. As a first example, we may note that the space C(X) of continuous, real valued functions on a topological space is a vector lattice with re- spect to the pointwise ordering, and comes equipped with a host of useful topologies, such as the topology of pointwise convergence. Recall that this topology has a basis at 0 consisting of sets of the form (1.2) B(ϵ, F ) = ff 2 C(X): jf(x)j < ϵ, x 2 F g; 2010 Mathematics Subject Classification. Primary 46A19, 46A40; Secondary 46A40, 46A20. Key words and phrases. Vector lattices, convergence spaces. The author wishes to thank the reviewer of the paper for his useful comments and suggestions. ⃝c 2012 Topology Proceedings. 311 312 JAN HARM VAN DER WALT where F is any finite subset of X. Note that the sets B(ϵ, F ) all satisfy 8 f 2 B(ϵ, F ); g 2 C(X): (1.3) jgj ≤ jfj ) g 2 B(ϵ, F ): Therefore the topology of pointwise convergence is an example of a locally solid topology on C(X). Similarly, the spaces Lp(Ω), with Ω a σ-finite measure space and 1 ≤ p ≤ 1, are locally solid Riesz spaces with respect to their natural norm topologies. More generally, any Banach lattice is a locally solid Riesz space. On the other hand, a number of fundamental topological processes on a vector lattice cannot be described in terms of HKB topology, see Examples 1.2 and 1.4 in the following. Recall [8] that topological convergence of nets is characterized by the so called Moore-Smith Axioms (MS1) to (MS4) given below in terms of sequences. Consider a mapping σ : S 3 s 7! σ(s) 2 P(X) from the set S of sequences in X to the powerset P(X) of X. Here σ(s) is interpreted as the set of limits of s. Note that if σ(s) = ;, then s does not converge to any member of X. If there is a topology τ on X so that σ(s) is the set of τ-limits of s for all s 2 S, then the following conditions are satisfied: (MS1) If s is the sequence with all terms equal to x, then x 2 σ(s). (MS2) If x 2 σ(s), then x 2 σ(s0) for all subsequences of s0 of s. (MS3) If every subsequence s0 of s contains a further subsequence s00 so that x 2 σ(s00), then x 2 σ(s). n n 2 n 2 N 2 (MS4) Assume that s = (xm), xn σ(s ) for every n , and x σ(s) where s = (xn). Then there exists a strictly increasing mapping N ! N 2 0 0 n δ : so that x s where s = (xδ(n)). Condition (MS3) is usually referred to as the Urysohn property, while (MS4) is, for obvious reasons, called the Diagonal property. A more general notion of topology, which may describe notions of convergence that do not satisfy the Moore-Smith Axioms, is that of a convergence structure and a convergence space, see for instance [3]. Definition 1.1. A convergence structure on a set X is a mapping λ from X into powerset of the set of all filters on X that satisfies the following: (i) For all x 2 X, the filter [x] = fF ⊆ X : x 2 F g belongs to λ(x). (ii) If F 2 λ(x) and G ⊇ F, then G 2 λ(x). (iii) If F; G 2 λ(x), then F \ G 2 λ(x). If λ is a convergence structure on X, then the pair (X; λ) is called a convergence space. If there is no ambiguity, we will denote the pair (X; λ) by X alone. If F 2 λ(x) we say that F converges to x. CONVERGENCE VECTOR LATTICES 313 Every topology τ on a set X is identified with a convergence structure λτ on X given by F 2 λτ (x) ,F contains the τ−neighborhood filter at x: However, not all convergence structures can be defined in this way. In connection with the aforementioned non-topological modes of convergence on a vector lattice L, we recall the following well known topological notion of convergence. Example 1.2. Let X be a compact topological space. A sequence (fn) in C(X) converges to f 2 C(X) in the uniform topology whenever 8 ϵ > 0 : 9 N 2 N : (1.4) ϵ 8 x 2 X; n 2 N; n ≥ Nϵ : jf(x) − fn(x)j < ϵ. Note that the condition (1.4) in Example 1.5 is equivalent to 9 u 2 C(X); u ≥ 0 : 8 ϵ > 0 : 9 Nϵ 2 N : 8 n 2 N; n ≥ Nϵ : −ϵu ≤ f − fn ≤ ϵu, which is formulated entirely in terms of the vector lattice structure of C(X). Thus one may generalize the notion of uniform convergence in C(X) to an arbitrary vector lattice L, see for instance [6, Theorem 16.2]. Definition 1.3. A sequence (fn) in L converges relatively uniformly to f 2 L whenever 9 u 2 L; u ≥ 0 : 8 ϵ > 0 : (1.5) 9 Nϵ 2 N : 8 n 2 N; n ≥ Nϵ : jf − fnj ≤ ϵu. In contradistinction with uniform convergence on C(X), relatively uniform convergence on an arbitrary vector lattice is, in general, not induced by a topology. In this regard, we denote by C0(R) the space of continuous, real valued functions defined on R with compact carrier, that is, each f 2 C0(R) satisfies f(x) = 0 for all x outside some compact set K ⊂ R. The space C0(R) is a vector lattice with respect to the usual pointwise order. In this space, relatively uniform convergence does not satisfy the Diagonal Property, see [6, Excercise 16.9]. 314 JAN HARM VAN DER WALT Example 1.4. Let (Ω; M; m) be a σ-finite measure space and 1 ≤ p ≤ p 1. A sequence (fn) in L (Ω) converges boundedly almost everywhere to f 2 Lp(Ω) whenever 9 u 2 Lp(Ω);E ⊂ Ω; m(E) = 0 : 8 x 2 Ω n E : (1.6) 1) jfn(x)j ≤ u(x); n 2 N 2) (fn(x)) converges to f(x) in R: The relevance of the concept of boundedly almost everywhere conver- gence is demonstrated by its role in the Lebesgue Dominated Convergence Theorem: If a sequenceZ (fn) converges boundedlyZ almost everywhere to 1 f in L (Ω), then fn(x)dx converges to f(x)dx. We may note that (1.6) is equivalent to the condition p 9 (un) ⊂ L (Ω) : 1) −u ≤ f − f ≤ u ; n 2 N (1.7) n n n 2) un+1 ≤ un; n 2 N 3) inffun : n 2 Ng = 0: Since (1.7) is formulated only in terms of the vector lattice structure of Lp(Ω), we may generalize the notion of boundedly almost everywhere convergence to an arbitrary vector lattice L, see for instance [6, Theorem 16.1]. Definition 1.5. A sequence (fn) in L order converges to f 2 L whenever 9 (un) ⊂ L : 1) −u ≤ f − f ≤ u ; n 2 N (1.8) n n n 2) un+1 ≤ un; n 2 N 3) inffun : n 2 Ng = 0: In general there is no topology on L that induces order convergence of sequences. Indeed, even on L1(Ω) order convergence is not induced by a topology. In fact, while order convergence in this space satisfies the Diagonal Property [6, Theorem 71.8], the Urysohn property may fail, see [9]: There exists a sequence (fn) in L, which does not order converge to 0, with the property that every subsequence of (fn) contains a further subsequence that order converges to 0. Thus order convergence in L1(Ω) is not induced by a topology, since convergent sequences in a topological space satisfy the Urysohn Property. Note that, in general, order conver- gence satisfies neither the Urysohn property, nor the Diagonal property, see for instance [2].
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