Unbounded Order Convergence in Ordered Vector Spaces

Ebrahimzadeh, Masoumeh; Haghnejad Azar, Kazem

doi:https://doi.org/10.1155/2024/9960246

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Research Article | Open Access

Volume 2024 | Article ID 9960246 | https://doi.org/10.1155/2024/9960246

Unbounded Order Convergence in Ordered Vector Spaces

Masoumeh Ebrahimzadeh¹and Kazem Haghnejad Azar²

Academic Editor: Ömer Kişi

Received29 Sept 2023

Revised21 Mar 2024

Accepted28 Mar 2024

Published29 Apr 2024

Abstract

We consider an ordered vector space . We define the net to be unbounded order convergent to (denoted as ). This means that for every , there exists a net (potentially over a different index set) such that , and for every , there exists such that whenever . The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every -convergent net implies -Cauchy, and vice versa. Let be an order dense subspace of the directed ordered vector space . If is a -band in , then we establish that is a -band in .

1. Introduction

In the domain of vector lattices, it is common to encounter partially ordered vector spaces that lack lattice structure, particularly in spaces of operators between vector lattices. To address this, one approach is to assume the Dedekind completeness of the codomain. Another strategy involves extending the necessary concepts from lattice theory to a broader class of partially ordered vector spaces, thereby yielding intrinsic definitions of various notions. In many instances, a lattice concept in a vector lattice can be expressed in multiple ways. When reformulated in partially ordered vector spaces, it may give rise to different concepts. The most valuable generalizations often align these two approaches: direct reformulation and the utilization of embeddings. In this paper, we adopt this perspective to further explore “vector lattice notions” in pre-Riesz spaces. In addition, this topic pertains to notations that involve unbounded order convergence (uo-convergence). Unbounded order convergence, as investigated in [1, 2], has found various applications in economics, particularly within the realms of economic theory, decision-making processes, and optimization problems. In addition, the authors delved into the introduction and exploration of positive operators on unbounded vector spaces in their paper [3].

The relevance of unbounded order convergence can be observed in various specific areas. We have extensively explored convergence in ordered vector spaces, which bears resemblance to the concept of unbounded order convergence examined in prior works, such as those mentioned in references [1, 4]. The main objective of this article is to examine the convergence properties in ordered vector spaces and extend the concepts and problems that have been previously explored in vector lattice spaces. While there exist numerous vector spaces that possess an ordered structure, not all of them exhibit lattice properties (see [5]). Therefore, the study of unbounded order in these spaces holds significant importance. Our research is motivated by the exploration of these convergences and the establishment of their generalizations within the framework of ordered vector spaces. The authors in [6] demonstrated that in vector lattices, -convergence does not arise from a topology. Consequently, it can be concluded that -convergence is not necessarily topological in ordered vector spaces in general. Thus, by [6], we see that there is no inherent relationship between the partial-order topology and their newly introduced concept of convergence in general. On the other hand, by Theorem 7.5 of [7], -convergence in a vector lattice that agrees with the convergence of a locally convex-solid topology on if and only if is atomic.

For information on bounded and unbounded order (or norm) convergence in vector lattices, we recommend referring to [1, 6, 8]. In the following sections, we presented several fundamental definitions pertaining to ordered vector space.

We consider a real vector space and a cone in . Here, is defined as a wedge, meaning that if and belong to , and and are nonnegative scalars, then also belongs to . In addition, , indicating that the only element common to both and its negation is the zero vector. In , we define a partial order denoted by , where if and only if lies in . Consequently, the space , or simply , is referred to as an ordered vector space (or partially ordered vector space).

A subspace is said to be majorizing in if, for every , there exists an element such that (or, equivalently, if for each , there exists an element with ).

A subspace is called directed if for every , there is an element such that and . An ordered vector space is directed if and only if is generating in , that is, . An ordered vector space is called Archimedean if for every with for every , one has . The ordered vector space has the Riesz decomposition property if for every with , there exist such that with and . We call a linear subspace of an ordered vector space order dense in if for every we havethat is, the greatest lower bound of set exists in and equals to (see page 360 of [?]). Clearly, if is order dense in , then is majorizing in . We denote for a subset of , the set of all upper bounds (resp, down bounds) by (resp, ). It is clear that for every subset of , . Moreover, if has a Riesz decomposition property and , then .

The elements are called disjoint, in symbols , if . The disjoint complement of a subset is . A sequence is said to be disjoint, if for every , . A linear subspace of an ordered vector space is called a band in if . A subset of an ordered vector space is called solid if for every and the relation implies that . A solid subspace of is called an ideal.

We recall that a linear map is said to be bipositive if for every , one has if and only if . A partially ordered vector space is called a pre-Riesz space if for every , the inclusion implies that . Clearly, each vector lattice is a pre-Riesz space, since the inclusion in the definition of pre-Riesz space reduces to inequality , so , which implies that . By Theorem 4.3 of [9], partially ordered vector space is a pre-Riesz space if and only if there exists a vector lattice and a bipositive linear map such that is order dense in . The pair (or, loosely ) is then called a vector lattice cover of . The theory of pre-Riesz spaces and their vector lattice covers is due to van Haandel (see [10]). Let be an ordered vector space. An ideal in is supremum closed (short -closed), if for every , the relation implies that (see [11]).

A net is said to be decreasing (in symbols, ), whenever implies that . For , the notation which means that and hold. The meanings of are analogous. We say that a net -converges (respectively, -converges) to (in symbols, , respectively) if there is a net (respectively, possibly over a different index set) such that (respectively, ) and for all (for every , there exists such that for all, ), one has , (respectively, ). For two elements, with denotes the according order interval by . A set is called order-bounded if there are such that . Let and be ordered vector spaces.

We recall that the net in vector lattice is said to be order convergent to (or, -convergent for short) if there is a net , possibly over a different index set, such that and for every , there exists such that , whenever . In this case, we write . In a vector lattice , a net is unbounded order convergent (or, -convergent for short) to if for all . In this case, we write .

2. Unbounded Order Convergence on Ordered Vector Spaces

Definition 1. Let be an ordered vector space. The net is unbounded order convergent (or, -convergent for short) to if for each , there is a net , possibly over a different index set, such that and for every , there exists such that , whenever . In this case, we write .

Remark 2. The assertion that -convergence implies -convergence is valid, but the reverse is not generally true.
Let and . Therefore, there exists a net and , and for each , there is an so that for each , . Let for all , where . Since for , , hence . So, . Therefore, .
On the other hand, it is worth noting that the standard basis of , denoted by , exhibits -convergence but not -convergence. It should be emphasized, however, that for each order-bounded net, -convergence is equivalent to -convergence.

Proposition 3. If is a vector lattice, then the net is unbounded order convergent to (in the sense of Definition 1) if and only if it is an unbounded order convergent to .

Proof. Let and . For each , there exists a net such that and for each , there exists an such that for each we have . Let . It is clear that and therefore for each , . Therefore, . It means that .
Conversely, let (in the sense of Definition 1). It is clear that for each . Therefore, for each , there exists such that for each , . Hence, for each whenever . It means that .

Lemma 4. Let be an ordered vector space and , then(1) iff .(2)if for each , and , then .(3)if , then . Moreover, if has the property, then(4)if and , then for each scalar .(5)if , , and for all , then .(6)if and , then .(7)if , then for each subnet of , .

Proof. (1)By Definition 1, it is established.(2)Based on the assumption made, it is evident that is order-bounded. Consequently, we can conclude that . As a result, the proof becomes clear by applying Lemma 2.1 from [12].(3)Let , therefore for each , there exists net such that and for every , there exists such that whenever . We know that for all . Therefore, it is clear that for all , . So, we have for all . Therefore, for , for each . It is clear that for each . Hence, . So, .(4)Let and . Then, we have . Note that, if , then . We have . Therefore, for each . Since has the property, . Hence, .(5)For each , we have . It is clear that and therefore . Since has the property, the rest of the proof is clear.(6)Similar to the proof of Proposition 17, since for each , we have , therefore and . Therefore, .(7)We have . Therefore, and so for each . Hence, the proof is complete.

Proposition 5. Let be a projection band of an ordered Banach space , and let denote the corresponding band projection. If is -null in , then is -null in .

Proof. Let in , therefore for each , there is a net , possibly over a different index set, such that and for every , there exists such that , whenever . We know that for all . Therefore, . So, for each fixed , we have . Note that, for all and in . We suppose that . Hence, . It means that . Therefore, and so .

Definition 6. A net in ordered vector space is said to be -Cauchy, if -converges to 0 in .

Proposition 7. Let be an ordered vector space with the property . Then,(1)if an -Cauchy net has an -convergent subnet whose -limit is , then .(2)each -convergent net is a -Cauchy net.

Proof. (1)Let be a -Cauchy net in and be a subnet of such that . We have and , and . Therefore, for each . Since has the property, . So, by assumption, and . Hence, the proof is complete.(2)Without loss of generality, by Lemma 4, we assume that is an unbounded order convergent to 0 in . Then, there is a net such that and for every , there exists such that , whenever and for each , we haveLet . Then, by property , we haveIt follows that , whenever , and so the proof follows.

Theorem 8. Let be an order dense subspace of an ordered vector space and . in if in .

Proof. Let in and for fixed . Since is order dense in , therefore there exists such that and . By assumption, there is a net such that in and . By Proposition 5.1 of [9], in . Therefore, for each , there is a net such that and for each , there is an such that for each , . So, in .
Consequently, let in . Therefore, for each , there exists a net such that and for every , there exists such that , whenever . Since is order dense in , therefore is majorizing in . Hence, for each , there is a such that . We define . Hence, in . It is obvious that . Therefore, for each and for each , there is an such that for each , .

Corollary 9. Let be a pre-Riesz space with order complete vector lattice cover and is a -null in . By Theorem 8, in . If has a weak unit and , then by Lemma 3.2 of [1], in if there is a net , possibly over a different index set, such that and for every , there exists such that , whenever .

Theorem 10. If is a pre-Riesz space with a vector lattice cover , then the following assertions are true:(1) is -null if is -null in .(2)If the sequence is disjoint, then in . Moreover, if is normed space and(3) have an order continuous norm and is norm bounded and -null, then is -null.(4) is order-bounded, in and is a Banach lattice with order continuous norm, then is norm-null.

Proof. (1)Let be -null in . Then, for each , there is a net , possibly over a different index set, such that and for every , there exists such that , whenever . By Lemma 4 of [13], in . It is obvious that is -null in . Since is order dense in , then by Proposition 3, is -null in . Conversely, let and in . Since in , therefore it is -null in . Therefore, for each , there is a net , possibly over a different index set, such that and for every , there exists such that whenever . It follows that Since is a bipositive operator, therefore and so in .(2)Since is an order dense subspace of , then by Proposition 5.9 of [9], is disjoint in . By Corollary 3.6 of [4], in . Therefore, in . By 1, in .(3)Let be -null. Then by Proposition 3, in . Since and have order continuous norm, then by Theorem 5 of [2], in . By Theorem 3.6 of [14], in .(4)We have in . Since is a Banach lattice with order continuous norm and the net is almost order-bounded in , therefore by Proposition 3.7 of [1], is norm-null in and so it is norm-null in .

Definition 11. A subset of ordered vector space is said to be -closed, if with implies that .

Let be a solid subset of a vector lattice . By Lemma 8.1 from [7], is (sequentially) -closed if and only if it is (sequentially) -closed.

Remark 12. Let be a -closed set and such that . It is obvious that . Since is -closed, therefore . So, is -closed.

Theorem 13. The following assertions are true.(1)Let be an order dense subspace of an ordered vector space . If is -closed in , then is -closed in .(2)Let be a pre-Riesz space with vector lattice cover . is -closed if is -closed in .

Proof. (1)Let and in . Hence, for each , there exists a net such that and for each , there exists such that , whenever . Since is order dense in , then for each , and in . Since and is -closed in , therefore .(2)Let and . It is obvious that . Since is -closed, therefore . So, .Conversely, let and . So, for each , there exists a net such that and for each , there exists such that , whenever . By Lemma 4 of [13], in . Therefore, . Since, and is -closed, hence and therefore .

Theorem 14. Let be a -closed subspace in Archimedean ordered vector space and . Then in if in .

Proof. Let and in . Since is -closed it is obvious that in .
Conversely, let and in . Therefore, for each , there is a net , possibly over a different index set, such that in and for every , there exists such that , whenever . It is obvious that in . Since is Archimedean, there exists a such that in . It is clear that in and so in . Therefore, in . Since by Lemma 4, -limits are unique, therefore . Hence, in . Let for each . It is clear that for each . So, for each . Therefore, in .

Proposition 15. A solid subset of an ordered vector space is -closed if and imply that .

Proof. Let and . It is clear that and therefore . Since is -closed, it implies that .
Conversely, let a net be and . Then, for each , there is a net , possibly over a different index set, such that in and for every , there exists such that , whenever . It is clear that . Since is solid, we have . It is obvious that . It follows that . Hence, is -closed.
Pre-Riesz spaces are precisely defined as the order dense linear subspaces of vector lattices. The research paper [9] explores the restriction and extension properties of ideals, solvex ideals, and bands within this context. The authors specifically investigate these properties in Archimedean-directed partially ordered vector spaces, as they are all considered pre-Riesz spaces. For ordered vector spaces, in the subsequent discussion, we will introduce these concepts for the unbounded case while highlighting their relevant properties.
Let be an ideal in ordered vector space . Then, is said to be a -band in if is -closed in . Based on this definition, we will obtain the following outcome.

Corollary 16. (1)Let be an order dense subspace of directed ordered vector space . If is -band in , then by Theorem 13, is -closed in and by Proposition 5.3 of [9], is an ideal in . Therefore, is a -band in .(2)Let be a pre-Riesz space with vector lattice cover . It is clear that is an ideal in if is an ideal in and by Theorem 13, is -closed in if is -closed in . So, is a -band in if is -band in .

Proposition 17. Let be a vector lattice. If an ideal is a -band in , then it is a band in .

Proof. Let an ideal be a -band in and be a net such that . Therefore, in . Since is a vector lattice, then by Proposition 3, in . Given the assumption that belongs to , we can conclude that forms a band in .
Malinowski [11] demonstrated that in an Archimedean pre-Riesz space, a directed band may not be -closed. However, in the following, by Proposition 15, every -band is indeed -closed.

Corollary 18. Let be an Archimedean pre-Riesz space. Then, every -band is -closed.

Data Availability

No data were used to support this study.

Disclosure

This article has already been pre-printed in [15].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2024 Masoumeh Ebrahimzadeh and Kazem Haghnejad Azar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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