Abstract and Applied Analysis

Volume 2014 (2014), Article ID 596384, 8 pages

http://dx.doi.org/10.1155/2014/596384

Research Article

A Characterization of Completeness via Absolutely Convergent Series and the Weierstrass Test in Asymmetric Normed Semilinear Spaces

N. Shahzad¹ and O. Valero²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

²Departamento de Ciencias Matemáticas e Informática, Universidad de las Islas Baleares, Carretera de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain

Received 4 April 2014; Revised 9 June 2014; Accepted 13 June 2014; Published 10 July 2014

Academic Editor: J. J. Font

Copyright © 2014 N. Shahzad and O. Valero. 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.

Abstract.

Asymmetric normed semilinear spaces are studied. A description of biBanach, left K-sequentially complete, and Smyth complete asymmetric normed semilinear spaces is provided and three appropriate notions of absolute convergence in the asymmetric normed framework are introduced. Some characterizations of completeness are also obtained via absolutely convergent series. Moreover, as an application, a Weierstrass test for the convergence of series is derived.

1. Introduction

Asymmetric normed linear spaces were applied to solve extremal problems arising in a natural way in mathematical programming first by Duffin and Karlovitz in 1968 [1] and then by Krein and Nudelman in 1977 [2]. Since then, the interest in such asymmetric structures has been growing. Asymmetric functional analysis has become a research branch of analysis nowadays. In 2013, Cobzas [3] published a monograph entitled “Functional Analysis in Asymmetric Normed Spaces,” which collects a large number of results in the aforesaid research line. On account of [3], a systematic and deep study of asymmetric normed linear spaces and other related structures such as asymmetric normed semilinear spaces has been made by Romaguera and some of his collaborators. Many of the aforesaid results can be found in [4–20].

Inspired by the intense research activity in the field under consideration, our purpose is to study some properties of asymmetric normed semilinear spaces. The main goal of this paper is to delve into the relationship between completeness of asymmetric normed semilinear spaces and the absolute convergence of series in such a way that the classical context of Banach normed linear spaces can be recovered as a particular case. In particular, we introduce three absolute convergence notions which are appropriate to describe biBanach asymmetric normed semilinear spaces, left -sequentially complete asymmetric normed semilinear spaces, and Smyth complete asymmetric normed semilinear spaces. Moreover, as an application of the developed theory, we derive a criterion, inspired by the celebrated Weierstrass -test, for the convergence of series of asymmetric normed semilinear space valued bounded mappings.

2. Preliminaries

Throughout, we will denote the set of real numbers, the set of nonnegative real numbers, and the set of positive integer numbers by , , and , respectively.

According to [3], a semilinear space (or cone) is a subset of a linear space such that and for all and . Clearly, every linear space can be considered also as a semilinear space.

Following [3], an asymmetric norm on a linear space is a function satisfying the following conditions for all and :

(i);(ii);(iii).

A pair is called an asymmetric normed linear space provided that is a linear space and is an asymmetric norm on . Observe that if is an asymmetric normed linear space, then the function defined on by for all is a norm on .

On account of [11], an asymmetric normed semilinear space (also called normed cone in [3]) is a pair where is a semilinear space of an asymmetric normed linear space . The restriction of to is also denoted by .

In our context, by a quasimetric space, we mean a pair such that is a nonempty set and is a function satisfying the following conditions for all :

(i);(ii).

Furthermore, every quasimetric allows defining a metric on by for all .

It is well known that, given a quasimetric space , a topology can be induced on which has as a base the family of open -balls ,  , where for all and . We will say that a subset of is -closed whenever is closed with respect to the topology . Moreover, if is a sequence which converges to a point with respect to the topology , then we will say that   -converges to . Furthermore, a quasimetric space is said to be bicomplete provided that the associated metric space is complete. For a fuller treatment of quasimetric spaces, we refer the reader to [21].

Each asymmetric norm on a linear space induces a quasimetric on which is defined by for all . According to [11] (see also [3]), an asymmetric normed linear space is biBanach whenever the induced quasimetric space is bicomplete. In addition, if is a semilinear space of a linear space , then is called a biBanach asymmetric normed semilinear space provided that the quasimetric space is bicomplete where the restriction of to is also denoted by .

Notice that, given an asymmetric normed semilinear space , for all .

In what follows, we will work with asymmetric normed semilinear spaces in such a way that is assumed to be a -closed semilinear space of an asymmetric normed linear space (where again the restriction of to is denoted by ). Of course, in order to work with asymmetric normed linear spaces, we only need to take .

3. The Absolute Convergence of Series in Asymmetric Normed Linear Spaces

3.1. The Notion of Absolute Convergence of Series

Let us recall that a series in a normed linear space is a sequence of the form where is a sequence in . However, it is customary to denote the sequence by . Of course, when is convergent, we will say that the series is convergent and its limit will be also denoted by [22].

In the classical framework of normed linear spaces, the notion of absolutely convergent series is stated as follows.

If is a sequence in a normed linear space , then the series is absolutely convergent provided that the series of nonnegative real numbers is convergent.

The preceding notion, among other things, allows characterizing completeness of normed linear spaces [23].

Theorem 1. Let be a normed linear space. Then, the following assertions are equivalent.

(1) is Banach.(2)Every absolutely convergent series is -convergent.

It is clear that the notion of series can be extended to the framework of asymmetric normed semilinear spaces in the following obvious way [3].

A series in an asymmetric normed semilinear space is a sequence of the form where is a sequence in . Following the notation used for series in normed linear spaces, a series in an asymmetric normed semilinear space will be also denoted by . Of course, when is -convergent, we will say that is -convergent and its limit will be also denoted by .

A natural attempt to extend the notion of absolute convergence to the asymmetric normed semilinear spaces would be as follows.

Definition 2. Let be a sequence in asymmetric normed semilinear space . Then, we will say that the series is absolutely convergent provided that the series is convergent. Moreover, will be said to have the absolute convergence property provided that every absolutely convergent series is -convergent.

We must point out that the absolute convergence was introduced by Cobzas in [3] in the framework of asymmetric normed linear spaces (in fact in the context of asymmetric seminormed linear spaces). So, in Definition 2, we extend the absolute convergence to the case of asymmetric normed semilinear spaces.

Clearly, the notion of absolutely convergent series in normed linear spaces is retrieved as a particular case of the preceding one whenever the asymmetric norm in Definition 2 is exactly a norm.

The example below shows that there exists an asymmetric normed semilinear space without the absolute convergence property.

Example 3. Consider the asymmetric normed linear space where denotes the space of all real sequences with only a finite number of nonzero terms, is the asymmetric norm defined by

for all , and is the asymmetric norm defined by for all . Now, take the sequence in given by

for all . It is clear that the series converges. However, the series does not -converge.

In Example 5, we will show that there are asymmetric semilinear spaces with the absolute convergence property.

Now, it seems natural to wonder whether the characterization provided by Theorem 1 remains valid when we replace in its statement “Banach normed linear space” with “biBanach asymmetric normed semilinear space.” Thus, the desired result could be stated as follows.

“Let be an asymmetric normed semilinear space. Then, the following assertions are equivalent:

(1) is biBanach;(2) has the absolute convergence property.”

Nevertheless, Example 21 in Section 4 shows that such a result is not true. Accordingly, we can conclude that the fact that the asymmetric normed semilinear space is biBanach does not guarantee that the absolute convergence property holds. Since biBanach completeness implies that the metric is complete, it seems obvious that the characterization of biBanach completeness must require an additional property more restrictive than the absolute convergence property which involves and, thus, in some sense, the norm . Inspired by this fact, we propose the following notion which also retrieves as a particular case the classical one.

Definition 4. An asymmetric normed semilinear space will be said to have the strong absolute convergence property provided that every absolutely convergent series is -convergent.

Clearly asymmetric normed semilinear spaces that hold the strong absolute convergence property form a subclass of those satisfying the absolute convergence property.

The next example shows that there are asymmetric normed semilinear spaces which do not have the strong absolute convergence property.

Example 5. Consider the asymmetric normed linear space where is the asymmetric norm introduced in Example 3. Then, the metric is exactly the Euclidean metric on . It follows that the quasimetric space is bicomplete and, thus, the asymmetric normed linear space is biBanach. Define the sequence in by for all . It is clear that is absolutely convergent, since for all . However, the series is not -convergent. So, does not have the strong absolute convergence property.

The following is an example of an asymmetric normed semilinear space with the strong absolute convergence property.

Example 6. Consider the asymmetric normed linear space introduced in Example 5. It is clear that the pair is an asymmetric normed semilinear space. Since for all , we have that every absolutely convergent series is -convergent.

In the remainder of this section, we consider a few notions of completeness that arise in a natural way in the asymmetric context. Thus, we focus our efforts on characterizing those asymmetric normed semilinear spaces that enjoy the (strong) absolute convergence property in terms of the aforementioned notions of completeness.

3.2. The Characterization

In order to achieve our goal, let us recall two notions of completeness for quasimetric spaces, the so-called left -sequential completeness and the Smyth completeness.

A quasimetric space is said to be left -sequentially complete (Smyth complete) provided that every left -Cauchy sequence is -convergent (-convergent), where a sequence is left -Cauchy whenever for each there exists such that for all .

According to [11, 24], an asymmetric normed semilinear space is left -sequentially complete (Smyth complete) if the quasimetric space is left -sequentially complete (Smyth complete).

3.2.1. Absolute Convergence

In this subsection, we provide a description of asymmetric normed semilinear spaces that have the absolute convergence property.

The following result will be crucial for our purpose, whose proof can be found in [25].

Lemma 7. Let be a quasimetric space and let be a left -Cauchy sequence. If there exists a subsequence of which -converges to , then   -converges to .

From the preceding result, we immediately obtain the following one for asymmetric normed semilinear spaces.

Lemma 8. Let be an asymmetric normed semilinear space and let be a left -Cauchy sequence. If there exists a subsequence of which -converges to , then   -converges to .

Taking into account the preceding lemma, we characterize asymmetric normed semilinear spaces that enjoy the absolute convergence property in the result below. It must be pointed out that the equivalence between assertions and in Theorem 9 is given in [3] for asymmetric normed linear spaces. However, here, we prove the equivalence following a technique that, although related to the one used in [3], allows us to provide a bit more information, in the spirit of [23], about the spaces under study (assertion in the statement of Theorem 9).

Theorem 9. Let be an asymmetric normed semilinear space. Then, the following assertions are equivalent.

(1) is -convergent for every sequence in with for all .(2) is left -sequentially complete.(3) has the absolute convergence property.

Proof. . Assume that is a left -Cauchy sequence in . Our aim is to show that has a -convergent subsequence . Indeed, we can consider an increasing sequence in such that

for all . It follows that

for all with . Hence, we have that

for all with , where . Next, define a sequence in as follows:

It is obvious that for all . Then, we obtain that is -convergent. Since

for all , we conclude that is -convergent. By Lemma 8, the sequence is -convergent. It follows that the asymmetric normed semilinear space is left -sequentially complete.

. Consider a sequence such that the induced series is absolutely convergent. Then, given , there exists such that for all . Thus, taking such that , we have that

It follows that is a left -Cauchy sequence. Since is left -sequentially complete, we obtain that is -convergent.

. Consider a sequence such that for all . Then, we show that the series is absolutely convergent. Indeed, for all . It follows that the series is convergent. Thus, by hypothesis, we obtain that the series is -convergent. This concludes the proof.

Next, we give a few examples of asymmetric normed semilinear spaces having the absolute convergence property.

Example 10. In [24], the following asymmetric normed semilinear spaces were proved to be left -sequentially complete and, thus, by Theorem 9 all have the absolute convergence property.

(1) and , where is the set of real number sequences such that , , and is the asymmetric norm introduced in Example 3.(2) and , where , is the set of real number sequences such that , , and is the asymmetric norm defined on by

for all .(3) and , where is the set of all functions which are continuous from into , , and is the asymmetric norm on given by

for all .(4) and , where is the set of real number sequences which converge to and .(5) and , where , , , and is the asymmetric norm on defined by

for all .

3.2.2. Strong Absolute Convergence

In this subsection, we provide a description of asymmetric normed semilinear spaces that have the strong absolute convergence property.

In the following, the well-known result below plays a central role [3].

Lemma 11. Let be a quasimetric space and let be a left -Cauchy sequence. If there exists a subsequence of which -converges to , then   -converges to .

As a direct consequence, we obtain the next result.

Lemma 12. Let be an asymmetric normed semilinear space and let be a left -Cauchy sequence in . If there exists a subsequence of which -converges to , then   -converges to .

Next, we are able to provide a description of asymmetric normed semilinear spaces having the strong absolute convergence property.

Theorem 13. Let be an asymmetric normed semilinear space. Then, the following assertions are equivalent.

(1) is -convergent for every sequence in with for all .(2) is Smyth complete.(3) has the strong absolute convergence property.

Proof. . Assume is a left -Cauchy sequence in . Our aim is to show that has a -convergent subsequence . By the same method as in the proof of of Theorem 9, we can show the existence of a sequence such that and for all . Then, by hypothesis, the series is -convergent and, hence, as a consequence the sequence is -convergent. By Lemma 12, we obtain that the sequence is -convergent. So, is Smyth complete.

. Consider a sequence such that the induced series is absolutely convergent. Then, as in the proof of of Theorem 9, we have that the sequence is left -Cauchy. Since is Smyth complete, we deduce that is -convergent.

. Consider a sequence such that for all . Then, according to the proof of of Theorem 9, we have that the series is absolutely convergent. Since has the strong absolute convergence property, we conclude that the series is -convergent.

Among all asymmetric normed semilinear spaces given in Example 10, the only ones that are Smyth complete are , , and (see [26]). Hence, by Theorem 13, the aforesaid asymmetric normed semilinear spaces have the strong absolute convergence property.

Since every Cauchy sequence is a left -Cauchy sequence and every Smyth complete asymmetric normed semilinear space is, at the same time, left -sequential complete and biBanach, we immediately deduce the next result.

Corollary 14. Let be an asymmetric normed semilinear space which has the strong absolute convergence property. Then, the following assertions hold:

(1) is left -sequentially complete;(2) is biBanach.

Remark 15. Observe that Example 10 yields instances of asymmetric normed semilinear spaces that hold the absolute convergence property but not the strong one. Therefore, left -sequential completeness is not equivalent to the strong absolute convergence property. Moreover, although the strong absolute convergence property has been introduced in Definition 4 with the aim of describing biBanach completeness, next, we show that both the aforesaid notions are not equivalent. Indeed, let be the biBanach asymmetric normed linear space given in Example 5. In the aforementioned example, it was shown that, for the sequence defined by for all , the series is absolutely convergent; however, it is not -convergent. So does not have the strong absolute convergence property.

In the light of the preceding remark, it seems clear that we will need a new subclass of absolutely convergent series for providing a characterization of biBanach asymmetric normed semilinear spaces. To this end, we introduce the following notion.

Definition 16. If is a sequence in an asymmetric normed semilinear space , then the series is -absolutely convergent provided that the series is convergent. Moreover, we will say that the asymmetric normed semilinear space has the -absolute convergence property whenever every -absolutely convergent series is -convergent.

Obviously, every -absolutely convergent series is absolutely convergent and, besides, all spaces with the strong absolute convergence property have, at the same time, the -absolute convergence property. However, the converse does not hold. Indeed, is an example of asymmetric normed semilinear space which enjoys the -absolute convergence property but not the strong absolute convergence property.

We end the section yielding the characterization in the case of biBanach asymmetric normed semilinear spaces through the next result whose proof runs as the proof of Theorem 1.

Theorem 17. Let be an asymmetric normed semilinear space. Then, the following assertions are equivalent.

(1) is -convergent for every sequence in with for all .(2) is biBanach.(3) has the -absolute convergence property.

4. The Weierstrass Test in Asymmetric Normed Semilinear Spaces

The relevance of Theorem 1 is given by its wide number of applications. Among others, it allows to state a criterion, known as Weierstrass -test, for the uniform convergence of series of bounded mappings [27]. For the convenience of the reader and with the aim of making our exposition self-contained, let us recall the notion of bounded mapping [28].

Let be a metric space and let be a nonempty set. Then, a mapping is said to be bounded if for some there exists such that for all , where stands for the set of positive real numbers.

Next, consider a normed linear space . Then, the set becomes a normed linear space endowed with the norm defined by for all . Observe that the boundness of guarantees that . Furthermore, the normed linear space is Banach provided that the normed linear space is so.

The following theorem contains the Weierstrass -test in the normed case.

Theorem 18. Let be a Banach normed linear space and let be a nonempty set. If is a sequence in and there exists a sequence in such that the series is convergent and for all , then the series is -convergent.

Our main goal in this section is to prove a version of Theorem 18 in the realm of asymmetric normed semilinear spaces. To this end, Theorems 9 and 13 will play a crucial role. Of course, the notion of bounded mapping can be formulated in our context in two different ways. Indeed, given a quasimetric space and a nonempty set , a mapping will be said to be bounded from the left if for some there exists such that for all . A mapping is bounded from the right if for some there exists such that for all .

It is routine to check that the pair is an asymmetric normed semilinear space whenever is an asymmetric normed semilinear space, where and the asymmetric norm is defined as in the normed case. The same occurs with which can be defined dually.

It is clear that the notions of bounded mapping in the quasimetric case allow us to recover the bounded notion for metric spaces. Moreover, when is a normed linear space and, in addition, we impose the constraint “” in the definition of bounded mapping.

The next example gives mappings which are bounded from both the left and the right.

Example 19. Let be the asymmetric normed linear space given in Example 5. Define the mapping by

Clearly,

for all . Thus, is a bounded mapping from the left.

Next, consider the asymmetric normed semilinear space and define the mapping by

Obviously,

for all . Hence, we have that is a bounded mapping from the right.

Of course, the sets and are different in general as shown in the preceding example. In fact, .

In the next result, we discuss the left -sequential completeness and the Smyth completeness of and .

Theorem 20. Let be an asymmetric normed semilinear space and let be a nonempty set. If is Smyth complete, then and are both left -sequentially complete.

Proof. We only prove that is left -sequentially complete. Similar arguments can be applied to prove that is left -sequentially complete.

Let be a left -Cauchy sequence in . Then, given , there exists such that for all . Hence, for all and for all . It follows that, for each , is a left -Cauchy sequence in . Since is Smyth complete, we deduce that there exists such that   -converges to .

Next, define the mapping by . Then, . Indeed, the fact that is bounded from the left yields the existence of and such that for all . Thus, we obtain that

for all . Moreover, the existence of is guaranteed such that and, in addition, and for all . Thus, we deduce that

for all and for all . So, we have shown that is -convergent as claimed.

The next example shows that there are left -sequentially complete asymmetric normed semilinear spaces whose asymmetric normed semilinear space of bounded mappings is not left -sequentially complete.

Example 21. Consider the left -sequentially asymmetric normed linear space introduced in Example 10. Of course, it is routine to check that is not Smyth complete. Let . Define the sequence given by . Clearly, for all . Hence, belongs to . Moreover, is convergent. Nevertheless, is not -convergent. Thus, by Theorem 9, we conclude that is not left -sequentially complete. Observe, in addition, that is biBanach and, hence, we have an example of a biBanach asymmetric normed linear space which does not have the absolute convergence property as announced in Section 3.1.

The following example provides a Smyth complete asymmetric normed semilinear space whose asymmetric normed semilinear space of bounded mappings is not Smyth complete.

Example 22. Consider the Smyth complete asymmetric normed linear space given (see Example 5) and let be as in Example 21. Take the sequence given in Example 21. Then, it is clear that belongs to for all and that the series is convergent. However, the series is not -convergent. Thus, by Theorem 13, we conclude that is not Smyth complete.

We end the paper proving the announced Weierstrass test in the asymmetric framework.

Theorem 23. Let be a Smyth complete asymmetric normed semilinear space and let be a nonempty set. If is a sequence in and there exists a sequence in such that the series is convergent and for all , then the series is -convergent.

Proof. Since and the series is convergent, we have that the series is convergent. The Smyth completeness of provides the left -sequential completeness of and, by Theorem 9, we deduce that the series is -convergent.

Using arguments similar to those in the proof of preceding result, one can get the following result.

Theorem 24. Let be a Smyth complete asymmetric normed semilinear space and let be a nonempty set. If is a sequence in and there exists a sequence in such that the series is convergent and for all , then the series is -convergent.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR for financial support. The second author acknowledges the support from the Spanish Ministry of Economy and Competitiveness, under grant no. MTM2012-37894-C02-01. The authors thank the referees for valuable comments and suggestions, which improved the presentation of this paper.

References

R. J. Duffin and L. A. Karlovitz, “Formulation of linear programs in analysis. I. Approximation theory,” SIAM Journal on Applied Mathematics, vol. 16, pp. 662–675, 1968.

M. G. Krein and A. A. Nudelman, The Markov Moment Problem and Extremal Problems, American Mathematical Society, 1977.

S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Birkhauser, Basel, Switzerland, 2013.

C. Alegre, “Continuous operators on asymmetric normed spaces,” Acta Mathematica Hungarica, vol. 122, no. 4, pp. 357–372, 2009.

C. Alegre and I. Ferrando, “Quotient subspaces of asymmetric normed linear spaces,” Boletín de la Sociedad Matemática Mexicana, vol. 13, no. 2, pp. 357–365, 2007.

C. Alegre, I. Ferrando, L. M. García-Raffi, and E. A. Sánchez Pérez, “Compactness in asymmetric normed spaces,” Topology and its Applications, vol. 155, no. 6, pp. 527–539, 2008.

C. Alegre, J. Ferrer, and V. Gregori, “Quasi-uniform structures in linear lattices,” The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 877–884, 1993.

C. Alegre, J. Ferrer, and V. Gregori, “On the Hahn-Banach theorem in certain linear quasi-uniform structures,” Acta Mathematica Hungarica, vol. 82, no. 4, pp. 325–330, 1999.

J. Ferrer, V. Gregori, and C. Alegre, “Quasi-uniform structures in linear lattices,” The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 877–884, 1993.

L. M. García-Raffi, “Compactness and finite dimension in asymmetric normed linear spaces,” Topology and its Applications, vol. 153, no. 5-6, pp. 844–853, 2005.

L. M. García-Raffi and S. Romaguera, “Sequence spaces and asymmetric norms in the theory of computational complexity,” Mathematical and Computer Modelling, vol. 36, no. 1-2, pp. 1–11, 2002.

L. M. García-Raffi, S. Romaguera, and E. A. Sánchez-Pérez, “The bicompletion of an asymmetric normed linear space,” Acta Mathematica Hungarica, vol. 97, no. 3, pp. 183–191, 2002.

L. M. García-Raffi and R. Sanchez-Pérez, “The dual space of an asymmetric normed linear space,” Quaestiones Mathematicae. Journal of the South African Mathematical Society, vol. 26, no. 1, pp. 83–96, 2003.

L. M. García-Raffi, S. Romaguera, and E. A. Sánchez-Pérez, “The supremum asymmetric norm on sequence spaces: a general framework to measure complexity distances,” Electronic Notes in Theoretical Computer Science, vol. 74, pp. 39–50, 2003.

L. M. García Raffi, S. Romaguera, and E. A. Sánchez Pérez, “Weak topologies on asymmetric normed linear spaces and non-asymptotic criteria in the theory of complexity analysis of algorithms,” Journal of Analysis and Applications, vol. 2, no. 3, pp. 125–138, 2004.

J. Rodríguez-López and S. Romaguera, “Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric,” Bulletin of the Belgian Mathematical Society: Simon Stevin, vol. 13, no. 3, pp. 551–562, 2006.

S. Romaguera, E. A. Sánchez-Pérez, and O. Valero, “A characterization of generalized monotone normed cones,” Acta Mathematica Sinica, vol. 23, no. 6, pp. 1067–1074, 2007.

S. Romaguera, E. A. Sánchez Pérez, and O. Valero, “Dominated extensions of functionals and -convex functions on cancellative cones,” Bulletin of the Australian Mathematical Society, vol. 67, no. 1, pp. 87–94, 2003.

S. Romaguera and M. Sanchis, “Semi-Lipschitz functions and best approximation in quasi-metric spaces,” Journal of Approximation Theory, vol. 103, no. 2, pp. 292–301, 2000.

S. Romaguera and M. Schellekens, “Duality and quasi-normability for complexity spaces,” Applied General Topology, vol. 3, no. 1, pp. 91–112, 2002.

H. P. A. Künzi, “Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology,” in Handbook of the History of General Topology, C. E. Aull and R. Lowen, Eds., vol. 3, pp. 853–968, Kluwer Academic, New York, NY, USA, 2001.

E. A. Ok, Real Analysis with Economic Applications, Princeton University Press, 2007.

G. J. O. Jameson, Topology and Normed Spaces, Chapman and Hall, London, UK, 1974.

J. J. Conradie and M. D. Mabula, “Convergence and left--sequential completeness in asymmetrically normed lattices,” Acta Mathematica Hungarica, vol. 139, no. 1-2, pp. 147–159, 2013.

I. L. Reilly, P. V. Subrahmanyam, and M. K. Vamanamurthy, “Cauchy sequences in quasi-pseudo-metric spaces,” Monatshefte für Mathematik, vol. 93, no. 2, pp. 127–140, 1982.

M. D. Mabula, Compactness in asymmetric normed lattices [Ph.D. thesis], Department of Mathematics and Applied Mathematics, University of Cape Town, 2012.

D. H. Griffel, Applied Functional Analysis, Dover, 1985.

J. L. Kelley, I. Namioka, and W. F. Donoghue Jr., Linear Topological Spaces, D. Van Nostrand, 1963.