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Asymmetric normed semilinear spaces are studied. A description of biBanach, left

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 [

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

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

According to [

Following [

A pair

On account of [

In our context, by a quasimetric space, we mean a pair

Furthermore, every quasimetric

It is well known that, given a quasimetric space

Each asymmetric norm

Notice that, given an asymmetric normed semilinear space

In what follows, we will work with asymmetric normed semilinear spaces

Let us recall that a series in a normed linear space

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

If

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

Let

Every absolutely convergent series is

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

A series in an asymmetric normed semilinear space

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

Let

We must point out that the absolute convergence was introduced by Cobzas in [

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

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

Consider the asymmetric normed linear space

In Example

Now, it seems natural to wonder whether the characterization provided by Theorem

“Let

Nevertheless, Example

An asymmetric normed semilinear space

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.

Consider the asymmetric normed linear space

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

Consider the asymmetric normed linear space

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.

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

A quasimetric space

According to [

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 [

Let

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

Let

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

Let

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

In [

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 [

Let

As a direct consequence, we obtain the next result.

Let

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

Let

Among all asymmetric normed semilinear spaces given in Example

Since every Cauchy sequence is a left

Let

Observe that Example

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.

If

Obviously, every

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

Let

The relevance of Theorem

Let

Next, consider a normed linear space

The following theorem contains the Weierstrass

Let

Our main goal in this section is to prove a version of Theorem

It is routine to check that the pair

It is clear that the notions of bounded mapping in the quasimetric case allow us to recover the bounded notion for metric spaces. Moreover,

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

Let

Next, consider the asymmetric normed semilinear space

Of course, the sets

In the next result, we discuss the left

Let

We only prove that

Let

Next, define the mapping

The next example shows that there are left

Consider the left

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

Consider the Smyth complete asymmetric normed linear space

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

Let

Since

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

Let

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

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.