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International Journal of Analysis
Volume 2013 (2013), Article ID 451762, 7 pages
http://dx.doi.org/10.1155/2013/451762
Research Article

-Convergence in the Topology Induced by Random 2-Normed Spaces

Department of Mathematics, Suleyman Demirel University, East Campus, 32260 Isparta, Turkey

Received 16 January 2013; Revised 20 June 2013; Accepted 26 June 2013

Academic Editor: Baruch Cahlon

Copyright © 2013 U. Yamancı and M. Gürdal. 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

We study -convergence which is common generalization of the -convergence of sequences in the topology induced by random 2-normed spaces and prove some important results.

1. Introduction

Kostyrko et al. (cf. [1]; a similar concept was presented in [2]) introduced the concept of -convergence of sequences in a metric space and studied some properties of such convergence. Note that -convergence is an interesting generalization of statistical convergence. The notion of statistical convergence of sequences of real numbers was introduced by Fast in [3] and Steinhaus in [4].

Motivated by a result of Šalát [5] and Fridy [6] about statistically convergent sequences, in [7] Kostyrko et al. also defined so-called -convergence and asked for which ideals the notions of -convergence and -convergence coincide. This question was answered in [1] where the authors showed that these notions coincide if and only if the ideal satisfies the property AP, which we call AP (, Fin) here (see also [8]).

Another important variant of ideal convergence is the notion of -convergence introduced by Maaj and Sleziak [9]. Recently, Eshaghi Gordji et al. [10] studied -convergence in -normed spaces.

The concept of 2-normed spaces was initially introduced by Gähler [11] in the 1960s. Since then, many researchers have studied these subjects and obtained various results [1217].

The theory of probabilistic normed (PN) spaces is an important area of research in functional analysis. Much work has been done in this theory and it has many important applications in real-world problems. PN spaces are the vector spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. A PN space is a generalization of an ordinary normed linear space. In a PN space, the norms are represented by distance distribution functions. If is an element of a PN space, then its norm is denoted by , and the value is interpreted as the probability that the norm of is smaller than . PN spaces were first introduced by Šerstnev in [18] by means of a definition that was closely modelled on the theory of normed spaces. In 1993, Alsina et al. [19] presented a new definition of a PN space which includes the definition of Šerstnev [20] as a special case. This new definition has naturally led to the definition of the principal class of PN spaces, the Menger spaces, and is compatible with various possible definitions of a probabilistic inner product space. It is based on the probabilistic generalization of a characterization of ordinary normed spaces by means of a betweenness relation and relies on the tools of the theory of probabilistic metric (PM) spaces (see [21]). This new definition quickly became the standard one and it has been adopted by many authors (e.g., [2229]), who have investigated several properties of PN spaces. A detailed history and the development of the subject up to 2006 can be found in [30].

The notion of -convergence of sequences has not been studied previously in the setting of random -normed spaces. Motivated by this fact, in this paper, as a variant of -convergence, the notion of -convergence of sequences is introduced in a random 2-normed space as a common generalization of all these types of -convergence, and some important results are established.

2. Preliminaries

In this section we recall some of the basic concepts, which will be used in this paper.

The notion of a statistically convergent sequence can be defined using the asymptotic density of subsets of the set of positive integers . A subset of is said to have density if exists, where is the characteristic function of the set [31]. The notion of statistical convergence was originally defined for sequences of numbers in the paper [3] and also in [32]. We say that a number sequence statistically converges to a point if for each we have , where and in such situation we will write -.

The notion of statistical convergence was further generalized in the paper [1, 8] using the notion of an ideal of subsets of the set . We say that a nonempty family of sets is an ideal on if (a) ; (b) imply ; (c) imply . An ideal on for which is called a proper ideal. A proper ideal is called admissible if contains all finite subsets of . If not otherwise stated in the sequel will denote an admissible ideal. Let be a nontrivial ideal. A class , called the filter associated with the ideal , is a filter on .

Recall the generalization of statistical convergence from [1, 8].

Let be an admissible ideal on and a sequence of points in a metric space . We say that the sequence is -convergent (or -converges) to a point , and we denote it by -, if for each we have This generalizes the notion of usual convergence, which can be obtained when we take for the ideal (Fin the ideal) of all finite subsets of . A sequence is statistically convergent if and only if it is -convergent, where is the admissible ideal of the sets of zero asymptotic density.

Definition 1 (see [11]). Let be a real vector space of dimension , where . A 2-norm on is a function which satisfies (i) if and only if and are linearly dependent; (ii)(iii)(iv). The pair is then called a 2-normed space.

As an example of a 2-normed space, we may take being equipped with the 2-norm the area of the parallelogram spanned by the vectors and , which may be given explicitly by the formula Observe that in any 2-normed space we have and for all and . Also, if , and are linearly dependent, then or . Given a 2-normed space , one can derive a topology for it via the following definition of the limit of a sequence: a sequence in is said to be convergent to in if for every .

All the concepts listed below are studied in depth in the fundamental book by Schweizer and Sklar [21].

Definition 2. Let denote the set of real numbers, , and the closed unit interval. A mapping is called a distribution function if it is nondecreasing and left continuous with and .

We denote the set of all distribution functions by such that . If , then , where It is obvious that for every .

Definition 3. A triangular norm (-norm) is a continuous mapping such that is an abelian monoid with unit one and if and for all . A triangle function is a binary operation on which is commutative, associative, and for every .

Definition 4. Let be a linear space of dimension greater than one, a triangle function, and . Then, is called a probabilistic 2-norm and a probabilistic 2-normed space if the following conditions are satisfied.(i) if and are linearly dependent, where denotes the value of at .(ii) if and are linearly independent.(iii) for all .(iv) for every and .(v) whenever , and .
If (v) is replaced with(vi) for all and ; then is called a random 2-normed space (for short, RTN space), where and are given by Definition 3.

As a standard example, we can give the following.

Example 5. Let be a 2-normed space, and let for all . For all and every , consider Then, observe that is a random 2-normed space.

Let be an RTN space. Since is a continuous -norm, the system of -neighborhoods of (the null vector in ) where determines a first countable Hausdorff topology on , called the -topology. Thus, the -topology can be completely specified by means of -convergence of sequences. It is clear that means and vice versa.

A sequence in is said to be -convergence to if for every , and any nonzero there exists a positive integer such that or equivalently, In this case, we write -.

We also recall that the concept of -convergence and -convergence of sequences in a random 2-normed space is studied in [24].

Definition 6. Let be an RTN space and a proper ideal in . The sequence in is said to be -convergent to (-convergent to with respect to -topology) if for each and any nonzero ,

In this case the vector is called the -limit of the sequence and we write -.

Definition 7. Let be an RTN space and an admissible ideal in . We say that a sequence in is said to be -convergent to with respect to the random 2-norm if there exists a subset such that i.e., and -.

In this case we write - and is called the -limit of the sequence .

3. -Convergence in RTNS

In this section, we aim to generalize the notion of -convergence of sequences in random -normed space.

We give a few basic facts concerning -convergence for future reference.

Lemma 8. Let  , be two RTN spaces and , be ideals on . Then,(i) if is not proper ideal, then every sequence in is -convergent to each point of with respect to the random -norm ;(ii) If , then for every sequence , one has

Proof. (i) Let be arbitrary element of . Then for each and any nonzero
(ii) Let -. Then we have for each and any nonzero Hence, -.

As we have previously mentioned, we aim to generalize the notion of -convergence of sequences, introduced in [24]. Therefore, we modify this definition in the following way.

Definition 9. Let be an RTN space and an ideal in . A sequence in is called -convergent to the point with respect to the random -norm if there exists a set such that the sequence defined by is Fin-convergent to with respect to the random 2-norm . If is -convergent to , we write -.

We introduce the definition of -convergence in ; we simply replace the ideal Fin with an arbitrary ideal on the set .

Definition 10. Let be an RTN space and let and be ideals on . The sequence in is called -convergent to the point with respect to the random 2-norm if there exists a set such that the sequence given by is -convergent to with respect to the random -norm . If is -convergent to , we write -.

Remark 11. The definition of -convergence can be reformulated in the form of decomposition theorem. A sequence is -convergent if and only if it can be written as , where is -convergent with respect to the random 2-norm and is nonzero only on a set from .

Example 12. (i) Put . is the minimal ideal in . A sequence is -convergent if and only if it is constant.
(ii) Let , . Let be a proper ideal in . Let . A sequence is -convergent if and only if it is constant on .
(iii) Let be an admissible ideal in and an arbitrary ideal. A sequence is -convergent if there exists a set and the sequence that is the usual -convergences.

Theorem 13. Let be an RTN space and let and be ideals on . -limit of any sequence if exists is unique.

Proof. Let be any sequence and suppose that -, -, where . Since , select , and any nonzero such that and are disjoint neighborhoods of and . Since -, and -, by the definition there exists such that the sequences given by having the following properties for each , and any nonzero belong to . This implies that sets and belong to . Since is a filter in , we have that is a nonempty in . In this way, we obtain a contradiction to the fact that the neighborhoods and of and are disjoints. Hence, we have . This completes the proof.

Lemma 14. Let be a RTN space and let and be ideals on . If and are two sequences in with for every ; then (i)if - and -, then -; (ii)if - and , then -; (iii)if - and -, then -.

Proof. (i) Let , and any nonzero . Since -, and -, there exists such that the sequences given by having the following conditions belong to . Let and have the condition . Since is an ideal, it is sufficient to show that . This is equivalent to showing that where and belong to . Let , that is, and , and we have Since , we have .
(ii) Let and any nonzero . Since -, there exists such that the sequence given by have the following condition This implies that . Let . For the case , we have and for the case So . Hence -.
(iii) The result follows from (i) and (ii).

One can show easily directly the definitions that -convergence with respect to the random -norm implies -convergence.

Lemma 15. Let and be ideals on . If is an RTN space and is a sequence in such that - with respect to the random -norm , then -.

Lemma 16. Let an RTN space and let , , , and be ideals on such that and . Then, for every sequence in , we have (i)--, (ii)--.

Proof. (i) Let and any nonzero and suppose that -. Then there exists such that the sequence given by has the following condition On the other hand, since , and the by Definition 10, -.
(ii) Let and any nonzero . Since -, there exists such that the sequence given by has the following condition Hence, and the proof is complete.

In the next theorem, we show the relationship between the -convergence and -convergence in random 2-normed spaces.

Theorem 17. Let be an RTN space and let and be ideals on . Let be a sequence in . (i)If - implies - holds for some , which has at least one neighborhood different from , then . (ii)If , then - implies -.

Proof. (i) Let , and any nonzero . Suppose that ; that is, there exists a set . Let be a point with neighborhood such that and . Let us define a sequence on by Clearly, - and thus by Lemma 15 we get -. As , the sequence is not -convergent to .
(ii) Let be an RTN space and and let be a sequence on . Assume that and -. By the definition of -convergence, there exists such that the sequence given by has the condition for every and any nonzero Hence, . Consequently, and thus -.

The following example shows that the converse of Theorem 17(ii) does need not to be true.

Example 18. Let , with the 2-norm where , and let for all . Now for all , and , let us define Then, is an RTN space. Let be a decomposition of such that for any each contains infinitely many ’s where and for . Denote by the class of all such that intersects only a finite numbers of . Let be the family of all finite subsets of . We define a sequence as follows: if . Then, for nonzero , we have as . Hence -.
Now assume that -. Then, there exists such that the sequence given by has the condition for every and any nonzero that is, -. Since , then there exists such that . Now, by the definition of , there exists a such that . But , and therefore for infinitely many ’s from which contradicts the assumption that -. Hence, the converse of the theorem does need not to be true.

Acknowledgment

The authors would like to express their appreciation to the referee for him/her valuable suggestions and corrections which help to provide better presentation of this paper.

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