ℐ^𝒦-Convergence in the Topology Induced by Random 2-Normed Spaces

Abstract

We study ℐ^𝒦-convergence which is common generalization of the ℐ^∗-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 ℐ-convergence of sequences in a metric space and studied some properties of such convergence. Note that ℐ-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 ℐ^∗-convergence and asked for which ideals the notions of ℐ-convergence and ℐ^∗-convergence coincide. This question was answered in [1] where the authors showed that these notions coincide if and only if the ideal ℐ satisfies the property AP, which we call AP (ℐ, Fin) here (see also [8]).

Another important variant of ideal convergence is the notion of ℐ^𝒦-convergence introduced by Mac̆‌aj and Sleziak [9]. Recently, Eshaghi Gordji et al. [10] studied ℐ^𝒦-convergence in 2-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 [12–17].

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 𝑥 is an element of a PN space, then its norm is denoted by 𝐹_𝑥, and the value 𝐹_𝑥(𝑡) is interpreted as the probability that the norm of 𝑥 is smaller than 𝑡. 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., [22–29]), 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 ℐ^𝒦-convergence of sequences has not been studied previously in the setting of random 2-normed spaces. Motivated by this fact, in this paper, as a variant of ℐ-convergence, the notion of ℐ^𝐹𝒦-convergence of sequences is introduced in a random 2-normed space as a common generalization of all these types of ℐ^𝐹∗-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 ℕ={1,2,…}. A subset 𝐾 of ℕ is said to have density 𝛿(𝐾) if 𝛿(𝐾)=lim_𝑛→∞𝑛⁻¹∑^𝑛_𝑘=1𝜒_𝐾(𝑘) exists, where 𝜒_𝐾 is the characteristic function of the set 𝐾 [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 𝑥=(𝑥_𝑘)_𝑘∈ℕ statistically converges to a point 𝐿 if for each 𝜀>0 we have 𝛿(𝐾(𝜀))=0, where 𝐾(𝜀)={𝑘∈ℕ∶|𝑥_𝑘−𝐿|≥𝜀} and in such situation we will write 𝐿=st-lim𝑥_𝑘.

The notion of statistical convergence was further generalized in the paper [1, 8] using the notion of an ideal of subsets of the set ℕ. We say that a nonempty family of sets ℐ⊂𝒫(ℕ) is an ideal on ℕ if (a) ∅∈ℐ; (b) 𝐴,𝐵∈ℐ imply 𝐴∪𝐵∈ℐ; (c) 𝐴∈ℐ,𝐵⊂𝐴 imply 𝐵∈ℐ. An ideal ℐ on ℕ for which ℐ≠𝒫(ℕ) is called a proper ideal. A proper ideal ℐ is called admissible if ℐ contains all finite subsets of ℕ. If not otherwise stated in the sequel ℐ will denote an admissible ideal. Let ℐ⊂𝒫(ℕ) be a nontrivial ideal. A class ℱ(ℐ)={𝑀⊂ℕ∶∃𝐴∈ℐ∶𝑀=ℕ⧵𝐴}, called the filter associated with the ideal ℐ, is a filter on ℕ.

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

Let ℐ be an admissible ideal on ℕ and 𝑥=(𝑥_𝑘)_𝑘∈ℕ a sequence of points in a metric space (𝑋,𝜌). We say that the sequence 𝑥 is ℐ-convergent (or ℐ-converges) to a point 𝜉∈𝑋, and we denote it by ℐ-lim𝑥=𝜉, if for each 𝜀>0 we have 𝐴(𝜀)=𝑘∈ℕ∶𝜌𝑥_𝑘,𝜉≥𝜀∈ℐ. (1) This generalizes the notion of usual convergence, which can be obtained when we take for ℐ the ideal ℐ_𝑓 (Fin the ideal) of all finite subsets of ℕ. A sequence is statistically convergent if and only if it is ℐ_𝛿-convergent, where ℐ_𝛿∶={𝐾⊂ℕ∶𝛿(𝐾)=0} is the admissible ideal of the sets of zero asymptotic density.

Definition 1 (see [11]). Let 𝑋 be a real vector space of dimension 𝑑, where 2≤𝑑<∞. A 2-norm on 𝑋 is a function ‖⋅,⋅‖∶𝑋×𝑋→ℝ which satisfies (i)‖𝑥,𝑦‖=0 if and only if 𝑥 and 𝑦 are linearly dependent; (ii)‖𝑥,𝑦‖=‖𝑦,𝑥‖;(iii)‖𝛼𝑥,𝑦‖=|𝛼|‖𝑥,𝑦‖,𝛼∈ℝ;(iv)‖𝑥,𝑦+𝑧‖≤‖𝑥,𝑦‖+‖𝑥,𝑧‖. The pair (𝑋,‖⋅,⋅‖) is then called a 2-normed space.

As an example of a 2-normed space, we may take 𝑋=ℝ² being equipped with the 2-norm ‖𝑥,𝑦‖∶= the area of the parallelogram spanned by the vectors 𝑥 and 𝑦, which may be given explicitly by the formula 𝑥,𝑦=𝑥₁𝑦₂−𝑥₂𝑦₁,𝑥=𝑥₁,𝑥₂,𝑦=𝑦₁,𝑦₂. (2) Observe that in any 2-normed space (𝑋,‖⋅,⋅‖) we have ‖𝑥,𝑦‖≥0 and ‖𝑥,𝑦+𝛼𝑥‖=‖𝑥,𝑦‖ for all 𝑥,𝑦∈𝑋 and 𝛼∈ℝ. Also, if 𝑥,𝑦, and 𝑧 are linearly dependent, then ‖𝑥,𝑦+𝑧‖=‖𝑥,𝑦‖+‖𝑥,𝑧‖ or ‖𝑥,𝑦−𝑧‖=‖𝑥,𝑦‖+‖𝑥,𝑧‖. Given a 2-normed space (𝑋,‖⋅,⋅‖), one can derive a topology for it via the following definition of the limit of a sequence: a sequence (𝑥_𝑛) in 𝑋 is said to be convergent to 𝑥 in 𝑋 if lim_𝑛→∞‖𝑥_𝑛−𝑥,𝑦‖=0 for every 𝑦∈𝑋.

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

Definition 2. Let ℝ denote the set of real numbers, ℝ₊={𝑥∈ℝ∶𝑥≥0}, and 𝑆=[0,1] the closed unit interval. A mapping 𝑓∶ℝ→𝑆 is called a distribution function if it is nondecreasing and left continuous with inf_𝑡∈ℝ𝑓(𝑡)=0 and sup_𝑡∈ℝ𝑓(𝑡)=1.

We denote the set of all distribution functions by 𝐷⁺ such that 𝑓(0)=0. If 𝑎∈ℝ₊, then 𝐻_𝑎∈𝐷⁺, where 𝐻_𝑎(𝑡)=1if0𝑡>𝑎,if𝑡≤𝑎. (3) It is obvious that 𝐻₀≥𝑓 for every 𝑓∈𝐷⁺.

Definition 3. A triangular norm (𝑡-norm) is a continuous mapping ∗∶𝑆×𝑆→𝑆 such that (𝑆,∗) is an abelian monoid with unit one and 𝑐∗𝑑≤𝑎∗𝑏 if 𝑐≤𝑎 and 𝑑≤𝑏 for all 𝑎,𝑏,𝑐,𝑑∈𝑆. A triangle function 𝜏 is a binary operation on 𝐷⁺ which is commutative, associative, and 𝜏(𝑓,𝐻₀)=𝑓 for every 𝑓∈𝐷⁺.

Definition 4. Let 𝑋 be a linear space of dimension greater than one, 𝜏 a triangle function, and 𝐹∶𝑋×𝑋→𝐷⁺. Then, 𝐹 is called a probabilistic 2-norm and (𝑋,𝐹,𝜏) a probabilistic 2-normed space if the following conditions are satisfied.(i)𝐹(𝑥,𝑦;𝑡)=𝐻₀(𝑡) if 𝑥 and 𝑦 are linearly dependent, where 𝐹(𝑥,𝑦;𝑡) denotes the value of 𝐹(𝑥,𝑦) at 𝑡∈ℝ.(ii)𝐹(𝑥,𝑦;𝑡)≠𝐻₀(𝑡) if 𝑥 and 𝑦 are linearly independent.(iii)𝐹(𝑥,𝑦;𝑡)=𝐹(𝑦,𝑥;𝑡) for all 𝑥,𝑦∈𝑋.(iv)𝐹(𝛼𝑥,𝑦;𝑡)=𝐹(𝑥,𝑦;𝑡/|𝛼|) for every 𝑡>0,𝛼≠0 and 𝑥,𝑦∈𝑋.(v)𝐹(𝑥+𝑦,𝑧;𝑡)≥𝜏(𝐹(𝑥,𝑧;𝑡),𝐹(𝑦,𝑧;𝑡)) whenever 𝑥,𝑦,𝑧∈𝑋, and 𝑡>0.
If (v) is replaced with(vi)𝐹(𝑥+𝑦,𝑧;𝑡)≥sup{𝐹(𝑥,𝑧;𝑡₁)∗𝐹(𝑦,𝑧;𝑡₂)∶𝑡₁+𝑡₂=𝑡} for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡₁,𝑡₂∈ℝ₊; then (𝑋,𝐹,∗) is called a random 2-normed space (for short, RTN space), where ∗ and 𝜏 are given by Definition 3.

As a standard example, we can give the following.

Example 5. Let (𝑋,‖⋅,⋅‖) be a 2-normed space, and let 𝑎∗𝑏=min{𝑎,𝑏} for all 𝑎,𝑏∈𝑆. For all 𝑥,𝑦∈𝑋 and every 𝑡>0, consider 𝐹𝑥,𝑦;𝑡=𝐻₀𝑡−𝑥,𝑦. (4) Then, observe that (𝑋,𝐹,∗) is a random 2-normed space.

Let (𝑋,𝐹,∗) be an RTN space. Since ∗ is a continuous 𝑡-norm, the system of (𝜀,𝜆)-neighborhoods of 𝜃 (the null vector in 𝑋) 𝒩_𝜃,𝑧(𝜀,𝜆)∶𝜀>0,𝜆∈(0,1),𝑧∈𝑋, (5) where 𝒩_𝜃,𝑧(𝜀,𝜆)=𝑥,𝑧∈𝑋×𝑋∶𝐹_𝑥,𝑧(𝜀)>1−𝜆 (6) determines a first countable Hausdorff topology on 𝑋×𝑋, called the 𝐹-topology. Thus, the 𝐹-topology can be completely specified by means of 𝐹-convergence of sequences. It is clear that 𝑥−𝑦∈𝒩_𝜃,𝑧 means 𝑦∈𝒩_𝑥,𝑧 and vice versa.

A sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be 𝐹-convergence to 𝐿∈𝑋 if for every 𝜀>0,𝜆∈(0,1), and any nonzero 𝑧∈𝑋 there exists a positive integer 𝑁 such that 𝑥_𝑘,𝑧−𝐿∈𝒩_𝜃,𝑧(𝜀,𝜆)foreach𝑘≥𝑁 (7) or equivalently, 𝑥_𝑘,𝑧∈𝒩_𝐿,𝑧(𝜀,𝜆)foreach𝑘≥𝑁. (8) In this case, we write 𝐹-lim𝑥_𝑘,𝑧=𝐿.

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

Definition 6. Let (𝑋,𝐹,∗) be an RTN space and ℐ a proper ideal in ℕ. The sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be ℐ^𝐹-convergent to 𝐿∈𝑋 (ℐ^𝐹-convergent to 𝐿∈𝑋 with respect to 𝐹-topology) if for each 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋, 𝑘∈ℕ∶𝑥_𝑘,𝑧∉𝒩_𝐿,𝑧(𝜀,𝜆)∈ℐ. (9)

In this case the vector 𝐿 is called the ℐ^𝐹-limit of the sequence 𝑥=(𝑥_𝑘) and we write ℐ^𝐹-lim𝑥,𝑧=𝐿.

Definition 7. Let (𝑋,𝐹,∗) be an RTN space and ℐ an admissible ideal in ℕ. We say that a sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be ℐ^𝐹∗-convergent to 𝐿∈𝑋 with respect to the random 2-norm 𝐹 if there exists a subset 𝑀=𝑚_𝑘∶𝑚₁<𝑚₂<⋯⊂ℕ (10) such that 𝑀∈ℱ(ℐ)(i.e., ℕ⧵𝑀∈ℐ) and 𝐹-lim𝑥_𝑚𝑘,𝑧=𝐿.

In this case we write ℐ^𝐹∗-lim𝑥,𝑧=𝐿 and 𝐿 is called the ℐ^𝐹∗-limit of the sequence 𝑥=(𝑥_𝑘).

3. ℐ^𝐹𝒦-Convergence in RTNS

In this section, we aim to generalize the notion of ℐ^𝐹∗-convergence of sequences in random 2-normed space.

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

Lemma 8. Let  (𝑋,𝐹,∗), (𝑌,𝐹,∗) be two RTN spaces and ℐ, ℐ₁,ℐ₂ be ideals on ℕ. Then,(i) if ℐ is not proper ideal, then every sequence 𝑥=(𝑥_𝑘) in 𝑋 is ℐ^𝐹-convergent to each point of 𝑋 with respect to the random 2-norm 𝐹;(ii) If ℐ₁⊂ℐ₂, then for every sequence 𝑥=(𝑥_𝑘), one has ℐ^𝐹₁-lim𝑥_𝑘,𝑧=𝐿⟹ℐ^𝐹₂-lim𝑥_𝑘,𝑧=𝐿. (11)

Proof. (i) Let 𝐿 be arbitrary element of 𝑋. Then for each 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋𝑘∈ℕ∶𝑥_𝑘,𝑧∉𝒩_𝐿,𝑧(𝜀,𝜆)∈ℐ. (12)
(ii) Let ℐ₁⊂ℐ₂,ℐ^𝐹₁-lim𝑥_𝑘,𝑧=𝐿. Then we have for each 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋𝑘∈ℕ∶𝑥_𝑘,𝑧∉𝒩_𝐿,𝑧(𝜀,𝜆)∈ℐ₁⊂ℐ₂. (13) Hence, ℐ^𝐹₂-lim𝑥_𝑘,𝑧=𝐿.

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

Definition 9. Let (𝑋,𝐹,∗) be an RTN space and ℐ an ideal in ℕ. A sequence (𝑥_𝑘)_𝑘∈ℕ in 𝑋 is called ℐ^𝐹∗-convergent to the point 𝐿∈𝑋 with respect to the random 2-norm 𝐹 if there exists a set 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ defined by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝐿,if𝑘∉𝑀 (14) is Fin-convergent to 𝐿 with respect to the random 2-norm 𝐹. If (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹∗-convergent to 𝐿, we write ℐ^𝐹∗-lim𝑥_𝑘,𝑧=𝐿.

We introduce the definition of ℐ^𝐹𝒦-convergence in (𝑋,𝐹,∗); we simply replace the ideal Fin with an arbitrary ideal on the set ℕ.

Definition 10. Let (𝑋,𝐹,∗) be an RTN space and let 𝒦 and ℐ be ideals on ℕ. The sequence (𝑥_𝑘)_𝑘∈ℕ in 𝑋 is called ℐ^𝐹𝒦-convergent to the point 𝐿∈𝑋 with respect to the random 2-norm 𝐹 if there exists a set 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝐿,if𝑘∉𝑀 (15) is 𝒦-convergent to 𝐿 with respect to the random 2-norm 𝐹. If (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹𝒦-convergent to 𝐿, we write ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝐿.

Remark 11. The definition of ℐ^𝐹𝒦-convergence can be reformulated in the form of decomposition theorem. A sequence (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹𝒦-convergent if and only if it can be written as (𝑥_𝑘)_𝑘∈ℕ=(𝑦_𝑘)_𝑘∈ℕ+(𝑧_𝑘)_𝑘∈ℕ, where (𝑦_𝑘)_𝑘∈ℕ is 𝒦-convergent with respect to the random 2-norm 𝐹 and (𝑧_𝑘)_𝑘∈ℕ is nonzero only on a set from ℐ.

Example 12. (i) Put ℐ₀=𝒦₀={∅}. ℐ₀ is the minimal ideal in ℕ. A sequence (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹𝒦0₀-convergent if and only if it is constant.
(ii) Let ∅≠𝑀⊂ℕ, 𝑀≠ℕ. Let 𝒦 be a proper ideal in ℕ. Let ℐ={∅}. A sequence (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹𝒦-convergent if and only if it is constant on ℕ⧵𝑀.
(iii) Let 𝒦 be an admissible ideal in ℕ and ℐ an arbitrary ideal. A sequence (𝑥_𝑘)_𝑘∈ℕ is ℐ^𝐹𝒦-convergent if there exists a set 𝑀∈ℱ(ℐ) and the sequence (𝑦_𝑘)_𝑘∈ℕ that is the usual 𝐹-convergences.

Theorem 13. Let (𝑋,𝐹,∗) be an RTN space and let 𝒦 and ℐ be ideals on ℕ. ℐ^𝐹𝒦-limit of any sequence if exists is unique.

Proof. Let (𝑥_𝑘)_𝑘∈ℕ be any sequence and suppose that ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉, ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜂, where 𝜉≠𝜂. Since 𝜉≠𝜂, select 𝜀>0, 𝜆∈(0,1) and any nonzero 𝑧∈𝑋 such that 𝒩_𝜉,𝑧(𝜀,𝜆) and 𝒩_𝜂,𝑧(𝜀,𝜆) are disjoint neighborhoods of 𝜉 and 𝜂. Since ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉, and ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜂, by the definition there exists 𝑀₁,𝑀₂∈ℱ(ℐ) such that the sequences (𝑦_𝑘)_𝑘∈ℕ,(𝑧_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀₁𝜉,if𝑘∉𝑀₁,𝑧_𝑘=𝑥_𝑘,if𝑘∈𝑀₂𝜂,if𝑘∉𝑀₂ (16) having the following properties for each 𝜀>0, 𝜆∈(0,1) and any nonzero 𝑧∈𝑋𝐴=𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆),𝐵=𝑘∈ℕ∶𝑧_𝑘,𝑧∉𝒩_𝜂,𝑧(𝜀,𝜆) (17) belong to 𝒦. This implies that sets 𝐴^𝑐={𝑘∈ℕ∶𝑦_𝑘,𝑧∈𝒩_𝜉,𝑧(𝜀,𝜆)} and 𝐵^𝑐={𝑘∈ℕ∶𝑧_𝑘,𝑧∈𝒩_𝜂,𝑧(𝜀,𝜆)} belong to ℱ(ℐ). Since ℱ(ℐ) is a filter in ℕ, we have that 𝐴^𝑐∩𝐵^𝑐 is a nonempty in ℱ(ℐ). In this way, we obtain a contradiction to the fact that the neighborhoods 𝒩_𝜉,𝑧(𝜀,𝜆) and 𝒩_𝜂,𝑧(𝜀,𝜆) of 𝜉 and 𝜂 are disjoints. Hence, we have 𝜉=𝜂. This completes the proof.

Lemma 14. Let (𝑋,𝐹,∗) be a RTN space and let 𝒦 and ℐ be ideals on ℕ. If (𝑥_𝑘) and (𝑦_𝑘) are two sequences in (𝑋,𝐹,∗) with 𝑎∗𝑎>𝑎 for every 𝑎∈(0,1); then (i)if ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉 and ℐ^𝐹𝒦-lim𝑦_𝑘,𝑧=𝜂, then ℐ^𝐹𝒦-lim(𝑥_𝑘+𝑦_𝑘),𝑧=𝜉+𝜂; (ii)if ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉 and 𝛼∈ℝ, then ℐ^𝐹𝒦-lim𝛼𝑥_𝑘,𝑧=𝛼𝜉; (iii)if ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉 and ℐ^𝐹𝒦-lim𝑦_𝑘,𝑧=𝜂, then ℐ^𝐹𝒦-lim(𝑥_𝑘−𝑦_𝑘),𝑧=𝜉−𝜂.

Proof. (i) Let 𝜀>0,𝜆∈(0,1), and any nonzero 𝑧∈𝑋. Since ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉, and ℐ^𝐹𝒦-lim𝑦_𝑘,𝑧=𝜂, there exists 𝑀₁,𝑀₂∈ℱ(ℐ) such that the sequences (𝑧_𝑘)_𝑘∈ℕ,(𝑡_𝑘)_𝑘∈ℕ given by 𝑧_𝑘=𝑥_𝑘,if𝑘∈𝑀₁𝜉,if𝑘∉𝑀₁,𝑡_𝑘=𝑦_𝑘,if𝑘∈𝑀₂𝜂,if𝑘∉𝑀₂ (18) having the following conditions 𝐴=𝑘∈ℕ∶𝑧_𝑘,𝑧∉𝒩_𝜉,𝑧𝜀2,𝜆,𝐵=𝑘∈ℕ∶𝑡_𝑘,𝑧∉𝒩_𝜂,𝑧𝜀2,𝜆 (19) belong to 𝒦. Let 𝑀=𝑀₁∩𝑀₂ and 𝑝_𝑘=𝑥_𝑘+𝑦_𝑘,if𝑘∈𝑀𝜉+𝜂,if𝑘∉𝑀 (20) have the condition 𝐶={𝑘∈ℕ∶𝑝_𝑘,𝑧∉𝒩_{𝜉+𝜂,𝑧}(𝜀,𝜆)}. Since 𝒦 is an ideal, it is sufficient to show that 𝐶⊂𝐴∪𝐵. This is equivalent to showing that 𝐶^𝑐⊃𝐴^𝑐∩𝐵^𝑐 where 𝐴^𝑐 and 𝐵^𝑐 belong to ℱ(ℐ). Let 𝑘∈𝐴^𝑐∩𝐵^𝑐, that is, 𝑘∈𝐴^𝑐 and 𝑘∈𝐵^𝑐, and we have 𝐹_{(𝑝𝑘)−(𝜉+𝜂),𝑧}(𝜀)=𝐹_{(𝑧𝑘+𝑡𝑘)−(𝜉+𝜂),𝑧}(𝜀)≥sup𝐹_{𝑧𝑘−𝜉,𝑧}𝜀2∗𝐹_{𝑡𝑘−𝜂,𝑧}𝜀2∶𝜀2+𝜀2)∗(>().=𝜀>sup{(1−𝜆1−𝜆)}1−𝜆 (21) Since 𝑘∈𝐶^𝑐⊃𝐴^𝑐∩𝐵^𝑐∈ℱ(ℐ), we have 𝐶⊂𝐴∪𝐵∈𝒦.
(ii) Let 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋. Since ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉, there exists 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝜉,if𝑘∉𝑀 (22) have the following condition 𝐴=𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)∈𝒦. (23) This implies that 𝐴^𝑐={𝑘∈ℕ∶𝑦_𝑘,𝑧∈𝒩_𝜉,𝑧(𝜀,𝜆)}∈ℱ(ℐ). Let 𝑘∈𝐴^𝑐. For the case 𝛼=0, we have 𝐹_{0𝑦𝑘−0𝜉,𝑧}(𝜀)=𝐹_0,𝑧(𝜀)>(1−𝜆) (24) and for the case 𝛼≠0𝐹_{𝛼𝑦𝑘−𝛼𝜉,𝑧}(𝜀)=𝐹_{𝑦𝑘−𝜉,𝑧}𝜀|𝛼|≥sup𝐹_{𝑦𝑘−𝜉,𝑧}(𝜀)∗𝐹_0,𝑧𝜀|𝛼|𝜀−𝜀∶𝜀+|𝛼|)}=().−𝜀=𝜀>sup{(1−𝜆∗11−𝜆 (25) So {𝑘∈ℕ∶𝛼𝑦_𝑘,𝑧∉𝒩_{𝛼𝜉,𝑧}(𝜀,𝜆)}∈𝒦. Hence ℐ^𝐹𝒦-lim𝛼𝑥_𝑘,𝑧=𝛼𝜉.
(iii) The result follows from (i) and (ii).

One can show easily directly the definitions that 𝒦-convergence with respect to the random 2-norm 𝐹 implies ℐ^𝐹𝒦-convergence.

Lemma 15. Let 𝒦 and ℐ be ideals on ℕ. If (𝑋,𝐹,∗) is an RTN space and (𝑥_𝑘)_𝑘∈ℕ is a sequence in 𝑋 such that 𝒦-lim𝑥_𝑘,𝑧=𝜉 with respect to the random 2-norm 𝐹, then ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉.

Lemma 16. Let (𝑋,𝐹,∗) an RTN space and let ℐ, ℐ₁,ℐ₂,𝒦, 𝒦₁, and 𝒦₂ be ideals on ℕ such that ℐ₁⊂ℐ₂ and 𝒦₁⊂𝒦₂. Then, for every sequence (𝑥_𝑘)_𝑘∈ℕ in 𝑋, we have (i)ℐ^𝐹𝒦₁-lim𝑥_𝑘,𝑧=𝜉⇒ℐ^𝐹𝒦₂-lim𝑥_𝑘,𝑧=𝜉, (ii)ℐ^𝐹𝒦1-lim𝑥_𝑘,𝑧=𝜉⇒ℐ^𝐹𝒦2-lim𝑥_𝑘,𝑧=𝜉.

Proof. (i) Let 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋 and suppose that ℐ^𝐹𝒦₁-lim𝑥_𝑘,𝑧=𝜉. Then there exists 𝑀∈ℱ(ℐ₁) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝜉,if𝑘∉𝑀 (26) has the following condition 𝐴=𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)∈𝒦. (27) On the other hand, since ℐ₁⊂ℐ₂, 𝑀∈ℱ(ℐ₁)⊂ℱ(ℐ₂) and the by Definition 10, ℐ^𝐹𝒦₂-lim𝑥_𝑘,𝑧=𝜉.
(ii) Let 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋. Since ℐ^𝐹𝒦1-lim𝑥_𝑘,𝑧=𝜉, there exists 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝜉,if𝑘∉𝑀 (28) has the following condition 𝐴=𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)∈𝒦₁⊂𝒦₂. (29) Hence, 𝐴∈𝒦₂ and the proof is complete.

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

Theorem 17. Let (𝑋,𝐹,∗) be an RTN space and let 𝒦 and ℐ be ideals on ℕ. Let (𝑥_𝑘)_𝑘∈ℕ be a sequence in 𝑋. (i)If ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉 implies ℐ^𝐹-lim𝑥_𝑘,𝑧=𝜉 holds for some 𝜉∈𝑋, which has at least one neighborhood different from 𝑋, then 𝒦⊂ℐ. (ii)If 𝒦⊂ℐ, then ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉 implies ℐ^𝐹-lim𝑥_𝑘,𝑧=𝜉.

Proof. (i) Let 𝜀>0,𝜆∈(0,1), and any nonzero 𝑧∈𝑋. Suppose that 𝒦⊆ℐ̸; that is, there exists a set 𝐴∈𝒦⧵ℐ. Let 𝜉∈𝑋 be a point with neighborhood 𝑈⊂𝑋 such that 𝑈≠𝑋 and 𝑦∈𝑋⧵𝑈. Let us define a sequence {𝑥_𝑘} on 𝑋 by 𝑥_𝑘=𝜉,if𝑘∉𝐴𝑦,otherwise. (30) Clearly, 𝒦-lim𝑥_𝑘,𝑧=𝜉 and thus by Lemma 15 we get ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉. As {𝑘∈ℕ∶𝑥_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)}=𝐴∉ℐ, the sequence (𝑥_𝑘)_𝑘∈ℕ is not ℐ^𝐹𝒦-convergent to 𝜉.
(ii) Let (𝑋,𝐹,∗) be an RTN space and 𝜉∈𝑋 and let (𝑥_𝑘)_𝑘∈ℕ be a sequence on 𝑋. Assume that 𝒦⊂ℐ and ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=𝜉. By the definition of ℐ^𝐹𝒦-convergence, there exists 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀𝜉,if𝑘∉𝑀 (31) has the condition for every 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋𝐴=𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜉,𝑧(=𝜀,𝜆)𝑘∈𝑀∶𝑥_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)∈𝒦. (32) Hence, 𝐴∩𝑀∈𝒦⊂ℐ. Consequently, 𝑘∈ℕ∶𝑥_𝑘,𝑧∉𝒩_𝜉,𝑧(𝜀,𝜆)⊆(𝑋⧵𝑀)∪(𝐴∩𝑀)∈ℐ (33) and thus ℐ^𝐹-lim𝑥_𝑘,𝑧=𝜉.

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

Example 18. Let 𝑋=ℝ², with the 2-norm ‖𝑥,𝑦‖∶=|𝑥₁𝑦₂−𝑥₂𝑦₁| where 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ², and let 𝑎∗𝑏=𝑎𝑏 for all 𝑎,𝑏∈𝑆. Now for all 𝑥,𝑦∈𝑋, and 𝑡>0, let us define 𝐹_𝑥,𝑦(𝑡)=𝑡.𝑡+𝑥,𝑦 (34) Then, (ℝ²,𝐹,∗) is an RTN space. Let ℕ=∪^∞_𝑖=1𝐷_𝑖 be a decomposition of ℕ such that for any 𝑛∈ℕ each 𝐷_𝑖 contains infinitely many 𝑖’s where 𝑖≥𝑗 and 𝐷_𝑖∩𝐷_𝑗=∅ for 𝑖≠𝑗. Denote by ℐ the class of all 𝐴⊆ℕ such that 𝐴 intersects only a finite numbers of 𝐷_𝑖. Let 𝒦 be the family of all finite subsets of ℕ. We define a sequence (𝑥_𝑘)_𝑘∈ℕ as follows: 𝑥_𝑘=(1/𝑖,0)∈ℝ² if 𝑘∈𝐷_𝑖. Then, for nonzero 𝑧∈𝑋, we have 𝐹_{𝑥𝑘,𝑧}(𝑡)=𝑡𝑡+𝑥_𝑘,𝑧⟶1 (35) as 𝑘→∞. Hence ℐ^𝐹-lim𝑥_𝑘,𝑧=𝜉.
Now assume that ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=0. Then, there exists 𝑀∈ℱ(ℐ) such that the sequence (𝑦_𝑘)_𝑘∈ℕ given by 𝑦_𝑘=𝑥_𝑘,if𝑘∈𝑀0,if𝑘∉𝑀 (36) has the condition for every 𝜀>0,𝜆∈(0,1) and any nonzero 𝑧∈𝑋𝑘∈ℕ∶𝑦_𝑘,𝑧∉𝒩_𝜃,𝑧(𝜀,𝜆)∈𝒦; (37) that is, 𝒦-lim𝑦_𝑘,𝑧=𝜉. Since 𝑀∈ℱ(ℐ), then there exists 𝐴∈ℐ such that 𝑀=ℕ⧵𝐴. Now, by the definition of ℐ, there exists a 𝑠∈ℕ such that 𝐴⊂∪^𝑠_𝑖=1𝐷_𝑖. But 𝐷_𝑠+1⊂𝑀, and therefore 𝑥_𝑘=(1/(𝑠+1),0) for infinitely many 𝑘’s from 𝑀 which contradicts the assumption that ℐ^𝐹𝒦-lim𝑥_𝑘,𝑧=0. 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.