# ℐ

Authors: U. Yamancı; M. Gürdal
Journal: International Journal of Analysis
 (2013)

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## Abstract

We studyℐ𝒦-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.


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## Body

## 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 ofℐ𝒦-convergence introduced by Mac˘aj and Sleziak [9]. Recently, Eshaghi Gordji et al. [10] studied ℐ𝒦-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. Ifx is an element of a PN space, then its norm is denoted by Fx, and the value Fx(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 ofℐ𝒦-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 ℐF𝒦-convergence of sequences is introduced in a random 2-normed space as a common generalization of all these types of ℐF*-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 integersℕ={1,2,…}. A subset K of ℕ is said to have density δ(K) if δ(K)=limn→∞n-1∑k=1nχK(k) exists, where χK 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=(xk)k∈ℕ statistically converges to a point L if for each ɛ>0 we have δ(K(ɛ))=0, where K(ɛ)={k∈ℕ:|xk-L|≥ɛ} and in such situation we will write L=st-limxk.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].Letℐ be an admissible ideal on ℕ and x=(xk)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 ɛ>0 we have
(1)A(ɛ)={k∈ℕ:ρ(xk,ξ)≥ɛ}∈ℐ.
This generalizes the notion of usual convergence, which can be obtained when we take for ℐ the ideal ℐf (Fin the ideal) of all finite subsets of ℕ. A sequence is statistically convergent if and only if it is ℐδ-convergent, where ℐδ:={K⊂ℕ:δ(K)=0} is the admissible ideal of the sets of zero asymptotic density.Definition 1 (see [11]).
LetX 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 takeX=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
(2)∥x,y∥=|x1y2-x2y1|,x=(x1,x2),y=(y1,y2).
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 (xn) in X is said to be convergent to x in X if limn→∞∥xn-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.
Letℝ 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 inft∈ℝf(t)=0 and supt∈ℝf(t)=1.We denote the set of all distribution functions byD+ such that f(0)=0. If a∈ℝ+, then Ha∈D+, where
(3)Ha(t)={1ift>a,0ift≤a.
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.
LetX 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
(4)F(x,y;t)=H0(t-∥x,y∥).
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 (ɛ,λ)-neighborhoods of θ (the null vector in X) (5){𝒩θ,z(ɛ,λ):ɛ>0,λ∈(0,1),z∈X},
where
(6)𝒩θ,z(ɛ,λ)={x,z∈X×X:Fx,z(ɛ)>1-λ}
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∈𝒩θ,z means y∈𝒩x,z and vice versa.A sequencex=(xk) in X is said to be F-convergence to L∈X if for every ɛ>0,λ∈(0,1), and any nonzero z∈X there exists a positive integer N such that
(7)xk,z-L∈𝒩θ,z(ɛ,λ)foreachk≥N
or equivalently,
(8)xk,z∈𝒩L,z(ɛ,λ)foreachk≥N.
In this case, we write F-limxk,z=L.We also recall that the concept ofℐF-convergence and ℐF*-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=(xk) in X is said to be ℐF-convergent to L∈X (ℐF-convergent to L∈X with respect to F-topology) if for each ɛ>0,λ∈(0,1) and any nonzero z∈X,
(9){k∈ℕ:xk,z∉𝒩L,z(ɛ,λ)}∈ℐ.In this case the vectorL is called the ℐF-limit of the sequence x=(xk) and we write ℐF-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=(xk) in X is said to be ℐF*-convergent to L∈X with respect to the random 2-norm F if there exists a subset
(10)M={mk:m1<m2<⋯}⊂ℕ
such that M∈ℱ(ℐ)(i.e., ℕ∖M∈ℐ) and F-limxmk,z=L.In this case we writeℐF*-limx,z=L and L is called the ℐF*-limit of the sequence x=(xk).


## 3.ℐF𝒦-Convergence in RTNS

In this section, we aim to generalize the notion ofℐF*-convergence of sequences in random 2-normed space.We give a few basic facts concerningℐF-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)
ifℐ is not proper ideal, then every sequence x=(xk) in X is ℐF-convergent to each point of X with respect to the random 2-norm F;(ii)
Ifℐ1⊂ℐ2, then for every sequence x=(xk), one has
(11)ℐ1F-limxk,z=L⟹ℐ2F-limxk,z=L.Proof.
(i) LetL be arbitrary element of X. Then for each ɛ>0,λ∈(0,1) and any nonzero z∈X(12){k∈ℕ:xk,z∉𝒩L,z(ɛ,λ)}∈ℐ.
(ii) Letℐ1⊂ℐ2,ℐ1F-limxk,z=L. Then we have for each ɛ>0,λ∈(0,1) and any nonzero z∈X(13){k∈ℕ:xk,z∉𝒩L,z(ɛ,λ)}∈ℐ1⊂ℐ2.
Hence, ℐ2F-limxk,z=L.As we have previously mentioned, we aim to generalize the notion ofℐF*-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 (xk)k∈ℕ in X is called ℐF*-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 (yk)k∈ℕ defined by
(14)yk={xk,ifk∈ML,ifk∉M
is Fin-convergent to L with respect to the random 2-norm F. If (xk)k∈ℕ is ℐF*-convergent to L, we write ℐF*-limxk,z=L.We introduce the definition ofℐF𝒦-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 (xk)k∈ℕ in X is called ℐF𝒦-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 (yk)k∈ℕ given by
(15)yk={xk,ifk∈ML,ifk∉M
is 𝒦-convergent to L with respect to the random 2-norm F. If (xk)k∈ℕ is ℐF𝒦-convergent to L, we write ℐF𝒦-limxk,z=L.Remark 11.
The definition ofℐF𝒦-convergence can be reformulated in the form of decomposition theorem. A sequence (xk)k∈ℕ is ℐF𝒦-convergent if and only if it can be written as (xk)k∈ℕ=(yk)k∈ℕ+(zk)k∈ℕ, where (yk)k∈ℕ is 𝒦-convergent with respect to the random 2-norm F and (zk)k∈ℕ is nonzero only on a set from ℐ.Example 12.
(i) Putℐ0=𝒦0={∅}. ℐ0 is the minimal ideal in ℕ. A sequence (xk)k∈ℕ is ℐ0F𝒦0-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 (xk)k∈ℕ is ℐF𝒦-convergent if and only if it is constant on ℕ∖M.
(iii) Let𝒦 be an admissible ideal in ℕ and ℐ an arbitrary ideal. A sequence (xk)k∈ℕ is ℐF𝒦-convergent if there exists a set M∈ℱ(ℐ) and the sequence (yk)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. ℐF𝒦-limit of any sequence if exists is unique.Proof.
Let(xk)k∈ℕ be any sequence and suppose that ℐF𝒦-limxk,z=ξ, ℐF𝒦-limxk,z=η, where ξ≠η. Since ξ≠η, select ɛ>0, λ∈(0,1) and any nonzero z∈X such that 𝒩ξ,z(ɛ,λ) and 𝒩η,z(ɛ,λ) are disjoint neighborhoods of ξ and η. Since ℐF𝒦-limxk,z=ξ, and ℐF𝒦-limxk,z=η, by the definition there exists M1,M2∈ℱ(ℐ) such that the sequences (yk)k∈ℕ,(zk)k∈ℕ given by
(16)yk={xk,ifk∈M1ξ,ifk∉M1,zk={xk,ifk∈M2η,ifk∉M2
having the following properties for each ɛ>0, λ∈(0,1) and any nonzero z∈X(17)A={k∈ℕ:yk,z∉𝒩ξ,z(ɛ,λ)},B={k∈ℕ:zk,z∉𝒩η,z(ɛ,λ)}
belong to 𝒦. This implies that sets Ac={k∈ℕ:yk,z∈𝒩ξ,z(ɛ,λ)} and Bc={k∈ℕ:zk,z∈𝒩η,z(ɛ,λ)} belong to ℱ(ℐ). Since ℱ(ℐ) is a filter in ℕ, we have that Ac∩Bc is a nonempty in ℱ(ℐ). In this way, we obtain a contradiction to the fact that the neighborhoods 𝒩ξ,z(ɛ,λ) and 𝒩η,z(ɛ,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 (xk) and (yk) are two sequences in (X,F,*) with a*a>a for every a∈(0,1); then(i)
ifℐF𝒦-limxk,z=ξ and ℐF𝒦-limyk,z=η, then ℐF𝒦-lim(xk+yk),z=ξ+η;(ii)
ifℐF𝒦-limxk,z=ξ and α∈ℝ, then ℐF𝒦-limαxk,z=αξ;(iii)
ifℐF𝒦-limxk,z=ξ and ℐF𝒦-limyk,z=η, then ℐF𝒦-lim(xk-yk),z=ξ-η.Proof.
(i) Letɛ>0,λ∈(0,1), and any nonzero z∈X. Since ℐF𝒦-limxk,z=ξ, and ℐF𝒦-limyk,z=η, there exists M1,M2∈ℱ(ℐ) such that the sequences (zk)k∈ℕ,(tk)k∈ℕ given by
(18)zk={xk,ifk∈M1ξ,ifk∉M1,tk={yk,ifk∈M2η,ifk∉M2
having the following conditions
(19)A={k∈ℕ:zk,z∉𝒩ξ,z(ɛ2,λ)},B={k∈ℕ:tk,z∉𝒩η,z(ɛ2,λ)}
belong to 𝒦. Let M=M1∩M2 and
(20)pk={xk+yk,ifk∈Mξ+η,ifk∉M
have the condition C={k∈ℕ:pk,z∉𝒩ξ+η,z(ɛ,λ)}. Since 𝒦 is an ideal, it is sufficient to show that C⊂A∪B. This is equivalent to showing that Cc⊃Ac∩Bc where Ac and Bc belong to ℱ(ℐ). Let k∈Ac∩Bc, that is, k∈Ac and k∈Bc, and we have
(21)F(pk)-(ξ+η),z(ɛ)=F(zk+tk)-(ξ+η),z(ɛ)≥sup{Fzk-ξ,z(ɛ2)*Ftk-η,z(ɛ2):ɛ2+ɛ2=ɛ}>sup{(1-λ)*(1-λ)}>(1-λ).
Since k∈Cc⊃Ac∩Bc∈ℱ(ℐ), we have C⊂A∪B∈𝒦.
(ii) Letɛ>0,λ∈(0,1) and any nonzero z∈X. Since ℐF𝒦-limxk,z=ξ, there exists M∈ℱ(ℐ) such that the sequence (yk)k∈ℕ given by
(22)yk={xk,ifk∈Mξ,ifk∉M
have the following condition
(23)A={k∈ℕ:yk,z∉𝒩ξ,z(ɛ,λ)}∈𝒦.
This implies that Ac={k∈ℕ:yk,z∈𝒩ξ,z(ɛ,λ)}∈ℱ(ℐ). Let k∈Ac. For the case α=0, we have
(24)F0yk-0ξ,z(ɛ)=F0,z(ɛ)>(1-λ)
and for the case α≠0(25)Fαyk-αξ,z(ɛ)=Fyk-ξ,z(ɛ|α|)≥sup{Fyk-ξ,z(ɛ)*F0,z(ɛ|α|-ɛ):ɛ+ɛ|α|-ɛ=ɛ}>sup{(1-λ)*1}=(1-λ).
So {k∈ℕ:αyk,z∉𝒩αξ,z(ɛ,λ)}∈𝒦. Hence ℐF𝒦-limαxk,z=αξ.
(iii) The result follows from (i) and (ii).One can show easily directly the definitions that𝒦-convergence with respect to the random 2-norm F implies ℐF𝒦-convergence.Lemma 15.
Let𝒦 and ℐ be ideals on ℕ. If (X,F,*) is an RTN space and (xk)k∈ℕ is a sequence in X such that 𝒦-limxk,z=ξ with respect to the random 2-norm F, then ℐF𝒦-limxk,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 (xk)k∈ℕ in X, we have(i)
ℐ1F𝒦-limxk,z=ξ⇒ℐ2F𝒦-limxk,z=ξ,(ii)
ℐF𝒦1-limxk,z=ξ⇒ℐF𝒦2-limxk,z=ξ.Proof.
(i) Letɛ>0,λ∈(0,1) and any nonzero z∈X and suppose that ℐ1F𝒦-limxk,z=ξ. Then there exists M∈ℱ(ℐ1) such that the sequence (yk)k∈ℕ given by
(26)yk={xk,ifk∈Mξ,ifk∉M
has the following condition
(27)A={k∈ℕ:yk,z∉𝒩ξ,z(ɛ,λ)}∈𝒦.
On the other hand, since ℐ1⊂ℐ2, M∈ℱ(ℐ1)⊂ℱ(ℐ2) and the by Definition 10, ℐ2F𝒦-limxk,z=ξ.
(ii) Letɛ>0,λ∈(0,1) and any nonzero z∈X. Since ℐF𝒦1-limxk,z=ξ, there exists M∈ℱ(ℐ) such that the sequence (yk)k∈ℕ given by
(28)yk={xk,ifk∈Mξ,ifk∉M
has the following condition
(29)A={k∈ℕ:yk,z∉𝒩ξ,z(ɛ,λ)}∈𝒦1⊂𝒦2.
Hence, A∈𝒦2 and the proof is complete.In the next theorem, we show the relationship between theℐF-convergence and ℐF𝒦-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 (xk)k∈ℕ be a sequence in X.(i)
IfℐF𝒦-limxk,z=ξ implies ℐF-limxk,z=ξ holds for some ξ∈X, which has at least one neighborhood different from X, then 𝒦⊂ℐ.(ii)
If𝒦⊂ℐ, then ℐF𝒦-limxk,z=ξ implies ℐF-limxk,z=ξ.Proof.
(i) Letɛ>0,λ∈(0,1), and any nonzero z∈X. Suppose that 𝒦⊈ℐ; 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 {xk} on X by
(30)xk={ξ,ifk∉Ay,otherwise.
Clearly, 𝒦-limxk,z=ξ and thus by Lemma 15 we get ℐF𝒦-limxk,z=ξ. As {k∈ℕ:xk,z∉𝒩ξ,z(ɛ,λ)}=A∉ℐ, the sequence (xk)k∈ℕ is not ℐF𝒦-convergent to ξ.
(ii) Let(X,F,*) be an RTN space and ξ∈X and let (xk)k∈ℕ be a sequence on X. Assume that 𝒦⊂ℐ and ℐF𝒦-limxk,z=ξ. By the definition of ℐF𝒦-convergence, there exists M∈ℱ(ℐ) such that the sequence (yk)k∈ℕ given by
(31)yk={xk,ifk∈Mξ,ifk∉M
has the condition for every ɛ>0,λ∈(0,1) and any nonzero z∈X(32)A={k∈ℕ:yk,z∉𝒩ξ,z(ɛ,λ)}={k∈M:xk,z∉𝒩ξ,z(ɛ,λ)}∈𝒦.
Hence, A∩M∈𝒦⊂ℐ. Consequently,
(33){k∈ℕ:xk,z∉𝒩ξ,z(ɛ,λ)}⊆(X∖M)∪(A∩M)∈ℐ
and thus ℐF-limxk,z=ξ.The following example shows that the converse of Theorem17(ii) does need not to be true.Example 18.
LetX=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
(34)Fx,y(t)=tt+∥x,y∥.
Then, (ℝ2,F,*) is an RTN space. Let ℕ=∪i=1∞Di be a decomposition of ℕ such that for any n∈ℕ each Di contains infinitely many i’s where i≥j and Di∩Dj=∅ for i≠j. Denote by ℐ the class of all A⊆ℕ such that A intersects only a finite numbers of Di. Let 𝒦 be the family of all finite subsets of ℕ. We define a sequence (xk)k∈ℕ as follows: xk=(1/i,0)∈ℝ2 if k∈Di. Then, for nonzero z∈X, we have
(35)Fxk,z(t)=tt+∥xk,z∥⟶1
as k→∞. Hence ℐF-limxk,z=ξ.
Now assume thatℐF𝒦-limxk,z=0. Then, there exists M∈ℱ(ℐ) such that the sequence (yk)k∈ℕ given by
(36)yk={xk,ifk∈M0,ifk∉M
has the condition for every ɛ>0,λ∈(0,1) and any nonzero z∈X(37){k∈ℕ:yk,z∉𝒩θ,z(ɛ,λ)}∈𝒦;
that is, 𝒦-limyk,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⊂∪i=1sDi. But Ds+1⊂M, and therefore xk=(1/(s+1),0) for infinitely many k’s from M which contradicts the assumption that ℐF𝒦-limxk,z=0. Hence, the converse of the theorem does need not to be true.



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*Source: 451762-2013-07-16.xml*