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Filomat 30:3 (2016), 557–567
DOI 10.2298/FIL1603557S
Published by Faculty of Sciences and Mathematics,
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Ideal Convergent Function Sequences in Random 2-Normed Spaces
Ekrem Savas¸a, Mehmet G ¨urdalb
aDepartment of Mathematics, Istanbul Ticaret University, ¨
Usk¨udar-Istanbul, Turkey
bDepartment of Mathematics, Suleyman Demirel University, 32260, Isparta, Turkey
Abstract. In the present paper we are concerned with I-convergence of sequences of functions in random
2-normed spaces. Particularly, following the line of recent work of Karakaya et al. [23], we introduce
the concepts of ideal uniform convergence and ideal pointwise convergence in the topology induced by
random 2-normed spaces, and give some basic properties of these concepts.
1. Introduction and Preliminaries
The theory of probabilistic normed (PN) spaces is 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 norms of the vectors are uncertain due to randomness. A PN
space is a generalization of an ordinary normed linear space. In a PN space, the norms of the vectors are
represented by probability distribution functions instead of nonnegative real numbers. If xis 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 xis smaller than t. PN spaces were first introduced by Sherstnev in [42] by means of a definition that was
closely modelled on the theory of normed spaces. In 1993, Alsina et al. [1] presented a new definition of a
PN space which includes the definition of Sherstnev [43] 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 [39, 40]). This new definition quickly became the
standard one and it has been adopted by many authors (for instance, [2, 17, 21, 25, 32–35, 38, 50]), who have
investigated several properties of PN spaces. A detailed history and the development of the subject up to
2006 can be found in [41].
In [15], G¨
ahler introduced an attractive theory of 2-normed spaces in the 1960’s. This notion which is
nothing but a two dimensional analogue of a normed space got the attention of a wider audience after the
publication of a paper by Albert George White [51]. Siddiqi [44] delivered a series of lectures on this theme
in various conferences. His joint paper with G¨
ahler and Gupta [16] also provided valuable results related
to the theme of this paper. Results up to 1977 were summarized in the survey paper by Siddiqi [44]. Since
then, many researchers have studied these subjects and obtained various results [9, 17–22, 30, 46, 50].
2010 Mathematics Subject Classification. Primary 40A35; Secondary 05D40, 46S50
Keywords. Ideal, filter, I-convergence, random 2-normed space, sequences of functions
Received: 22 June 2015; Revised: 05 August 2015; Accepted: 06 August 2015
Communicated by Ljubiˇ
sa D.R. Koˇ
cinac
Email addresses: ekremsavas@yahoo.com (Ekrem Savas¸), gurdalmehmet@sdu.edu.tr (Mehmet G ¨
urdal)
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 558
The concepts of statistical convergence for sequences of real numbers was introduced (independently)
by Steinhaus [45] and Fast [12]. The concept of statistical convergence was further discussed and developed
by many authors including [45]. There has been an effort to introduce several generalizations and variants
of statistical convergence in different spaces [5, 14, 19, 25, 29–32, 36]. One such very important generalization
of this notion was introduced by Kostyrko et al. [26] by using an ideal Iof subsets of the set of natural
numbers, which they called I-convergence. More recent applications of ideals can be seen in [8, 20, 21, 33–
35, 37, 46, 48–50] where more references can befound. Different types of statistical convergence of sequences
of real functions and related notions were first studied in [4], and some important results and references
on statistical convergence and function sequences can be found in [6, 7, 10, 11]. Recently, in [23], Karakaya
et al. studied the statistical convergence of sequences of functions with respect to the intuitionistic fuzzy
normed spaces. Recently, in [24], Karakaya et al. introduced the concept of λ-statistical convergence of
sequences of functions in the intuitionistic fuzzy normed spaces.
The notion of ideal convergence of sequences of functions 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 ideal convergence of sequences of functions was introduced in a random 2-normed space, and
some important results are established. Finally, the notions of I-pointwise convergence and I-uniform
convergence in a random 2-normed space are introduced and studied.
First 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, ...}.For any K⊆Nand n∈Nwe denote K(n):=cardK ∩{1,2, ..., n}
and we define lower and upper asymptotic density of the set Kby the formulas
δ(K):=lim inf
n→∞
K(n)
n;δ(K):=lim sup
n→∞
K(n)
n.
If δ(K)=δ(K)=:δ(K),then the common value δ(K) is called the asymptotic density of the set Kand
δ(K)=lim
n→∞
K(n)
n.
Obviously all three densities δ(K), δ(K) and δ(K) (if they exist) lie in the unit interval [0,1].
δ(K)=lim
n
1
n|Kn|=lim
n
1
n
n
X
k=1
χK(k),
if it exists, where χKis the characteristic function of the set K[13]. We say that a number sequence x=(xk)k∈N
statistically converges to a point Lif for each ε > 0 we have δ(K(ε)) =0,where K(ε)={k∈N:|xk−L|≥ε}
and in such situation we will write L=st-lim xk.
The notion of statistical convergence was further generalized in the paper [26, 27] using the notion of
an ideal of subsets of the set N. We say that a non-empty family of sets I⊂P(N)is an ideal on Nif Iis
hereditary (i.e. B⊆A∈ I ⇒ B∈ I) and additive (i.e. A,B∈ I ⇒ A∪B∈ I). An ideal Ion Nfor which
I,P(N)is called a proper ideal. A proper ideal Iis called admissible if Icontains all finite subsets of N.
If not otherwise stated in the sequel Iwill denote an admissible ideal. Let I⊂P(N)be a non-trivial 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 [26, 27].
Let Ibe an admissible ideal on Nand x=(xk)k∈Nbe a sequence of points in a metric space (X, ρ).We
say that the sequence xis I-convergent (or I-converges) to a point ξ∈X,and we denote it by I-lim x=ξ,
if for each ε > 0 we have
A(ε)=k∈N:ρ(xk, ξ)≥ε∈ I.
This generalizes the notion of usual convergence, which can be obtained when we take for Ithe ideal If
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.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 559
Definition 1.1. ([15]) Let Xbe a real vector space of dimension d,where 2 ≤d<∞.A 2-norm on Xis a
function k(·,·)k:X×X→Rwhich satisfies (i)
x,y
=0 if and only if xand yare linearly dependent; (ii)
x,y
=
y,x
; (iii)
αx,y
=|α|
x,y
, α ∈R; (iv)
x,y+z
≤
x,y
+k(x,z)k.The pair (X,k(·,·)k)
is then called a 2-normed space.
As an example of a 2-normed space we may take X=R2being equipped with the 2-norm
x,y
:=the
area of the parallelogram spanned by the vectors xand y, which may be given explicitly by the formula
x,y
=x1y2−x2y1,x=(x1,x2),y=y1,y2.
Observe that in any 2-normed space (X,k(·,·)k)we have
x,y
≥0 and
x,y+αx
=
x,y
for all
x,y∈Xand α∈R.Also, if x,yand zare linearly independent, then
x,y+z
=
x,y
+k(x,z)kor
x,y−z
=
x,y
+k(x,z)k.Given a 2-normed space (X,k(·,·)k),one can derive a topology for it via the
following definition of the limit of a sequence: a sequence (xn)in Xis said to be convergent to xin Xif
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 [39].
Definition 1.2. Let Rdenote the set of real numbers, R+={x∈R:x≥0}and S=[0,1] the closed unit
interval. A mapping f:R→Sis 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 by D+such that f(0)=0. If a∈R+,then Ha∈D+,where
Ha(t)=(1 if t>a,
0 if t≤a.
It is obvious that H0≥ffor all f∈D+.
Definition 1.3. A triangular norm (t-norm) is a continuous mapping ∗:S×S→Ssuch that (S,∗)is an
abelian monoid with unit one and c∗d≤a∗bif c≤aand d≤bfor all a,b,c,d∈S.A triangle function τis a
binary operation on D+which is commutative, associative and τf,H0=ffor every f∈D+.
Definition 1.4. Let Xbe a linear space of dimension greater than one, τbe a triangle function, and F:
X×X→D+.Then Fis 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 xand yare 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 xand yare 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 by
(vi)F(x+y,z;t1+t2)≥F(x,z;t1)∗F(y,z;t2) for all x,y,z∈Xand t1,t2∈R+;
then (X,F,∗) is called a random 2-normed space (for short, RTN space).
As a standard example, we can give the following:
Example 1.5. Let (X,k(., .)k) be a 2-normed space, and let a∗b=ab for all a,b∈S.For all x,y∈Xand every
t>0,consider
F(x,y;t)=t
t+
x,y
.
Then observe that (X,F,∗)is a random 2-normed space.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 560
Let (X,F,∗) be a RTN space. Since ∗is a continuous t-norm, the system of (ε, η)-neighborhoods of θ(the
null vector in X)
nN(θ,z)(ε, η) : ε > 0, η ∈(0,1),z∈Xo,
where
N(θ,z)(ε, η)=n(x,z)∈X×X:F(x,z)(ε)>1−ηo.
determines a first countable Hausdorfftopology 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=(xk)in Xis said to be F-convergence to L∈Xif for every ε > 0, λ ∈(0,1) and for each
nonzero z∈Xthere exists a positive integer Nsuch that
(xk,z−L)∈ N(θ,z)(ε, λ) for each k≥N
or equivalently,
(xk,z)∈ N(L,z)(ε, λ) for each k≥N.
In this case we write F-lim (xk,z)=L.
We also recall that the concept of convergence and Cauchy sequence in a random 2-normed space is
studied in [3].
Definition 1.6. Let (X,F,∗)be a RN space. Then, a sequence x={xk}is said to be convergent to L∈Xwith
respect to the random norm Fif, for every ε > 0 and λ∈(0,1),there exists k0∈Nsuch that Fxk−L(ε)>1−λ
whenever k≥k0.It is denoted by F-lim x=Lor xk→FLas k→ ∞.
Definition 1.7. Let (X,F,∗)be a RN space. Then, a sequence x={xk}is called a Cauchy sequence with
respect to the random norm Fif, for every ε > 0 and λ∈(0,1),there exists k0∈Nsuch that Fxk−xm(ε)>1−λ
for all k,m≥k0.
2. Kinds of I-Convergence for Functions in RTNS
In this section we are concerned with convergence in I-pointwise convergence and I-uniform conver-
gence of sequences of functions in a random 2-normed spaces. Particularly, we introduce the ideal analog
of the Cauchy convergence criterion for pointwise and uniform ideal convergence in a random 2-normed
space. Finally, we prove that pointwise and uniform ideal convergence preserves continuity.
2.1. I-pointwise convergence in RTNS
Fix an admissible ideal I⊂P(N)and a random 2-normed space (Y,F0,∗).Assume that (X,F,∗) is a RTN
space and that N0
(θ,z)(ε, η)=n(x,z)∈X×X:F0
(x,z)(ε)>1−ηo,called the F0-topology, is given.
Let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of functions. A sequence of functions fkk∈N(on X) is
said to be F-convergence to f(on X) if for every ε > 0, λ ∈(0,1) and for each nonzero z∈X,there exists a
positive integer N=N(ε, λ, x)such that
fk(x)−f(x),z∈ N0
(θ,z)(ε, η)=nx,z∈X×X:F0
(( fk(x)−f(x)),z)(ε)>1−ηo
for each k≥Nand for each x∈Xor equivalently,
fk(x),z∈ N0
(f(x),z)(ε, η) for each k≥Nand for each x∈X.
In this case we write fk→F2f.
First we define I-pointwise convergence in a random 2-normed space.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 561
Definition 2.1. Let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of functions. fkk∈Nis said to be I-
pointwise convergent to a function f(on X) with respect to F-topology if for every x∈X, ε > 0, λ ∈(0,1)
and each nonzero z∈Xthe set
nk∈N:fk(x),z<N0
(f(x),z)(ε, λ)obelongs to I.
In this case we write fk→I(F2)f.
Theorem 2.2. Let I⊂P(N)be an admissible ideal and let (X,F,∗),(Y,F0,∗)be RTN spaces. Assume that fkk∈N
is pointwise convergent (on X) with respect to F-topology where fk: (X,F,∗)→(Y,F0,∗),k∈N.Then fk→I(F2)f
(on X). But the converse of this is not true.
Proof. Let ε > 0 and λ∈(0,1).Suppose that fkk∈Nis F-convergent on X.In this case the sequence fk(x)is
convergent with respect to F0-topology for each x∈X.Then, there exists a number k0=k0(ε)∈Nsuch that
fk(x),z∈ N0
(f(x),z)(ε, λ) for every k≥k0,every nonzero z∈Xand for each x∈X.This implies that the set
A(ε, λ)=nk∈N:fk(x),z<N0
(f(x),z)(ε, λ)o⊆{1,2,3, ..., k0−1}.
Since the right hand side belongs to I, we have A(ε, λ)∈ I.That is, fk→I(F2)f(on X).
Example 2.3. Consider Xas in Example 1.5, we have (X,F,∗) is a RTN space induced by the random 2-norm
Fx,y(ε)=ε
ε+k(x,y)k.Define a sequence of functions fk:[0,1]→Rvia
fk(x)=
xk2+1 if k=m2(m∈N)and x∈[0,1
2)
0 if k,m2(m∈N)and x∈[0,1
2)
0 if k=m2(m∈N)and x∈[1
2,1)
xk+1
2if k,m2(m∈N)and x∈[1
2,1)
2 if x=1.
Then, for every ε > 0, λ ∈(0,1),x∈[0,1
2) and each nonzero z∈X,let An(ε, λ)=k≤n:fk(x),z<N0
(f(x),z)(ε, λ).
We observe that
fk(x),z<N0
(θ,z)(ε, λ)⇒F0
(fk(x),z)(ε)≤1−λ
⇒ε
ε+
fk(x),z
≤1−λ
⇒
fk(x),z
≤ελ
1−ε>0.
Hence, we have
An(ε, λ)=nk≤n:
fk(x),z
>0o
=nk≤n:fk(x)=xk2+1o
=nk≤n:k=m2and m∈No
which yields An(ε, λ)∈ I.Therefore, for each x∈[0,1
2),fkk∈Nis I-convergence to 0 with respect to
F-topology. Similarly, if we take x∈[1
2,1) and x=1,it can be seen easily that fkk∈Nis I-convergence
to 1
2and 2 with respect to F-topology, respectively. Hence fkk∈Nis pointwise convergent with respect to
F-topology (on X).
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 562
Theorem 2.4. Let (X,F,∗),(Y,F0,∗)be RTN spaces and let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of functions.
Then the following statements are equivalent:
(i) fk→I(F2)f.
(ii) k≤n:fk(x),z<N0
(f(x),z)(ε, λ)∈ I for every ε > 0, λ ∈(0,1),for each x ∈X and each nonzero z ∈X.
(iii) k≤n:fk(x),z∈ N0
(f(x),z)(ε, λ)∈ F (I)for every ε > 0, λ ∈(0,1),for each x ∈X and each nonzero
z∈X.
(iv) I-lim F0
(fk(x)−f(x),z)(ε)=1for every x ∈X and each nonzero z ∈X.
Proof is standard.
Theorem 2.5. Let fkk∈Nand 1kk∈Nbe two sequences of functions from (X,F,∗)to (Y,F0,∗)with a ∗a>a for every
a∈(0,1).If fk→I(F2)f and 1k→I(F2)1, then αfk+β1k→I(F2)αf+β1where α, β ∈R(or C).
Proof. Let ε > 0 and λ∈(0,1).Since fk→I(F2)fand 1k→I(F2)1for each x∈X,we have
A=k∈N:fk(x),z<N0
(f(x),z)(ε
2, λ)and B=k∈N:1k(x),z<N0
(1(x),z)(ε
2, λ)
belong to I.This implies that Ac=k∈N:fk(x),z∈ N0
(f(x),z)(ε
2, λ)and Bc=k∈N:1k(x),z∈ N0
(1(x),z)(ε
2, λ)
belong to F(I).Let
C=nk∈N:αfk(x)+β1k(x),z<N0
((αf(x)+β1(x)),z)(ε, λ)o.
Since Iis an ideal it is sufficient to show that C⊂A∪B.This is equivalent to show that Cc⊃Ac∩Bc.Let
k∈Ac∩Bc.For the case α, β =0,we have
F0
(0·fk(x)−0·1k(x),z)(ε)=F0
0(ε)=1>1−λ
and for the case α, β ,0,we have
F0
((αfk(x)+β1k(x)−αf(x)+β1(x)),z)(ε)≥F0
((αfk(x)−αf(x)),z)ε
2∗F0
((β1k(x)−β1(x)),z)ε
2
=F0
((fk(x)−f(x)),z)ε
2α∗F0
((1k(x)−1(x)),z) ε
2β!
>(1−λ)∗(1−λ)
>1−λ.
Hence, k∈Cc⊃Ac∩Bc∈ F (I)which implies C⊂A∪B∈ I and the result follows.
Definition 2.6. Let (X,F,∗),(Y,F0,∗)be RTN spaces and let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of
functions. Then a sequence fkk∈Nis called I-pointwise Cauchy sequence in RTN space if for every ε > 0,
λ∈(0,1) and each nonzero z∈Xthere exists M=M(ε, λ, x)∈Nsuch that
nk∈N:fk(x)−fM(x),z<N0
θ(ε, λ)o∈ I.
Theorem 2.7. Let (X,F,∗),(Y,F0,∗)be RTN spaces such that a ∗a>a for every a ∈(0,1)and let fk: (X,F,∗)→
(Y,F0,∗),k∈N,be a sequence of functions. If fkk∈Nis an I-pointwise convergent sequence with respect to
F-topology, then fkk∈Nis an I-pointwise Cauchy sequence with respect to F-topology.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 563
Proof. Suppose that fkk∈Nis an I-pointwise convergent to fwith respect to F-topology. Let ε > 0 and
λ∈(0,1) be given. We have
A=k∈N:fk(x),z<N0
(f(x),z)(ε
2, λ)∈ I.
This implies that Ac∈ F (I).Now, for every k,m∈Ac,
F0
(fk(x)−fm(x),z)(ε)≥F0
(fk(x)−f(x),z)ε
2∗F0
(fm(x)−f(x),z)ε
2
>(1−λ)∗(1−λ)
>1−λ.
So, nk∈N:fk(x)−fm(x),z∈ N0
(θ,z)(ε, λ)o∈ F (I).Therefore
nk∈N:fk(x)−fm(x),z<N0
(θ,z)(ε, λ)o∈ I,
i.e., fkk∈Nis an I-pointwise Cauchy sequence with respect to F-topology.
The next result is a modification of a well-known result.
Theorem 2.8. Let (X,F,∗),(Y,F0,∗)be RTN spaces such that a ∗a>a for every a ∈(0,1).Assume that fk→I(F2)f
(on X) where functions fk: (X,F,∗)→(Y,F0,∗),k∈N,are equi-continuous (on X) and f : (X,F,∗)→(Y,F0,∗).
Then f is continuous (on X) with respect to F-topology.
Proof. We will prove that fis continuous with respect to F-topology. Let x0∈Xand (x−x0,z)∈ N(θ,z)(ε, λ)
be fixed. By the equi-continuity of fk’s, for every ε > 0 and each nonzero z∈X,there exists a γ∈(0,1)with
γ<λsuch that fk(x)−fk(x0),z∈ N0
(θ,z)(ε
3, γ) for every k∈N.Since fk→I(F2)f,the set
k∈N:fk(x0),z<N0
(f(x0),z)(ε
3, γ)∪k∈N:fk(x),z<N0
(f(x),z)(ε
3, γ)
is in Iand different from N.So, there exists k∈ F (I)such that fk(x0),z∈ N0
(f(x0),z)(ε
3, γ) and fk(x),z∈
N0
(f(x),z)(ε
3, γ).We have
F0
(f(x0)−f(x),z)(ε)≥F0
(f(x0)−fk(x0),z)ε
3∗F0
(fk(x0)−fk(x),z)ε
2∗F0
(fk(x)−f(x),z)ε
3
>1−γ∗1−γ∗1−γ
>1−γ∗1−γ
>1−γ
>1−λ
and the contiunity of fwith respect to F-topology is proved.
2.2. I-uniform convergence in RTNS
Now we define I-uniform convergence in a random 2-normed space.
Definition 2.9. Let I⊂P(N)be an admissible ideal and let (X,F,∗),(Y,F0,∗)be RTN spaces. We say that a
sequence of functions fk: (X,F,∗)→(Y,F0,∗),k∈N,is I-uniform convergence to a function f(on X) with
respect to F-topology if and only if ∀ε > 0,∃M⊂N,M∈ F (I)and ∃k0=k0(ε, λ, x)∈M3 ∀k>k0and
k∈M,∀z∈Xand ∀x∈X, λ ∈(0,1) fk(x),z∈ N0
(f(x),z)(ε, λ).
In this case we write fk⇒I(F2)f.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 564
Theorem 2.10. Let (X,F,∗),(Y,F0,∗)be RTN spaces and let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of
functions. Then for every ε > 0and λ∈(0,1),the following statements are equivalent:
(i) fk⇒I(F2)f.
(ii) k≤n:fk(x),z<N0
(f(x),z)(ε, λ)∈ I for every x ∈X and each nonzero z ∈X.
(iii) k≤n:fk(x),z∈ N0
(f(x),z)(ε, λ)∈ F (I)for every x ∈X and each nonzero z ∈X.
(iv) I-lim F0
(fk(x)−f(x),z)(ε)=1for every x ∈X and each nonzero z ∈X.
Proof is standard, so omitted.
Definition 2.11. Let (X,F,∗) be a RTN space. A subset Yof Xis said to be bunded on RTN spaces if for
every λ∈(0,1)there exists ε > 0 such that (x,z)∈ N(θ,z)(ε, λ) for all x∈Yand every nonzero z∈X.
Definition 2.12. Let (X,F,∗),(Y,F0,∗)be RTN spaces and let fk: (X,F,∗)→(Y,F0,∗),k∈N,and f: (X,F,∗)→
(Y,F0,∗)be bounded functions. Then fk⇒I(F2)fif and only if I-lim infx∈XF0
(fk(x)−f(x),z)(ε)=1.
Example 2.13. Let (X,F,∗) be as in Example 1.5. Define a sequence of functions fk: [0,1) →Rby
fk(x)=(xk+1 if k,m2(m∈N)
2 otherwise.
Then, for every ε > 0, λ ∈(0,1) and each nonzero z∈X,let An(ε, λ)=nk≤n:fk(x),z<N0
(1,z)(ε, λ)o.For
all x∈X,we have An(ε, λ)∈ I.Since fk→I(F2)1 for all x∈X,fk⇒I(F2)1 (on [0,1)).
Remark 2.14. If fk⇒I(F2)fthen fk→I(F2)f.But the converse of this is not true.
We prove this with the following example.
Example 2.15. Let’s define the sequence of functions
fk(x)=(0 if k=n2
k2x
1+k3x2otherwise
on [0,1].Since fk1
k→I(F2)1 and fk(0)→I(F2)0,this sequence of functions is I-pointwise convergence to
0 with respect to F-topology. But by Definition 2.12, it is not I-uniformly by convergent with respect to
F-topology.
Theorem 2.16. Let I⊂P(N)be an admissible ideal and let (X,F,∗),(Y,F0,∗)be RTN spaces. Assume that fkk∈N
is uniformly convergent (on X) with respect to F-topology where fk: (X,F,∗)→(Y,F0,∗),k∈N.Then fk⇒I(F2)f
(on X).
Proof. Assume that fkk∈Nis uniformly convergent to fon Xwith respect to F-topology. In this case, for
every ε > 0, λ ∈(0,1)and every nonzero z∈X,there exists a positive integer k0=k0(ε, λ)such that ∀x∈X
and ∀k>k0,fk(x),z∈ N0
(f(x),z)(ε, λ).That is, for k≤k0
A(ε, λ)=k∈N:fk(x),z<N0
(f(x),z)(ε, λ)⊆{1,2,3, ..., k0}∈ I
and Ac=Ac(ε, λ)belongs to F(I).Hence for every ε > 0 and every nonzero z∈X,there exists Ac⊂N,
Ac∈ F (I)and ∃k0=k0(ε, λ)∈Acsuch that ∀k>k0and k∈Acand ∀x∈X,fk(x),z∈ N0
(f(x),z)(ε, λ).This
implies that fk⇒I(F2)f(on X).
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 565
Definition 2.17. Let (X,F,∗),(Y,F0,∗)be RTN spaces and let fk: (X,F,∗)→(Y,F0,∗),k∈N,be a sequence of
functions. Then a sequence fkk∈Nis called I-uniform Cauchy sequence in RTN space if for every ε > 0,
λ∈(0,1) and each nonzero z∈Xthere exists N=N(ε, λ)∈Nsuch that
nk∈N:fk(x)−fN(x),z<N0
(θ,z)(ε, λ)o∈ I.
Theorem 2.18. Let (X,F,∗),(Y,F0,∗)be RTN spaces such that a ∗a>a for every a ∈(0,1)and let fk: (X,F,∗)→
(Y,F0,∗),k∈N,be a sequence of functions. If fkk∈Nis an I-uniform convergence sequence with respect to
F-topology, then fkk∈Nis an I-uniform Cauchy sequence with respect to F-topology.
Proof. Suppose that fk⇒I(F2)f. Let A=k∈N:fk(x),z∈ N0
(f(x),z)(ε, λ).By Definition 2.9, for every
ε > 0, λ ∈(0,1) and each nonzero z∈X, there exists A⊂N,A∈ F (I)and ∃k0=k0(ε, λ)∈Asuch that
for all k>k0,k∈Aand for all x∈X,fk(x),z∈ N0
(f(x),z)(ε
2, λ).Choose N=N(ε, λ)∈A,N>k0.So,
fN(x),z∈ N0
(f(x),z)(ε
2, λ).For every k∈A,we have
F0
(fk(x)−fN(x),z)(ε)≥F0
(fk(x)−f(x),z)ε
2∗F0
(f(x)−fN(x),z)ε
2
>(1−λ)∗(1−λ)
>1−λ.
Hence, nk∈N:fk(x)−fN(x),z∈ N0
(θ,z)(ε, λ)o∈ F (I).Therefore
nk∈N:fk(x)−fN(x),z<N0
(θ,z)(ε, λ)o∈ I,
i.e., fkis an I-uniformly Cauchy sequence in RTN space.
The next result is a modification of a well-known result.
Theorem 2.19. Let (X,F,∗),(Y,F0,∗)be RTN spaces such that a∗a>a for every a ∈(0,1)and the map fk: (X,F,∗)→
(Y,F0,∗),k∈N,be continuous (on X) with respect to F-topology. If fk⇒I(F2)f (on X) then f : (X,F,∗)→(Y,F0,∗)
is continuous (on X) with respect to F-topology.
Proof. Let x0∈Xand (x0−x,z)∈ N(θ,z)(ε, λ) be fixed. By F-continuity of fk’s, for every ε > 0 and each
nonzero z∈X,there exists a γ∈(0,1)with γ<λsuch that fk(x0)−fk(x),z∈ N0
(θ,z)(ε
3, γ) for every k∈N.
Since fk⇒I(F2)f,for all x∈X,the set
k∈N:fk(x),z<N0
(f(x),z)(ε
3, γ)∪k∈N:fk(x0),z<N0
(f(x0),z)(ε
3, γ)
is in Iand different from N.So, there exists m∈ F (I)such that fm(x),z∈ N 0
(f(x),z)(ε
3, γ) and fm(x0),z∈
N0
(f(x0),z)(ε
3, γ).It follows that
F0
(f(x)−f(x0),z)(ε)≥F0
(f(x)−fm(x),z)ε
3∗F0
(fm(x0)−fm(x0),z)ε
2∗F0
(fm(x0)−f(x0),z)ε
3
>1−γ∗1−γ∗1−γ
>1−γ∗1−γ
>1−γ
>1−λ.
This implies that fis continuous (on X) with respect to F-topology.
E. Savas¸, M. G¨urdal /Filomat 30:3 (2016), 557–567 566
Acknowledgement
We would like to express our gratitude to the referees of the paper for their useful comments and
suggestions towards the quality improvement of the paper.
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