Ideal convergent function sequences in random 2-normed spaces

Filomat 30:3 (2016), 557–567 DOI 10.2298/FIL1603557S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Ideal Convergent Function Sequences in Random 2-Normed Spaces Ekrem Savaşa , Mehmet Gürdalb a Department b Department of Mathematics, Istanbul Ticaret University, Üsküdar-Istanbul, Turkey 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 x 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 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], Gähler 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 Gähler 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 Ljubiša D.R. Kočinac Email addresses: [email protected] (Ekrem Savaş), [email protected] (Mehmet Gürdal) E. Savaş, M. Gürdal / 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 I of 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 be found. 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 ⊆ N and n ∈ N we denote K (n) := cardK ∩ {1, 2, ..., n} and we define lower and upper asymptotic density of the set K by the formulas δ (K) := lim inf n→∞ K (n) K (n) ; δ (K) := lim sup . n n n→∞ If δ(K) = δ(K) =: δ(K), then the common value δ(K) is called the asymptotic density of the set K and 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]. n δ (K) = lim n 1 1X χK (k) , |Kn | = lim n n n k=1 if it exists, where χK is the characteristic function of the set K [13]. We say that a number sequence x = (xk )k∈N statistically converges to a point L if 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 N if I is hereditary (i.e. B ⊆ A ∈ I ⇒ B ∈ I) and additive (i.e. A, B ∈ I ⇒ A ∪ 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 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 I be an admissible ideal on N and x = (xk )k∈N be 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-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 I the ideal I f 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. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 559 Definition 1.1. ([15]) Let X be a real vector space of dimension d, where 2 ≤ d < ∞. A 2-norm on X is a
function k(·, ·)k : 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 + 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 = 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 = x1 y2 − x2 y1 , 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 ∈ X and α ∈ R. Also, if x, y and z are linearly independent, then x, y + z = x, y + k(x, z)k or

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 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 [39]. Definition 1.2. 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 inft∈R f (t) = 0 and supt∈R f (t) = 1. We denote the set of all distribution functions by D+ such that f (0) = 0. If a ∈ R+ , then Ha ∈ D+ , where ( 1 if t > a, Ha (t) = 0 if t ≤ a. It is obvious that H0 ≥ f for all f ∈ D+ . Definition 1.3. 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 1.4. Let X be a linear space of dimension greater than one, τ be 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; |α|t ) 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 ∈ X and 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 ∈ X and every t > 0, consider F(x, y; t) = t t+
. x, y Then observe that (X, F, ∗) is a random 2-normed space. E. Savaş, M. Gürdal / 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) n o N(θ,z) (ε, η) : ε > 0, η ∈ (0, 1), z ∈ X , where o n N(θ,z) (ε, η) = (x, z) ∈ X × X : F(x,z) (ε) > 1 − η . 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 = (xk ) in X is said to be F-convergence to L ∈ X if for every ε > 0, λ ∈ (0, 1) and for each nonzero z ∈ X there exists a positive integer N such 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 ∈ X with respect to the random norm F if, for every ε > 0 and λ ∈ (0, 1) , there exists k0 ∈ N such that Fxk −L (ε) > 1 − λ whenever k ≥ k0 . It is denoted by F-lim x = L or xk →F L as 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 F if, for every ε > 0 and λ ∈ (0, 1) , there exists k0 ∈ N such 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 convergence 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 In ⊂ P (N) and a random 2-normed o space (Y, F , ∗) . Assume that (X, F, ∗) is a RTN ′ ′ ′ space and that N(θ,z) (ε, η) = (x, z) ∈ X × X : F(x,z) (ε) > 1 − η , called the F -topology, is given.
Let fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be a sequence of functions. A sequence of functions fk k∈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 n o
′ fk (x) − f (x) , z ∈ N(θ,z) (ε, η) = x, z ∈ X × X : F′(( fk (x)− f (x)),z) (ε) > 1 − η for each k ≥ N and for each x ∈ X or equivalently,
fk (x) , z ∈ N(′f (x),z) (ε, η) for each k ≥ N and for each x ∈ X. In this case we write fk →F2 f. First we define I-pointwise convergence in a random 2-normed space. E. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 561
Definition 2.1. Let fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be a sequence of functions. fk k∈N is said to be Ipointwise 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 ∈ X the set n o
k ∈ N : fk (x) , z < N(′f (x),z) (ε, λ) belongs 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, F′ , ∗) be RTN spaces. Assume that fk k∈N is pointwise convergent (on X) with respect to F-topology where fk : (X, F, ∗) → (Y, F′ , ∗) , 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 fk k∈N is F-convergent on X. In this case the sequence fk (x) is convergent respect to F′ -topology for each x ∈ X. Then, there exists a number k0 = k0 (ε) ∈ N such that
with ′ fk (x) , z ∈ N( f (x),z) (ε, λ) for every k ≥ k0 , every nonzero z ∈ X and for each x ∈ X. This implies that the set n o
A (ε, λ) = k ∈ N : fk (x) , z < N(′f (x),z) (ε, λ) ⊆ {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 X as in Example 1.5, we have (X, F, ∗) is a RTN space induced by the random 2-norm ε Fx,y (ε) = ε+ (x,y) . Define a sequence of functions fk : [0, 1] → R via k k  k2  x + 1 if k = m2     0 if k , m2    fk (x) =  0 if k = m2    xk + 1 if k , m2   2   2 (m ∈ N) and x ∈ [0, 21 ) (m ∈ N) and x ∈ [0, 21 ) (m ∈ N) and x ∈ [ 12 , 1) (m ∈ N) and x ∈ [ 12 , 1) if x = 1.  
Then, for every ε > 0, λ ∈ (0, 1), x ∈ [0, 12 ) and each nonzero z ∈ X, let An (ε, λ) = k ≤ n : fk (x) , z < N(′f (x),z) (ε, λ) . We observe that
′ fk (x) , z < N(θ,z) (ε, λ) ⇒ F′( fk (x),z) (ε) ≤ 1 − λ ε ⇒
≤1−λ ε + fk (x) , z
ελ ⇒ fk (x) , z ≤ > 0. 1−ε Hence, we have o n
An (ε, λ) = k ≤ n : fk (x) , z > 0 n o 2 = k ≤ n : fk (x) = xk + 1 n o = k ≤ n : k = m2 and m ∈ N
which yields An (ε, λ) ∈ I. Therefore, for each x ∈ [0, 12 ), fk k∈N is I-convergence to 0 with respect to
F-topology. Similarly, if we take x ∈ [ 21 , 1) and x = 1, it can be seen easily that fk k∈N is I-convergence
to 12 and 2 with respect to F-topology, respectively. Hence fk k∈N is pointwise convergent with respect to F-topology (on X). E. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 562 Theorem 2.4. Let (X, F, ∗), (Y, F′ , ∗) be RTN spaces and let fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be a sequence of functions. Then the following statements are equivalent: (i) fk →I(F2 ) f.  
(ii) k ≤ n : fk (x) , z < N(′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 ∈ N(′f (x),z) (ε, λ) ∈ F (I) for every ε > 0, λ ∈ (0, 1), for each x ∈ X and each nonzero z ∈ X. (ε) = 1 for every x ∈ X and each nonzero z ∈ X. (iv) I-lim F′ ( fk (x)− f (x),z) Proof is standard.

(Y, ′ Theorem 2.5. Let fk k∈N and 1k k∈N be two sequences of
functions from (X,
F, ∗) to F , ∗) with a ∗ a > a for every a ∈ (0, 1) . If fk →I(F2 ) f and 1k →I(F2 ) 1, then α fk + β1k →I(F2 ) α f + β1 where α, β ∈ R (or C). Proof. Let ε > 0 and λ ∈ (0, 1) . Since fk →I(F2 ) f and 1k →I(F2 ) 1 for each x ∈ X, we have    

ε ε A = k ∈ N : fk (x) , z < N ′ f (x),z ( , λ) and B = k ∈ N : 1k (x) , z < N ′1(x),z ( , λ) ( ) 2 ( ) 2    

( 2ε , λ) belong to I. This implies that Ac = k ∈ N : fk (x) , z ∈ N ′ ( 2ε , λ) and Bc = k ∈ N : 1k (x) , z ∈ N ′ (1(x),z) ( f (x),z) belong to F (I) . Let n o

′ C = k ∈ N : α fk (x) + β1k (x) , z < N((α f (x)+β1(x)),z) (ε, λ) . Since I is 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 F′ 0· f (x)−0·1 (x),z (ε) = F′0 (ε) = 1 > 1 − λ ( k ) k and for the case α, β , 0, we have     ε ε ∗ F′ β1 (x)−β1(x) ,z F′ α f (x)+β1 (x)−α f (x)+β1(x) ,z (ε) ≥ F′ α f (x)−α f (x) ,z (( (( k ) ) (( ) ) ) ) k k k 2 2 !   ε ε ′ ′ ∗ F 1 (x)−1(x) ,z = F f (x)− f (x) ,z (( k ) ) 2β (( k ) ) 2α > (1 − λ) ∗ (1 − λ) > 1 − λ. Hence, k ∈ Cc ⊃ Ac ∩ Bc ∈ F (I) which implies C ⊂ A ∪ B ∈ I and the result follows. ′ (Y, F′ , ∗) , k ∈ N, be a sequence of Definition 2.6. Let (X, F, ∗), (Y, F
, ∗) be RTN spaces and let fk : (X, F, ∗) → functions. Then a sequence fk k∈N is called I-pointwise Cauchy sequence in RTN space if for every ε > 0, λ ∈ (0, 1) and each nonzero z ∈ X there exists M = M (ε, λ, x) ∈ N such that n o
k ∈ N : fk (x) − fM (x) , z < Nθ′ (ε, λ) ∈ I. (0, 1) and let fk : (X, F, ∗) → Theorem 2.7. Let (X, F, ∗), (Y, F′ , ∗) be RTN spaces such
that a ∗ a > a for every a ∈ (Y, F′ , ∗) , k ∈ N, be a sequence of functions. If fk k∈N is an I-pointwise convergent sequence with respect to
F-topology, then fk k∈N is an I-pointwise Cauchy sequence with respect to F-topology. E. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 563
Proof. Suppose that fk k∈N is an I-pointwise convergent to f with respect to F-topology. Let ε > 0 and λ ∈ (0, 1) be given. We have  
ε A = k ∈ N : fk (x) , z < N(′f (x),z) ( , λ) ∈ I. 2 This implies that Ac ∈ F (I) . Now, for every k, m ∈ Ac ,     ε ε ∗ F′ f (x)− f (x),z F′ f (x)− f (x),z (ε) ≥ F′ f (x)− f (x),z ) 2 (m (k ) (k ) 2 m > (1 − λ) ∗ (1 − λ) > 1 − λ. n o
′ So, k ∈ N : fk (x) − fm (x) , z ∈ N(θ,z) (ε, λ) ∈ F (I) . Therefore n o
′ (ε, λ) ∈ I, k ∈ N : fk (x) − fm (x) , z < N(θ,z) i.e., fk
k∈N is 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, F′ , ∗) 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, F′ , ∗) , k ∈ N, are equi-continuous (on X) and f : (X, F, ∗) → (Y, F′ , ∗) . Then f is continuous (on X) with respect to F-topology. Proof. We will prove that f is continuous with respect to F-topology. Let x0 ∈ X and (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 < λ such that fk (x) − fk (x0 ) , z ∈ N(θ,z) ( 3ε , γ) for every k ∈ N. Since fk →I(F2 ) f, the set    

ε ε k ∈ N : fk (x0 ) , z < N ′ f (x ),z ( , γ) ∪ k ∈ N : fk (x) , z < N ′ f (x),z ( , γ) ( ) 3 ( 0 ) 3

is in I and different from N. So, there exists k ∈ F (I) such that fk (x0 ) , z ∈ N(′f (x0 ),z) ( 3ε , γ) and fk (x) , z ∈ N(′f (x),z) ( 3ε , γ). We have        ε ε ε ∗ F′ f (x )− f (x),z ∗ F′ f (x)− f (x),z F′ f (x )− f (x),z (ε) ≥ F′ f (x )− f (x ),z (k 0 k ) 2 (k ) 3 ) ( 0 ( 0 k 0 ) 3


 > 1−γ ∗ 1−γ ∗ 1−γ

> 1−γ ∗ 1−γ >1−γ >1−λ and the contiunity of f with 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, F′ , ∗) be RTN spaces. We say that a sequence of functions fk : (X, F, ∗) → (Y, F′ , ∗) , 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) ∈ M ∋ ∀k > k0 and k ∈ M, ∀z ∈ X and ∀x ∈ X, λ ∈ (0, 1) fk (x) , z ∈ N ′ (ε, λ). ( f (x),z) In this case we write fk ⇒I(F2 ) f . E. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 564 Theorem 2.10. Let (X, F, ∗), (Y, F′ , ∗) be RTN spaces and let fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be a sequence of functions. Then for every ε > 0 and λ ∈ (0, 1), the following statements are equivalent: (i) fk ⇒I(F2 ) f. 
(ii) k ≤ n : fk (x) , z < N ′ (ε, λ) ∈ I for every x ∈ X and each nonzero z ∈ X. ( f (x),z)  
′ (x) (iii) k ≤ n : fk ,z ∈ N (ε, λ) ∈ F (I) for every x ∈ X and each nonzero z ∈ X. ( f (x),z) (ε) = 1 for every x ∈ X and each nonzero z ∈ X. (iv) I-lim F′ ( fk (x)− f (x),z) Proof is standard, so omitted. Definition 2.11. Let (X, F, ∗) be a RTN space. A subset Y of X is 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 ∈ Y and every nonzero z ∈ X. Definition 2.12. Let (X, F, ∗), (Y, F′ , ∗) be RTN spaces and let fk : (X, F,∗) → (Y, F′ , ∗) , k ∈ N,and f : (X, F, ∗) → (ε) = 1. (Y, F′ , ∗) be bounded functions. Then fk ⇒I(F2 ) f if and only if I-lim infx∈X F′ ( fk (x)− f (x),z) Example 2.13. Let (X, F, ∗) be as in Example 1.5. Define a sequence of functions fk : [0, 1) → R by ( xk + 1 if k , m2 (m ∈ N) fk (x) = 2 otherwise. n o
′ Then, for every ε > 0, λ ∈ (0, 1) and each nonzero z ∈ X, let An (ε, λ) = k ≤ n : fk (x) , z < N(1,z) (ε, λ) . 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 ) f then 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 ( 0 if k = n2 2 fk (x) = k x otherwise 1+k3 x2   on [0, 1] . Since fk 1k →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, F′ , ∗) be RTN spaces. Assume that fk k∈N is uniformly convergent (on X) with respect to F-topology where fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N. Then fk ⇒I(F2 ) f (on X).
Proof. Assume that fk k∈N is uniformly convergent to f on X with 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 ∈ N ′ (ε, λ). That is, for k ≤ k0 ( f (x),z)  
A (ε, λ) = k ∈ N : fk (x) , z < N ′ 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) ∈ Ac such that ∀k > k0 and k ∈ Ac and ∀x ∈ X, fk (x) , z ∈ N ′ (ε, λ). This ( f (x),z) implies that fk ⇒I(F2 ) f (on X). E. Savaş, M. Gürdal / Filomat 30:3 (2016), 557–567 565 Definition 2.17. Let (X, F, ∗), (Y,
F′ , ∗) be RTN spaces and let fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be a sequence of functions. Then a sequence fk k∈N is called I-uniform Cauchy sequence in RTN space if for every ε > 0, λ ∈ (0, 1) and each nonzero z ∈ X there exists N = N (ε, λ) ∈ N such that o n
′ (ε, λ) ∈ I. k ∈ N : fk (x) − fN (x) , z < N(θ,z) Theorem 2.18. Let (X, F, ∗), (Y, F′ , ∗) be RTN spaces such that a ∗ a > a for every a ∈ (0, 1) and let fk : (X, F, ∗) →
(Y, F′ , ∗) , k ∈ N, be a sequence of functions. If fk k∈N is an I-uniform convergence sequence with respect to
F-topology, then fk k∈N is an I-uniform Cauchy sequence with respect to F-topology.  
Proof. Suppose that fk ⇒I(F2 ) f . Let A = k ∈ N : fk (x) , z ∈ N ′ (ε, λ) . By Definition 2.9, for every ( f (x),z) ε > 0, λ ∈ (0, 1) and each nonzero z ∈ X, there exists A ⊂ N, A ∈ F (I) and ∃k0 = k0 (ε, λ) ∈ A such that
for all k > k0 , k ∈ A and for all x ∈ X, fk (x) , z ∈ N ′ ( 2ε , λ). Choose N = N (ε, λ) ∈ A, N > k0 . So, ( f (x),z)
( 2ε , λ). For every k ∈ A, we have fN (x) , z ∈ N ′ ( f (x),z)     ε ε F′ f (x)− f (x),z (ε) ≥ F′ f (x)− f (x),z ∗ F′ f (x)− f (x),z (k ) (k ) 2 ( ) 2 N N > (1 − λ) ∗ (1 − λ) > 1 − λ. n o
′ Hence, k ∈ N : fk (x) − fN (x) , z ∈ N(θ,z) (ε, λ) ∈ F (I) . Therefore n o
′ k ∈ N : fk (x) − fN (x) , z < N(θ,z) (ε, λ) ∈ I,
i.e., fk is 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, F′ , ∗) be RTN spaces such that a∗a > a for every a ∈ (0, 1) and the map fk : (X, F, ∗) → (Y, F′ , ∗) , k ∈ N, be continuous (on X) with respect to F-topology. If fk ⇒I(F2 ) f (on X) then f : (X, F, ∗) → (Y, F′ , ∗) is continuous (on X) with respect to F-topology. Proof. Let x0 ∈ X and (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 < λ such that fk (x0 ) − fk (x) , z ∈ N(θ,z) ( 3ε , γ) for every k ∈ N. Since fk ⇒I(F2 ) f, for all x ∈ X, the set    

ε ε k ∈ N : fk (x) , z < N ′ f (x),z ( , γ) ∪ k ∈ N : fk (x0 ) , z < N ′ f (x ),z ( , γ) ( 0 ) 3 ( ) 3

( ε , γ) and fm (x0 ) , z ∈ is in I and different from N. So, there exists m ∈ F (I) such that fm (x) , z ∈ N ′ ( f (x),z) 3 ( ε , γ). It follows that N′ ( f (x0 ),z) 3        ε ε ε ∗ F′ f (x )− f (x ),z ∗ F′ f (x )− f (x ),z F′ f (x)− f (x ),z (ε) ≥ F′ f (x)− f (x),z ) 3 (m 0 m 0 ) 2 (m 0 ) ( ) 3 ( 0 0 m


 > 1−γ ∗ 1−γ ∗ 1−γ

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