On I-convergent sequence spaces defined by a compact operator and a modulus function

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1 PURE MATHEMATICS RESEARCH ARTICLE On I-convergent sequence spaces defined by a compact operator and a modulus function Vaeel A. Khan 1 *, Mohd Shafiq 1 and Rami Kamel Ahmad Rababah 1 Received: 8 October 014 Accepted: 11 March 015 Published: 0 May 015 *Corresponding author: Vaeel A. Khan, Department of Mathematics, Aligarh Muslim University, Aligarh 000, India vahanmaths@gmail.com Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article Abstract: In this article, we introduce and study I-convergent sequence spaces I (f, I (f, and I 0 (f with the help of compact operator T on the real space R and a modulus function f. We study some topological and algebraic properties, and prove some inclusion relations on these spaces. Subjects: Science; Pure Mathematics; Advanced Mathematics; Mathematics & Statistics Keywords: compact operator; ideal; filter; I-convergent sequence; I-null sequence; I-bounded sequence; solid and monotone space; symmetric space; modulus function AMS subject classifications: 41A10; 41A5; 41A36; 40A30 1. Introduction and preliminaries Let N, R, and C be the sets of all natural, real, and complex numbers, respectively. We denote ω =x =(x : x R orc the space of all real or complex sequences. ABOUT THE AUTHORS Vaeel A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is a senior assistant professor in the same university. He is a vigorous researcher in the area of Sequence Spaces, and has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor s and Francis, Information Sciences (Elsevier, Applied Mathematics A Journal of Chinese Universities and Springer- Verlag (China, Rendiconti del Circolo Matematico di Palermo Springer-Verlag (Italy, Studia Mathematica (Poland, Filomat (Serbia, Applied Mathematics & Information Sciences (USA. Mohd Shafiq did his MSc in Mathematics from University of Jammu, Jammu and Kashmir, India. Currently, he is a PhD scholar at Aligarh Muslim University, Aligarh, India. His research interests are Functional Analysis, sequence spaces, I-convergence, invariant means, zweier sequences, and interval numbers theory. Rami Kamel Ahmad Rababah is a research scholar in the Department of Mathematics, Aligarh Muslim University, Aligarh, India. PUBLIC INTEREST STATEMENT The term sequence has a great role in Analysis. Sequence spaces play an important role in various fields of Real Analysis, Complex Analysis, Functional Analysis, and Topology Convergence of sequences has always remained a subject of interest to the researchers. Several new types of convergence of sequences were studied by the researchers and named them as usual convergence, uniform convergence, strong convergence,wee convergence, etc. Later, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of Mathematics lie Number Theory, Trigonometric Series, Summability Theory, Probability Theory, Measure Theory, Optimization, and Approximation Theory. The notion of Ideal convergence (I-convergence is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception. 015 The Author(s. This open access article is distributed under a Creative Commons Attribution (CC-BY 4.0 license. Page 1 of 13

2 Let l, c, and c 0 be denote the Banach spaces of bounded, convergent, and null sequences of reals, respectively with norm x = sup x Any subspace λ of ω is called a sequence space. A sequence space λ with linear topology is called a K-space provided each of maps p i C defined by p i (x =x i is continuous, for all i N. A space λ is called an FK-space provided λ is complete linear metric space. An FK-space whose topology is normable is called a BK-space. Definition 1.1 Let X and Y be two normed linear spaces and T : (T Y be a linear operator, where (T X. Then, the operator T is said to be bounded, if there exists a positive real such that Tx x, for all x (T The set of all bounded linear operators (X, Y is a normed linear space normed by (see Kreyszig, 1978 T = sup x X, x =1 Tx and (X, Y is a Banach space if Y is Banach space. Definition 1. Let X and Y be two normed linear spaces. An operator T : X Y is said to be a compact linear operator (or completely continuous linear operator, if (1 T is linear, ( T maps every bounded sequence (x in X onto a sequence T(x in Y which has a convergent subsequence. The set of all compact linear operators (X, Y is closed subspace of (X, Y and (X, Y is a Banach space if Y is Banach space. Following Basar and Altay (003 and Sengönül (009, we introduce the sequence spaces and 0 with the help of compact operator T on the real space R as follows. = x =(x : T(x c and 0 = x =(x : T(x c 0 Definition 1.3 A function f : [0, [0, is called a modulus if (1 f (t =0 if and only if t = 0, ( f (t + u f (t+f (u for all t, u 0, (3 f is increasing, and (4 f is continuous from the right at zero. For any modulus function f, we have the inequalities f (x f (y f ( x y and f (nx nf (x, for all x, y [0, ] A modulus function f is said to satisfy Δ Condition for all values of u if there exists a constant K > 0 such that f (Lu KLf (u for all values of L > 1. Page of 13

3 The idea of modulus was introduced by Naano (1953. Rucle (1967, 1968, 1973 used the idea of a modulus function f to construct the sequence space X(f = x =(x : f ( x < =1 This space is an FK-space and Rucle (1967, 1968, 1973 proved that the intersection of all such X(f spaces is φ, the space of all finite sequences. The space X(f is closely related to the space l 1 which is an X(f space with f (x =x for all real x 0. Thus Rucle (1967, 1968, 1973 proved that, for any modulus f. X(f l 1 and X(f α = l where X(f α = y =(y ω: f ( y x < =1 Spaces of the type X(f are a special case of the spaces structured by Gramsch (1967. From the point of view of local convexity, spaces of the type X(f are quite pathological. Symmetric sequence spaces, which are locally convex have been frequently studied by Garling (1966, Köthe (1970, and Rucle (1967, 1968, The sequence spaces by the use of modulus function was further investigated by Maddox (1969, 1986, Khan (005, 006, Bhardwaj (003, and many others. As a generalization of usual convergence, the concept of statistical convergent was first introduced by Fast (1951 and also independently by Buc (1953 and Schoenberg (1959 for real and complex sequences. Later on, it was further investigated from sequence space point of view and lined with the Summability Theory by Fridy (1985, Šalát (1980, Tripathy (1998, Khan (007, Khan and Sabiha (01, Khan, Shafiq, and Rababah (015, and many others. Definition 1.4 A sequence x =(x ω is said to be statistically convergent to a limit L C if for every ε >0, we have 1 lim n N : x n L ε, n = 0 where vertical lines denote the cardinality of the enclosed set. That is, if δ(a(ε = 0, where A(ε = N : x L ε The notation of ideal convergence (I-convergence was introduced and studied by Kostyro, Mačaj, Salǎt, and Wilczyńsi (000. Later on, it was studied by Šalát, Tripathy, and Ziman (004, 005, Tripathy and Hazaria (009, 011, Khan and Ebadullah (011, Khan, Ebadullah, Esi, and Shafiq (013, and many others. Now, we recall the following definitions: Definition 1.5 Let N be a non-empty set. Then a family of sets I N (power set of N is said to be an ideal if (1 I is additive i.e A, B I A B I ( I is hereditary i.e A I and B A B I. Page 3 of 13

4 Definition 1.6 A non-empty family of sets (I N is said to be filter on N if and only if (1 Φ (I, ( A, B (I we have A B (I, (3 A (I and A B B (I. Definition 1.7 An Ideal I N is called non-trivial if I N. Definition 1.8 A non-trivial ideal I N is called admissible if x : x N I Definition 1.9 A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J I containing I as a subset. Remar 1.10 K c = N K. For each ideal I, there is a filter (I corresponding to I. i.e (I = K N : K c I, where Definition 1.11 A sequence x =(x ω is said to be I-convergent to a number L if for every ε >0, the set N : x L ε I. In this case, we write I lim x = L. Definition 1.1 A sequence x =(x ω is said to be I-null if L = 0. In this case, we write I lim x = 0. Definition 1.13 A sequence x =(x ω is said to be I-Cauchy if for every ε >0 there exists a number m = m(ε such that N : x x m ε I. Definition 1.14 A sequence x =(x ω is said to be I-bounded if there exists some M > 0 such that N : x M I. Definition 1.15 A sequence space E is said to be solid (normal if (α x E whenever (x E and for any sequence (α of scalars with α 1, for all N. Definition 1.16 A sequence space E is said to be symmetric if (x π( E whenever x E. where π is a permutation on N. Definition 1.17 A sequence space E is said to be sequence algebra if (x (y =(x.y E whenever (x, (y E. Definition 1.18 A sequence space E is said to be convergence free if (y E whenever (x E and x = 0 implies y = 0, for all. Definition 1.19 Let K = 1 < < 3 < 4 < 5 N and E be a Sequence space. A K-step space of E is a sequence space λ E = (x K ω:(x n E. Definition 1.0 A canonical pre-image of a sequence (x n λ E is a sequence (y K ω defined by y = x, if K, 0, otherwise A canonical preimage of a step space λ E is a set of preimages all elements in K λe K. i.e. y is in the canonical preimage of λ E iff y is the canonical preimage of some x K λe. K Definition 1.1 A sequence space E is said to be monotone if it contains the canonical preimages of its step space. Page 4 of 13

5 Definition 1. (see, Khan et al., 015; Kostyro et al., 000. If I = I f, the class of all finite subsets of v. Then, I is an admissible ideal in v and I f convergence coincides with the usual convergence. Definition 1.3 (see, Khan et al., 015; Kostyro et al., 000. If I = I δ =A N : δ(a =0. Then, I is an admissible ideal in N and we call the I δ -convergence as the logarithmic statistical convergence. Definition 1.4 (see, Khan et al., 015; Kostyro et al., 000. If I = I d =A N : d(a =0. Then, I is an admissible ideal in N and we call the I d -convergence as the asymptotic statistical convergence. Remar 1.5 If I δ lim x = l, then I d lim x = l Definition 1.6 A map ħ defined on a domain D X i.e ħ : D X R is said to satisfy Lipschitz condition if ħ(x ħ(y K x y where K is nown as the Lipschitz constant. The class of K-Lipschitz functions defined on D is denoted by ħ (D, K. Definition 1.7 A convergence field of I-covergence is a set F(I =x =(x : there exists I lim x R The convergence field F(I is a closed linear subspace of l with respect to the supremum norm, F(I =l c I (see Šalát et al., 004, 005. Definition 1.8 Let X be a linear space. A function g : X R is called paranorm, if for all x, y X, (P 1 g(x =0 if x = θ, (P g( x =g(x, (P 3 g(x + y g(x+g(y, (P 4 If (λ n is a sequence of scalars with λ n λ (n and x n, a X with x n a (n in the sense that g(x n a 0 (n, then g(λ n x n λa 0 (n. The notation of paranorm sequence spaces was studied at the initial stage by Naano (1953. Later on, it was further investigated by Maddox (1969, Tripathy and Hazaria (009, Khan et al. (013, and the references therein. Throughout the article, we use the same techniques as used in Tripathy and Hazaria (009, 011. We used the following lemmas for establishing some results of this article. Lemma 1 (see, Tripathy & Hazaria, 009, 011. Every solid space is monotone. Lemma (see, Tripathy & Hazaria, 009, 011. If I N and M N. If M I, then M N I. Lemma 3 (see, Tripathy & Hazaria, 009, 011. Let K (I and M N. If M I, then M K I. Throughout the article, I, I 0, I, I, and I represent the I-convergent, I-null, I-Bounded, bounded I-convergent, and bounded I-null Sequences spaces defined by a compact operator T on the real space R, respectively.. Main results In this article, we introduce the following classes of sequences. ( I (f = x =(x : N : f T(x L ε I, for some L C ( I (f = 0 x =(x : N : f (Tx ε I (.1 (. Page 5 of 13

6 ( I (f = x =(x : N : K < 0 such that f T(x K I ( (f = x =(x : sup f T(x < (.3 (.4 where f is a modulus function. We also denote Theorem.1 Let f be a modulus function. Then, the classes of sequences I (f, I (f, 0 I (f, and I (f are linear spaces. Proof We shall prove the result for I (f. The proof for the other spaces will follow similarly. For, let x =(x, y =(y I (f and α, β be scalars. Then, for a given ε >0, we have ( N : f T(x L 1 ε, for some L C 1 I ( N : f T(x L ε, for some L C 1 I (.5 (.6 Let A 1 = N : f ( T(x L 1 < ε, for some L 1 C (I A = N : f ( T(y L < ε, for some L C (I (.7 (.8 be such that A c 1, Ac I. Since f is a modulus function, we have A 3 = N : f ( (αt(x +βt(y (αl 1 + βl <ε [ N : f ( α T(x L 1 < ε N : f ( β T(y L < ε ] [ N : f ( T(x L 1 < ε N : f ( T(y L < ε ] Therefore, A 3 = N : f ( (αt(x +βt(y (αl 1 + βl <ε [ N : f ( T(x L 1 < ε N : f ( T(y L < ε ] (.9 implies that A 3 (I. Thus, A c 3 = Ac 1 Ac I. Therefore, αx + βy I (f, for all scalars α, β, and (x, (y I (f. Page 6 of 13

7 Hence, I (f is a linear space. Theorem. The classes of sequences I (f and I (f are paranormed spaces, paranormed by g(x =g(x =sup f ( T(x Proof Let x =(x, y =(y I (f. (P 1 It is clear that g(x =0 if x = θ, a zero vector. (P g(x =g( x is obvious. (P 3 For x =(x, y =(y I (f, we have g(x + y =g(x + y =sup f ( T(x + y Therefore, g(x + y g(x + g(y (P 4 Let (λ be a sequence of scalars with (λ λ ( and (x, L I (f such that in the sense that Then, since the inequality holds by subadditivity of g, the sequence g(x is bounded. Therefore, = sup x L ( f ( T(x +T(y sup f ( T(x + sup f ( T(y = g(x+g(y g(x L 0 ( g(x g(x L+g(L g [ (λ x λl ] = g [ (λ x λx + λx λl ] = g [ (λ λx + λ(x L ] g [ (λ λx ]+g [ λ(x L ] (λ λ g(x + λ g(x L 0 as (. That is to say that scalar multiplication is continuous. Hence, I (f is a paranormed space. For I (f, the result is similar. Theorem.3 A sequence x =(x I-converges if and only if for every ε >0, there exists N ε N such that ( N : f T(x T(x Nε <ε (I (.10 Proof Let x =(x. Suppose that L = I lim x. Then, the set ( B ε = N : f T(x L < ε (I for all ε<0 Fix an N ε B ε. Then we have, Page 7 of 13

8 ( ( ( f T(x T(x Nε f T(x L + f T(x Nε L < ε + ε = ε ( which holds for all B ε. Hence N : f T(x T(x Nε <ε (I Conversely, suppose that ( N:f T(x T(x Nε <ε (I That is N: f ( T(x ( f T(x Nε < ε (I, for all ε >0. Then, the set C ε = N : f ( T(x [f ( T(x Nε ε, f ( T(x Nε + ε] (I for all ε<0 [ Let J ε = f ( T(x Nε ε, f ( T(x Nε ] + ε. If we fix an ε >0 then we have C ε (I as well as C ε (I. Hence C ε C ε (I. This implies that J = J ε J ε φ That is N : f ( T(x J (I That is diamj diamj ε where the diam of J denotes the length of interval J. In this way, by induction, we get the sequence of closed intervals J ε = I 0 I 1 I with the property that diami 1 diami 1 for ( =, 3, 4, and N : f ( T(x I (I for ( = 1,, 3, 4,. Then, there exists a ξ I where N such that ξ = I lim f ( T(x showing that x =(x is I-convergent. Hence the result. Theorem.4 Let f 1 and f be two modulus functions and satisfying Δ Condition, then (a (f (f 1 f, (b (f 1 (f (f 1 + f for = I, I, I and I. Proof (a Let x =(x I (f be any arbitrary element. Then, the set ( N : f T(x ε I (.11 Let ε >0 and choose δ with 0 <δ<1 such that f 1 (t <ε,0 t δ. Let us denote ( y = f T(x and consider ( lim f 1 y = lim f ( y δ, N 1 y + lim f ( y >δ, N 1 y Page 8 of 13

9 Now, since f 1 is an modulus function, we have lim f ( ( y δ, N 1 y f1 lim (y y δ, N (.1 For y >δ, we have y < y δ < 1 + y δ Now, since f 1 is non-decreasing and modulus, it follows that ( f 1 (y < f y < 1 δ f (+ 1 ( 1 f y 1 δ Again, since f 1 satisfies Δ Condition, we have ( 1 f 1 y < K (y δ f ( K (y δ f ( 1 ( (y Thus, f 1 y < K ( Hence, δ f 1 lim f ( ( y >δ, N 1 y max1, Kδ 1 f 1 lim (y y >δ, N (.13 Therefore, from Equations.11.13, we have (x I (f f Thus, I (f I (f f 1 1. Hence, (f (f 1 f for = I. For = I, I, and I the inclusions can be established similarly. (b Let x =(x I (f I (f 1. Let ε >0 be given. Then, the sets ( N : f 1 T(x ε I and ( N : f T(x ε I (.14 (.15 Therefore, from Equations.14 and.15 the set ( N : (f 1 + f T(x ε I Thus, x =(x I (f 1 + f. Hence, I (f 1 I (f I (f 1 + f. For = I, I, and I the inclusions are similar. For f (x =x and f 1 (x =f (x, x [0,, we have the following corollary. Corollary.5 (f for = I, I, I and I Theorem.6 For any modulus function f, the spaces I (f and I (f are solid and monotone. Proof we prove the result for the space I (f. For I (f, the proof can be obtained similarly. For, let (x I (f be any arbitrary element. Then, the set ( N : f T(x ε I (.16 Let (α be a sequence of scalars such that Page 9 of 13

10 α 1, for all N Then the result follows from Equation.16 and the following inequality. ( ( ( ( f T(α x = f α T(x α f T(x f T(x, for all N That the space I (f is monotone follows from the Lemma (I. Hence I (f is solid and monotone. Theorem.7 The spaces I (f and I (f are not neither solid nor monotone. Proof Here we give a counter example for the proof of this result. Counter example. Let I = I f and f (x =x for all x [0,. Consider the K-step K of defined as follows. Let (x and let (y K be such that y = x, if is even, 0, otherwise Consider the sequence (x defined as by x = 1 for all N. Then (x I (f and I (f but its K-step preimage does not belong to I (f and I (f. Thus, I (f and I (f are not monotone. Hence, I (f and I (f are not solid by Lemma(I. Theorem.8 If (x = x and (y = y be two sequences with T(x y =T(xT(y. Then, the spaces I (f and I (f are sequence algebra. Proof Let (x = x and (y = y be two elements of I (f with T(x y =T(xT(y. Then, the sets ( N : f T(x ε I (.17 and ( N:f T(y ε I (.18 Therefore, ( N : f T(x.T(y ε I Thus, (x.(y I (f. Hence, I (f is sequence algebra. For I (f, the result can be proved similarly. Theorem.9 Let f be a modulus function. Then, I (f I (f I (f. Proof The inclusion I (f I (f is obvious. Next, let (x I (f. Then there exists some L such that N : f ( T(x L ε I Page 10 of 13

11 We have f ( T(x 1 f ( T(x L + f ( 1 L Taing supremum over on both sides, we get (x I (f Hence, I (f I (f I (f Theorem.10 If f (x =x for all x [0, ]. Then, the function ħ : I (f R defined by ħ(x =I lim f ( T(x, where I (f = (f I (f is a Lipschitz function and hence uniformly continuous. Proof Clearly, the function ħ is well defined. Let x =(x, y =(y I (f, x y. Then, the sets A x = N : f ( T(x ħ(x x y I A y = N : f ( T(y ħ(y x y I where x y = sup f ( T(x T(y Thus, the sets B x = N : T(x ħ(x < x y (I B y = N : T(y ħ(y < x y (I Hence, B = B x B y (I, so that B Now, taing B, we have ħ(x ħ(y ħ(x T(x + T(x T(y + T(y ħ(y 3 x y Therefore, ħ is Lipschitz function and hence uniformly continuous. Theorem.11 If f (x =x for all x [0, ] and if x =(x, y =(y I (f with T(x y =T(xT(y. Then (x y I (f and ħ(xy =ħ(xħ(y where ħ : I (f R is defined by ħ(x =I lim f ( T(x. Proof For ε >0, the sets B x = N : T(x ħ(x <ε (I B y = N : T(y ħ(y <ε (I (.19 (.0 where x y = ε Now, T(x y ħ(xħ(y = T(x T(y T(x ħ(y+t(x ħ(y ħ(xħ(y T(x y ħ(y + ħ(y x ħ(x (.1 As I (f (f, there exists an M R such that T(x < M and ħ(y < M. Therefore, from Equations.19.1, we have T(x y ħ(xħ(y Mε + Mε = Mε Page 11 of 13

12 for all B x B y (I. Hence (x y I (f and ħ(xy =ħ(xħ(y. Acnowledgements The authors would lie to record their gratitude to the reviewer for his careful reading and maing some useful corrections which improved the presentation of the paper. Funding The authors received no direct funding for this research. Author details Vaeel A. Khan 1 vahanmaths@gmail.com Mohd Shafiq 1 shafiqmaths7@gmail.com Rami Kamel Ahmad Rababah 1 rami013r@gmail.com 1 Department of Mathematics, Aligarh Muslim University, Aligarh 000, India. Citation information Cite this article as: On I-convergent sequence spaces defined by a compact operator and a modulus function, Vaeel A. Khan, Mohd Shafiq & Rami Kamel Ahmad Rababah, Cogent Mathematics (015, : References Basar, F., & Altay, B. (003. On the spaces of sequences of p-bounded variation and related matrix mapping. Urainian Mathematical Journal, 55, Bhardwaj, V. K. (003. A generalization of a sequence space of Rucle. Bulletin of Calcutta Mathematical Society, 95, Buc, R. C. (1953. Generalized asymptotic density. American Journal of Mathematics, 75, Fast, H. (1951. Sur la convergence statistique [About statistical convergence]. Colloquium Mathematicum,, Fridy, J. A. (1985. On statistical convergence. Analysis, 5, Garling, D. J. H. (1966. On symmetric sequence spaces. Proceedings of the London Mathematical Society, 16, Gramsch, B. (1967. Die Klasse metrisher linearer Raume L(ϕ [The class of metric linear spaces L(ϕ]. Mathematische Annalen, 171, Khan, V. A. (005. On a sequence space defined by modulus function. Southeast Asian Bulletin of Mathematics, 9, 1 7. Khan, V. A. (006. Difference sequence spaces defined by a sequence modulii. Southeast Asian Bulletin of Mathematics, 30, Khan, V. A. (007. Statistically pre-cauchy sequences and Orlicz function. Southeast Asian Bulletin of Mathematics, 6, Khan, V. A., & Ebadullah, K. (011. On some I-convergent sequence spaces defined by a modullus function. Theory and Applications of Mathematics and Computer Science, 1, 30. Khan, V. A., Ebadullah, K., Esi, A., & Shafiq, M. (013. On some Zweier I-convergent sequence spaces defined by a modulus function. Africa Mathematia, Journal of the African Mathematical Society (Springer Verlag Berlin Heidelberg, 6, Khan, V. A., Shafiq, M., & Rababah, R. K. A. (015. On BVs I-convergent sequence spaces defined by an Orlicz function. Theory and Applications of Mathematics and Computer Science, 5, Khan, V. A., & Tabassum, S. (01. Statistically pre-cauchy double sequences. Southeast Asian Bulletin of Mathematics, 36, Kostyro, P., Šalát, T., & Wilczyńsi, W. (000. I-convergence. Raal Analysis Analysis Exchange, 6, Köthe, G. (1970. Topological vector spaces (Vol. 1. Berlin: Springer. Kreyszig, E. (1978. Introductory functional analysis with applications. New Yor, NY: Wiley. Maddox, I. J. (1969. Some properties of paranormed sequence spaces. Journal of the London Mathematical Society, 1, Maddox, I. J. (1986. Sequence spaces defined by a modulus. Mathematical Proceedings of the Cambridge Philosophical Society, 100, Naano, H. (1953. Concave modulars. Journal of the Mathematical Society of Japan, 5, Rucle, W. H. (1967. Symmetric coordinate spaces and symmetric bases. Canadian Journal of Mathematics, 19, Rucle, W. H. (1968. On perfect symmetric BK-spaces. Mathematische Annalen, 175, Rucle, W. H. (1973. FK-spaces in which the sequence of coordinate vectors is bounded. Canadian Journal of Mathematics, 5, Šalát, T. (1980. On statistical convergent sequences of real numbers. Mathematica Slovaca, 30, Šalát, T., Tripathy, B. C., & Ziman, M. (004. On some properties of I-convergence. Tatra Mountains Mathematical Publications, 8, Šalát, T., Tripathy, B. C., & Ziman, M. (005. On I-convergence field. Italian Journal of Pure and Applied Mathematics, 17, Schoenberg, I. J. (1959. The integrability of certain functions and related summability methods. American Mathematical Monthly, 66, Sengönül, M. (009. The Zweier sequence space. Demonstratio Mathematica, XL, Tripathy, B. C. (1998. On statistical convergence. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics Analysis, 47, Tripathy, B. C., & Hazaria, B. (009. Paranorm I-convergent sequence spaces. Mathematica Slovaca, 59, Tripathy, B. C., & Hazaria, B. (011. Some I-convergent sequence spaces defined by Orlicz function. Acta Mathematicae Applicatae Sinica, 7, Page 1 of 13

13 015 The Author(s. This open access article is distributed under a Creative Commons Attribution (CC-BY 4.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoe these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a lin to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics (ISSN: is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page 13 of 13