λ-statistical convergence in n-normed spaces

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1 DOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 2(2),203, statistical convergence in n-normed spaces Bipan Hazarika and Ekrem Savaş 2 Abstract In this paper, we introduce the concept of -statistical convergence in n-normed spaces. Some inclusion relations between the sets of statistically convergent and -statistically convergent sequences are established. We find its relations to statistical convergence, (C,)-summability and strong (V, )-summability in n-normed spaces. Introduction The notion of statistical convergence was introduced by Fast [8] and Schoenberg [28] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from various points of view. For example, statistical convergence has been investigated in summability theory by ( Fridy [0], Salát [26]), topological groups (Çakalli [], [2]), topological spaces (Di Maio and Ko cinac[20]), function spaces (Caserta and Ko cinac [3], Caserta, Di Maio and Ko cinac [4]), locally convex spaces (Maddox[9]), measure theory (Cheng et al., [5], Connor and Swardson [6], Millar[2]), fuzzy mathematics (Nuray and Savaş [24], Savaş [27]). In the recent years, generalization of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions [6]. Mursaleen [23], introduced the -statistical convergence for real sequences. In this article, we consider only sequences of real numbers, so Key Words: Statistical convergence; -statistical convergence; n-norm. 200 Mathematics Subject Classification: 40A05; 40B50; 46A9; 46A45. Received: November 202 Revised: March 203 Accepted: May 203 Corresponding author 4

2 42 Bipan Hazarika and Ekrem Savaş that a sequence means a sequence of real numbers. The notion of statistical convergence depends on the (natural or asymptotic) density of subsets of N. A subset of N is said to have natural density δ (E) if n δ (E) = lim χ E (k) exists. n n k= Definition.. A sequence x = (x k ) is said to be statistically convergent to l if for every ε > 0 δ ({k N : x k l ε}) = 0. S In this case, we write S lim x = l or (x k ) l and S denotes the set of all statistically convergent sequences. Let = ( ) be a non-decreasing sequence of positive numbers tending to such that + +, =. The collection of such sequences will be denoted by. The generalized de la Vallée-Poussin mean is defined by where I m = [m +, m]. t m (x) = x k, Definition.2.[7] A sequence x = (x k ) is said to be (V, )-summable to a number l if t m (x) l, as m. If = m, then (V, )-summability reduces to (C, )-summability. write { } m [C, ] = x = (x k ) : l R, lim x k l = 0 m and [V, ] = { x = (x k ) : l R, lim k= x k l = 0 for the sets of sequences x = (x k ) which are strongly Cesàro summable (see [9]) and strongly (V, )-summable to l, i.e. (x k ) [C,] l and (x k ) [V,] l, respectively. } We

3 -STATISTICAL CONVERGENCE IN n-normed SPACES 43 Definition.3. [23] A sequence x = (x k ) is said to be -satistically convergent or S -convergent to l if for every ε > 0 lim {k I m : x k l ε} = 0. In this case we write S lim x = l or (x k ) S l and S = {x = (x k ) : l R, S lim x = l}. It is clear that if = m, then S is same as S. The concept of 2-normed space was initially introduced by Gähler[2], in the mid of 960 s, while that of n-normed spaces can be found in Misiak [22]. Since then, many others authors have studied this concept and obtained various results (see, for instance, Gunawan[4],Gähler[], Gunawan and Mashadi ([3], [5]), Lewandowska[8], Dutta [7]). 2 Definitions and Preliminaries Let n be a non negative integer and X be a real vector space of dimension d n (d may be infinite). A real-valued function.,...,. from X n into R satisfying the following conditions: () x, x 2,..., x n = 0 if and only if x, x 2,..., x n are linearly dependent, (2) x, x 2,..., x n is invariant under permutation, (3) αx, x 2,..., x n = α x, x 2,..., x n, for any α R, (4) x + x, x 2,..., x n x, x 2,..., x n + x, x 2,..., x n is called an n-norm on X and the pair (X,.,...,. ) is called an n-normed space. A trivial example of an n-normed space is X = R n, equipped with the Euclidean n-norm x, x 2,..., x n E = the volume of the n-dimensional parallelepiped spanned by the vectors x, x 2,..., x n which may be given expicitly by the formula x, x 2,..., x n E = det(x ij ) = abs (det(< x i, x j >)), where x i = (x i, x i2,..., x in ) R n for each i =, 2, 3..., n. Let (X,.,...,. ) be an n-normed space of dimension d n 2 and {a, a 2,..., a n } be a linearly independent set in X. Then the function.,...,.

4 44 Bipan Hazarika and Ekrem Savaş from X n into R defined by x, x 2,..., x n = max i n { x, x 2,..., x n, a i } defines an (n )-norm on X with respect to {a, a 2,...a n } and this is known as the derived (n )-norm (for details see [3]). The standard n-norm on a real inner product space of dimension d n is as follows: x, x 2,..., x n S = [det(< x i, x j >)] 2, where <, > denotes the inner product on X. If we take X = R n then this n- norm is exactly the same as the Euclidean n-norm x, x 2,..., x n E mentioned earlier. For n = this n-norm is the usual norm x = < x, x >(for further details see [3]). Definition 2.. A sequence (x k ) in an n-normed space (X,.,...,. ) is said to be convergent to l X with respect to the n-norm if for each ε > 0 there exists an positive integer n 0 such that x k l, z, z 2,..., z n < ε, for all k n 0 and for every z, z 2,..., z n X. Definition 2.2. A sequence (x k ) in an n-normed space (X,.,...,. ) is said to be Cauchy with respect to the n-norm if for each ε > 0 there exists a positive integer n 0 = n 0 (ε) such that x k x m, z, z 2,..., z n < ε, for all k, m n 0 and for every z, z 2,..., z n X. If every Cauchy sequence in X converges to some l X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be an n-banach space. Definition 2.3. A sequence (x k ) in an n-normed space (X,.,...,. ) is said to be statistically-convergent to some l X with respect to the n-norm if for each ε > 0 the set {k N : x k l, z, z 2,..., z n ε} has natural density zero, for every z, z 2,..., z n X. In other words the sequence (x k ) statistical converges to l an n-normed space X if lim m {k N : x k l, z, z 2,..., z n ε} = 0, for each z, z 2,..., z n X. Let S nn (X) denotes the set of all statistically convergent sequences in n-normed space X.

5 -STATISTICAL CONVERGENCE IN n-normed SPACES 45 Recently, Gürdal and Pehlivan [6] studied statistical convergence in 2- normed spaces. B.S. Reddy [25] extended this idea to n-normed space and studied some properties. In the present paper we study -statistical convergence in n-normed spaces. We show that some properties of -statistical convergence of real numbers also hold for sequences in n-normed spaces. We find some relations related to statistical convergent, -statistical convergent sequences, (C,)-summability and strong (V, )-summability in n-normed spaces. 3 -statistical convergent sequences in n-normed space X In this section we define -statistically convergent sequences in n-normed linear space X. Also, we obtained some basic properties of this notion in n-normed spaces. Definition 3.. A sequence x = (x k ) in an n-normed space (X,.,...,. ) is said to be -satistically convergent or S -convergent to l X with respect to the n-norm if for every ε > 0 lim {k I m : x k l, z, z 2,..., z n ε} = 0, for each z, z 2,..., z n X. In this case we write S nn lim x = l or (x k) SnN l and S nn (X) = {x = (x k ) : l R, S nn lim x = l}. Let S nn (X) denotes the set of all -statistically convergent sequences in the n-normed space X. Definition 3.2. A sequence x = (x k ) in an n-normed space (X,.,...,. ) is said to be (V, )-summable to l X with respect to the n-norm if t m (x) l, as m. If = m, then (V, )-summability reduces to (C, )-summability with respect to the n-norm. We write { } [C, ] nn m (X) = x = (x k ) : l R, lim x k l, z,..., z n = 0 m k=

6 46 Bipan Hazarika and Ekrem Savaş and [V, ] nn (X) = { x = (x k ) : l R, lim x k l, z,..., z n = 0 for the sets of X-valued sequences x = (x k ) which are strongly Cesàro summable and strongly (V, )-summable to l with respect to the n-norm, i.e., (x k ) [C,]nN l and (x k ) [V,]nN l, respectively. Theorem 3.. Let X be an n normed space and = ( n ). If (x k ) is a sequence in X such that lim x k = l exists, then it is unique. S nn Proof. Suppose that there exist elements l, l 2 (l l 2 ) in X such that S nn lim x k = l ; S nn lim x k = l 2. k k Since l l 2, then l l 2 0, so there exist z, z 2,..., z n X such that l l 2 and z, z 2,..., z n are linearly independent. Therefore, Since S nn and l l 2, z, z 2,...z n = 2ε > 0. lim k x k = l and S nn lim lim There is k I m such that lim k x k = l 2 it follows that {k I m : x k l, z, z,...z n ε} = 0 {k I m : x k l 2, z, z 2,...z n ε} = 0. x k l, z, z,...z n < ε and x k l 2, z, z,...z n < ε. Further, for this k we have l l 2, z, z,...z n x k l, z, z,...z n + x k l 2, z, z,...z n < 2ε which is a contradiction. This completes the proof. The next theorem gives the algebraic characterization of -statistical convergence on n-normed spaces. }

7 -STATISTICAL CONVERGENCE IN n-normed SPACES 47 Theorem 3.2. Let X be an n-normed space, = ( n ), x = (x k ) and y = (y k ) be two sequences in X. (a) If S nn lim k x k = l and c( 0) R, then S nn lim k cx k = cl. (b) If S nn lim k x k = l and S nn lim k y k = l 2, then S nn lim k (x k + y k ) = l + l 2. Proof of the theorem is straightforward, thus omitted. Theorem 3.3. S nn (X) l (X) is a closed subset of l (X), if X is an n-banach space. Proof. Suppose that (x i ) i N, x i = (x i k ) k N, is a convergent sequence in S nn (X) l (X) converging to x = (x k ) l (X). We need to prove that x S nn (X) l (X). Assume that (x i k ) S nn k l i, for all i N. Take a positive decreasing convergent sequence (ε i ) i N, where ε i = ε 2, for a given i ε > 0. Clearly (ε i ) i N converges to 0. Choose a positive integer i such that x x i, z, z 2,..., z n < εi 4, for every z, z 2,..., z n X. Then we have lim {k I m : x i k l i, z, z 2,..., z n ε i 4 } = 0 and lim {k I m : x i+ k l i+, z, z 2,..., z n ε i+ } = 0. 4 Since, and for m N { k I m : x i k l i, z, z 2,..., z n ε i 4 x i+ k l i+, z, z 2,..., z n ε i+ 4 { k I m : x i k l i, z, z 2,..., z n ε } i { 4 k I m : x i+ k l i+, z, z 2,..., z n ε } i+ 4 } < is infinite. Hence there must exists a k I m for which we have simultaneously, x i k l i, z, z 2,..., z n < ε i 4 and xi+ k l i+, z, z 2,..., z n < ε i+ 4.

8 48 Bipan Hazarika and Ekrem Savaş Then it follows that l i l i+, z, z 2,..., z n l i x i k, z, z 2,...z n + x i k x i+ k, z, z 2,...z n + + x i+ k l i+, z, z 2,...z n x i k l i, z, z 2,...z n + x i+ k l i+, z, z 2,...z n + x x i, z, z 2,...z n + x x i+, z, z 2,...z n < ε i 4 + ε i+ 4 + ε i 4 + ε i+ < ε i. 4 This implies that (l i ) is a Cauchy sequence in X and there is an element l X such that l i l as i. We need to prove that (x k ) SnN l. For any ε > 0, choose i N such that ε i < ε 4, Then x k x i k, z, z 2,..., z n < ε 4, l i l, z, z 2,..., z n < ε 4. {k I m : x k l, z, z 2,..., z n ε} {k I m : x i k l i, z, z 2,..., z n + x k x i k, z, z 2,..., z n + l i l, z, z 2,..., z n ε} {k I m : x i k l i, z, z 2,..., z n + ε 4 + ε 4 ε} {k I m : x i k l i, z, z 2,..., z n ε } 0 as m. 2 This gives that (x k ) SnN l, which completes the proof. Theorem 3.4. Let X be an n-normed space and let = ( ). Then (i) (x k ) [V,]nN l (x k ) SnN l, (ii) [V, ] nn (X) is a proper subset of S nn (X), (iii) x l (X) and (x k ) SnN l, then (x k ) [V,]nN provided x = (x k ) is not eventually constant, (iv)s nn (X) l (X) = [V, ] nn (X) l (X). Proof. (i) If ε > 0 and (x k ) [V,]nN l, we can write l and hence (x k ) [C,]nN l,

9 -STATISTICAL CONVERGENCE IN n-normed SPACES 49 x k l, z, z 2,..., z n x k l, z, z 2,..., z n, x k l,z,z 2,...,z n ε ε {k I m : x k l, z, z 2,...z n ε} and so x k l, z, z 2,..., z n ε {k I m : x k l, z, z 2,..., z n ε}. This proves the result. (ii) In order to establish that the inclusion [V, ] nn (X) S nn (X) is proper, we define a sequence x = (x k ) by { k, if m [ m ] + k m; x k = 0, otherwise. Then x / l and for every ε > 0(0 < ε < ), x k 0, z, z 2,..., z n [ m ] 0, as m, i.e. (x k ) SnN 0. On the other hand, {k I m : x k 0, z, z 2,..., z n ε}, as m, i.e. (x k ) does not converge to 0 in [V, ] nn (X). (iii) Suppose that (x k ) SnN l and (x k ) l (X). Then there exists a M > 0 such that x k l, z, z 2,...z n M for all k N. Given ε > 0, we have x k l, z, z 2,..., z n =, x k l,z,z 2,...,z n ε 2 x k l, z, z 2,..., z n

10 50 Bipan Hazarika and Ekrem Savaş +, x k l,z,z 2,...,z n < ε 2 x k l, z, z 2,..., z n M {k I m : x k l, z, z 2,..., z n ε 2 } + ε 2, This shows that (x k ) [V,]nN l. Again, we have m x k l, z, z 2,..., z n m m m m k= k= x k l, z, z 2,..., z n + x k l, z, z 2,..., z n m m m x k l, z, z 2,..., z n + x k l, z, z 2,..., z n k= 2 x k l, z, z 2,..., z n. Hence (x k ) [C,]nN l, because (x k ) [V,]nN l. (iv) This is an immediate consequence of (i), (ii) and (iii). Theorem 3.5. Let X be an n-normed space and let = ( ). Then S nn (X) S nn (X) if and only if lim inf m m m > 0. Proof. Suppose first that lim inf m m > 0. Then a given ε > 0, we have m {k m : x k l, z, z 2,..., z n ε} m {k I m : x k l, z, z 2,..., z n ε} m. {k I m : x k l, z, z 2,..., z n ε}. It follows that (x k ) SnN l (x k ) SnN l. Hence S nn (X) S nn (X). Conversely, suppose that lim inf mm m = 0. Then we can select a subsequence (m(j)) j= such that (j) m(j) < j.

11 -STATISTICAL CONVERGENCE IN n-normed SPACES 5 We define a sequence x = (x k ) as follows: x k = {, if k Im(j), j =, 2, 3,...; 0, otherwise. Then x is statistically convergent, so x S nn (X). But x / [V, ] nn (X). Theorem 3.4(iii) implies that x / S nn (X). This completes the proof. Theorem 3.6. Let X be an n-normed space and let = ( ) such that lim mm m =. Then S nn (X) SnN (X). Proof. Since lim m m =, then for ε > 0, we observe that m {k m : x k l, z, z 2,..., z n ε} m {k m : x k l, z, z 2,..., z n ε} + m {k I m : x k l, z, z 2,..., z n ε} m m = m m + m {k I m : x k l, z, z 2,..., z n ε} + m {k I m : x k l, z, z 2,..., z n ε}. This implies that (x k ) is statistically convergent, if (x k ) is -statistically convergent. Hence S nn (X) SnN (X). Remark: We do not know whether the condition lim mm m Theorem 3.6 is necessary and leave it as an open problem. = in the Acknowledgements: The authors thank the referees for their comments which improved the presentation of the paper. References [] H. Çakalli, On statistical convergence in topological groups, Pure Appl. Math. Sci.,43(996), [2] H. Çakalli, A study on statistical convergence, Funct. Anal. Approx. Comput., (2)(2009), 9-24, MR

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13 -STATISTICAL CONVERGENCE IN n-normed SPACES 53 [9] I. J. Maddox, Statistical convergence in a locally convex spaces, Math. Proc. Cambridge Philos. Soc., 04()(988), [20] G. Di. Maio, Lj.D.R. Ko cinac, Statistical convergence in topology, Topology Appl. 56, (2008), [2] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(5)(995), [22] A. Misiak, n-inner product spaces, Math. Nachr. 40(989) [23] M. Mursaleen, -statistical convergence, Math. Slovaca, 50()(2000),- 5. [24] F. Nuary, E. Savaş, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca, 45(995), [25] B. S. Reddy, Statistical convergence in n-normed spaces, Int. Math. Forum, 24(200), [26] T. Salát, On statistical convergence of real numbers, Math. Slovaca, 30(980), [27] E. Savaş, Statistical convergence of fuzzy numbers, Inform. Sci., 37(200), [28] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66(959) Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-792, Arunachal Pradesh, India bh rgu@yahoo.co.in 2 Department of Mathematics, Istanbul Ticaret University, Uskudar-Istanbul, Turkey ekremsavas@yahoo.com

14 54 Bipan Hazarika and Ekrem Savaş