On Ideal Convergent Sequences in 2 Normed Spaces

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1 Thai Journal of Mathematics Volume Number 1 : On Ideal Convergent Sequences in Normed Spaces M Gürdal Abstract : In this paper, we investigate the relation between I-cluster points and ordinary limit points of sequence in -normed space Keywords : I-convergence, -normed space, I-cluster point 000 Mathematics Subject Classification : 40A05, 46A70 1 Introduction The notion of ideal convergence was introduced first by P Kostyrko et al [6] as an interesting generalization of statistical convergence [1, 11] Note that the concept of an I-cluster point and I-limit point of a sequence in a metric space was introduced and some results for the set of I-cluster points and I-limit points obtained in [7] The concept of -normed spaces was initially introduced by Gähler [] in the 1960 s Since then, this concept has been studied by many authors, see for instance [3, 10] In a natural way, one may unite these two concepts, and study the relation between I-cluster point set and ordinary limit point set of a given sequence in -normed spaces This is actually what we offer in this article Throughout this paper N will denote the set of positive integers Let X, be a normed space Recall that a sequence n n N of elements of X is called to be statistically convergent to ξ X if the set A ε = n N : n ξ ε} has natural density zero for each ε > 0 A family I Y of subsets a nonempty set Y is said to be an ideal in Y if i I; ii A, B I imply A B I; iii A I, B A imply B I, while an admissible ideal I of Y further satisfies } I for each Y [7, 8] Given I N be a nontrivial ideal in N The sequence n n N in X is said to be I-convergent to ξ X, if for each ε > 0 the set A ε = n N : n ξ ε} belongs to I [6, 7]

2 86 Thai J Math 4006/ M Gürdal An admissible ideal I N is said to have the property AP if for any sequence A 1, A, } of mutually disjoint sets of I there is a sequence B 1, B, } of subsets of N such that each symmetric difference A i B i i = 1,, is finite and B = B i I, [7] i=1 Let X be a real vector space of dimension d, where d < A -norm on X is a function, : X X R which satisfies i, y = 0 if and only if and y are linearly dependent; ii, y = y, ; iii α, y = α, y, α R; iv, y + z, y +, z The pair X,, is then called a -normed space [] As an eample of a -normed space we may take X = R being equipped with the -norm, y := the area of the parallelogram spanned by the vectors and y, which may be given eplicitly by the formula, y = 1 y y 1, = 1,, y = y 1, y Recall that X,, is a -Banach space if every Cauchy sequence in X is convergent to some in X Let u = u 1,, u d } be a basis of X We know the norm on X is defined by := ma, u i : i = 1,, d} Associated to the derived norm, we can define the open balls B u, ε centered at having radius ε by B u, ε := y : y < ε }, where y := ma y, u j, j = 1,, d } The Relation Between I-cluster Points and Ordinary Limit Points of -Normed Spaces It is known that there is a strong connection between statistical cluster points and ordinary limit points of a given sequence We will prove that these facts are satisfied for I-cluster I-limit point sets of a given sequences in -normed spaces Definition 1 Let I N be a nontrivial ideal in N The sequence n of X is said to be I-convergent to ξ, if for each ε > 0 and nonzero z in X the set A ε = n N : n ξ, z ε } belongs to I If n is I-convergent to ξ then we write I- lim n n ξ, z = 0 or I- lim n n, z = ξ, z The number ξ is I-limit of the sequence n

3 On Ideal Convergent Sequences in -Normed Spaces 87 Remark If n } is any sequence in X and ξ is any element of X, then the set n N : n ξ, z ε, for every z X } =, since if z = 0 0 vector, n ξ, z = 0 ε so the above set is empty Further we will give some eamples of ideals and corresponding I-convergences I Let I f be the family of all finite subsets of N Then I f is an admissible ideal in N and I f convergence coincides with usual convergence in [] II Put I δ = A N : δ A = 0} Then I δ is an admissible ideal in N and I δ convergence coincides with the statistical convergence in [4] Now we give an eample of I-convergence in -normed spaces Eample 3 Let I = I δ Define the n in -normed space X,, by n = 0, n, n = k, k N, 0, 0, otherwise and let ξ = 0, 0 and z = z 1, z Then for every ε > 0 and z X n N : n ξ, z ε } 1, 4, 9, 16,, n, } We have that δ n N : n ξ, z ε} = 0, for every ε > 0 and nonzero z X This implies that st lim n n, z = ξ, z But, the sequence n is not convergent to ξ Definition 4 Let I N be an admissible ideal and = n n N in linear -normed space X,, be a sequence i A number ξ is called to be an I-limit point of provided that there is set M = m 1 < m < } N such that M / I and lim m k ξ, z = 0 for k each nonzero z in X The set of all I-limit points of is denoted by I ii A number ξ is said to be an I-cluster point of provided that n N : n ξ, z < ε} / I for each ε > 0 and nonzero z X The set of all I-cluster points of is denoted by I Proposition 5 Let I N be an admissible ideal Then for each sequence = n n N of X we have I I and the set I is a closed set

4 88 Thai J Math 4006/ M Gürdal Proof Let ξ I Then there eists a set M = m1 < m < } / I such that lim m k ξ, z = 0 1 k for each nonzero z in X Take δ > 0 According to 1 there eists k 0 N such that for k > k 0 and each nonzero z X we have mk ξ, z < δ Hence and so n N : n ξ, z < δ} M\ m 1,, m k0 } n N : n ξ, z < δ} / I, which means that ξ I Let y I Take ε > 0 There eists ξ 0 I Bu y, ε Choose δ > 0 such that B u ξ 0, δ B u y, ε We obviously have n N : y n, z < ε} n N : ξ 0 n, z < δ} Hence n N : y n, z < ε} / I and y I Definition 6 Let I N be an admissible ideal and = n n N be a sequence in linear -normed space X,, If K = k 1 < k < } I, then the subsequence k = k n N is called I-thin subsequence of the sequence If M = m 1 < m < } / I, then the sequence M = m n N is called I-nonthin subsequence of It is clear that if ξ is a I-limit point of, then there is a I-nonthin subsequence M that converges to ξ Let L be the set of all ordinary limit points of sequence It is obvious I L : Take ξ / L, then there is ε > 0 such that the interval ξ ε, ξ + ε contains only a finite number of elements of Then n N : n ξ, z < ε } I, but it contradicts to ξ I Hence I Hence L, so I L Lemma 7 Let I N be an admissible ideal and = n n N be a sequence in linear -normed space X,, If is I-convergent in -normed space, then I } and are both equal to the singleton set I- lim n, z for each n nonzero z in X Proof Let I- lim n, z for each nonzero z in X Show that ξ I By n definition of I-convergence we have A ε = n N : n ξ, z ε } I for each ε > 0 and nonzero z in X Since I is an admissible ideal we can choose the set M = n 1 < n < } N such that n k / A 1 k and nk ξ, z < 1 k for

5 On Ideal Convergent Sequences in -Normed Spaces 89 all k N and nonzero z in X That is lim n n k ξ, z = 0 Suppose M I Since M n N : n ξ, z < 1} for each nonzero z in X, then N\M n N : n ξ, z < 1} =, but N\M F I and n N : n ξ, z < 1} F I for each nonzero z in X This contradiction gives M / I Hence we get M = m 1 < m < } N and M / I such that lim n n k ξ, z = 0 ie ξ I Since I I, then ξ I Now suppose there is η I such that η ξ It is clear that A = B = } η ξ n N : n ξ, z I and } η ξ n N : n ξ < / I for each nonzero z in X On the other hand, since n ξ, z n η η ξ, z > η ξ for each n B nonzero z in X, we have B A I This contradiction shows I = ξ} we have I = I = ξ} The Lemma 1 is proved Theorem 8 Let I N be an admissible ideal and = n n N, y = y n n N are sequences in linear -normed space X,, such that M = n N : n y n } I Then I = I y and I = I y Proof Let M = n N : n y n } I If ξ I, then there is a set K = k 1 < k < } / I such that I- lim n k n ξ, z = 0 Since K 1 = n N : n K n y n } M I, then K = n N : n K n = y n } / I indeed, if K I, then K = K 1 K I, but K / I Hence the sequence y K = y n n K is a I-nonthin subsequence of y = y n n N and y K converges to ξ in -normed space, ie ξ I y Now let ξ I Then K3 = n N : n ξ, z < ε} / I for each ε > 0 and nonzero z in X and K 4 = n N : n K 3 n = y n } / I Therefore, K 4 n N : y n ξ, z < ε} for each nonzero z in X It shows that, for each ε > 0 and nonzero z in X, n N : y n ξ, z < ε} / I ie ξ I y Theorem 1 is proved The net theorem proves a strong connection between I-cluster and ordinary limit points of a given sequence

6 90 Thai J Math 4006/ M Gürdal Theorem 9 Let I N be an admissible ideal with property AP and = n n N be a sequence in linear -normed space X,, Then there is a sequence y = y n n N such that L y = I, and n N : n y n } I, where L y is ordinary limit points set of the sequence y = y n n N Moreover y n : n N} n : n N} Proof If I = L, then y = and this case is trivial Let I is a proper subset of L : I L Then L \I and for each ξ L \I there is an open interval E ξ = ξ δ, ξ + δ such that I- lim j n k ξ, z = 0 Hence, there is an open interval E ξ = ξ δ, ξ + δ such that } k N : k E ξ I It is obvious that the collection of all intervals E ξ is an open cover of L \I,so by Covering Theorem there is a countable and mutually disjoint subcover E ξ } j=1 such that each E j contains an I-thin subsequence of n n N Now let A j = n N : n E j, jεn} It is clear that A j I j = 1,, and A i A j = Then by AP property of I there is a countable collection B j } j=1 of subsets of N such that B = j=1 B j and A j \B is a finite set for each jεn Let M = N\B = m 1 < m < } N Now the sequence y = y k is defined by y = y k if k B and y k = k if k M Obviously, k N : k y k } B I, so by Theorem 1 I y = I Since A j \B is a finite set then the subsequence y B = y k kεb has no limit point that is not also an I-limit point of y ie L y = I y Therefore, we have proved L y = I Moreover, the construction of the sequence y = y n n N shows y n : n N} n : n N} Theorem 9 is proved References [1] H Fast, Sur la convergence statistique, Colloq Math, 1951, [] J A Fridy, Statistical limit points, Proc Amer Math Soc, , [3] S Gähler, -metrische Räume und ihre topologische Struktur, Math Nachr, 61963, [4] H Gunawan and Mashadi, On Finite Dimensional -normed spaces, Soochow J Math, 73001, [5] M Gürdal and S Pehlivan, The Statistical Convergence in -Banach Spaces, Thai J Math, 1004,

7 On Ideal Convergent Sequences in -Normed Spaces 91 [6] J L Kelley, General Topology, Springer-Verlag, New York, 1955 [7] P Kostyrko, M Macaj and T Salat, I-Convergence, Real Anal Echange, 6000, [8] P Kostyrko, M Macaj, T Salat and M Sleziak, I-Convergence and Etremal I-Limit Points, Math Slovaca, 55005, [9] C Kuratowski, Topology I, PWN, Warszawa, 1958 [10] S Pehlivan, A Güncan and MA Mamedov, Statistical cluster points of sequences in finite-dimensional spaces, Czech Math J, , [11] W Raymond, Y Freese and J Cho, Geometry of linear -normed spaces, NY Nova Science Publishers, Huntington, 001 [1] H Steinhaus, Sur la convergence ordinarie et la convergence asymptotique, Colloq Math, 1951, M Gürdal Department of Mathematics University of Suleyman Demirel 360, Isparta, Turkey gurdal@fefsduedutr Received 1 November 005