I K - convergence in 2- normed spaces

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1 Functional Analysis, Approximation and Computation 4:1 (01), 1 7 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: I K - convergence in - normed spaces Madjid Eshaghi Gordji a, Saeed Sarabadan b, Fatemeh Amouei Arani c a Department of Mathematics, Semnan University, P. O. Box , Semnan, Iran b Department of Mathematics, Islamic Azad University South Tehran Branch, Tehran, Iran c Department of Mathematics, Payame Noor University, Tehran, Iran Abstract. In this paper, we introduce the concept of I K convergence of sequences in normed spaces. This notion can be regard as an extension of I-convergence of sequences in normed spaces. 1. Introduction The idea of I convergence was informally introduced by Kostyrko et al (001) and also independently by Nuray and Ruckle (000), as a generalization of statistical convergence. Ideal convergence provides a general framework to study the properties of various types of convergence. There are several works in recent years on I convergence of sequences (see [4,9,1]). The notion of linear normed spaces has been investigated by Gâhler in1960 s [7,8] and has been developed extensively in different subjects by others [,11,18,3]. It seems therefore reasonable to investigate the concepts of I and I convergence in normed spaces. Throughout this paper N will denote the set of positive integers. Definition 1.1. Let X be a real linear space of dimension greater than 1, and.,. be a non-negative real valued function on X X satisfying the following conditions: G1) x, y = 0 if and only if x and y are linearly dependent vectors, G) x, y = y, x for all x,y in X, G3) αx, y = α x, y where α is real number, G4) x + y, z x, z + y, z for all x, y, z in X..,. is called a norm on X and the pair (X,.,. ) is called a linear normed space. Every linear normed space (X,.,. ) of dimension different from one is a locally convex topological vector space. In fact, for a fixed b X, p b (x) = x, b, x X, is a seminorm and the family P = p b : b X}of seminorms generates a locally convex topology on X. In addition, we have the following properties: 1).,. is nonnegative. 010 Mathematics Subject Classification. Primary 40A05; Secondary 40C99, 40A99, 54A0. Keywords. I convergence; I convergence; Filter; Double sequences; normed space. Received: January, 01; Accepted: January 4, 01 Communicated by Dragan S. Djordjević addresses: madjid.eshaghi@gmail.com (Madjid Eshaghi Gordji), s.sarabadan@yahoo.com (Saeed Sarabadan), f.amoee@yahoo.com ( Fatemeh Amouei Arani)

2 M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), 1 7 ) x, y = x, y + αx 3) x y, y z = x y, x z for all scalars α and all x, y, z X. Some of the basic properties of norms studied in [18]. As an example of a -normed space we may take X = R being equipped with the -norm x, y := the area of the parallelogram spanned by the vectors x and y, which may be given clearly by the formula x, y = x 1 y x y 1, x = (x 1, x ) y = (y 1, y ). Given a -normed space (X,.,. ). One can derive a topology for it via the following definition of the limit of a sequence: A sequence (x n ) n N in X is said to be convergent to x in X if lim n x n x, z = 0 for all z X. This can be written by the formula:.,. X x n x. ( z Y)( ɛ > 0)( n 0 N)( n n 0 ) x n x, z < ɛ.. Preliminary Notes We recall the following definition, where Y represents an arbitrary set. Definition.1. A family I P(Y) of subsets a nonempty set Y is said to be an ideal in Y if: i) I, ii) A, B I implies A B I, iii) A I, B A implies B I, I is called a proper ideal if Y I and I is not proper ideal if I = P(Y). Definition.. The ideal of all finite subsets of a given set Y is called Fin. Definition.3. Let Y. A non empty family F of subsets of Y is said to be a filter in Y provided: i) F. ii) A, B F implies A B F. iii)a F, A B implies B F. If I is a nontrivial ideal in Y,Y, then the class is a filter on Y, called the filter associated with I. F(I) = M Y : ( A I)M = Y A} Definition.4. A nontrivial ideal I in Y is called admissible if x} I for each x Y. Definition.5. A nontrivial ideal I in N N is called strongly admissible if i} N and N i} belong to I for all i N. It is evident that a strongly admissible ideal is admissible also. Let I 0 =A N N : ( m(a) N)(i, j m(a) (i, j) N N A)}. Then I 0 is a nontrivial strongly admissible ideal and clearly an ideal I is strongly admissible if and only if I 0 I [3]. Definition.6. Let Y be a set and K be an ideal on Y. Let M Y, M. K M := A B : A K} is called trace of K on M and K M is an ideal on Y. Definition.7. Let I P(N) be a nontrivial ideal in N. The sequence (x n ) n N in X is said to be I-convergent to x X, if for each ɛ > 0, the set A(ɛ)=n N : x n x ɛ} belongs to I [1,1,14].

3 3. I K - convergence in -normed spaces M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), In [9,14,1], the concepts of I and I -convergence introduced in -normed space. concepts and introduce the I K -convergence for sequences in -normed spaces. First, we introduce the definition of I and I -convergence by the other manner. We extend this Definition 3.1. Let (X,.,. ) be a normed space and I be an ideal on a set A. The function f : A X is said to be I-convergent to x X if for all non zero z in X and for all ε > 0, we have: A(ɛ) = a A : f (a) x, z ɛ} I. I lim f = x. Remark 3.. If A = N, we obtain the usual definition I-convergent of sequence (x n ) n N to x X in -normed space X [9]. Lemma 3.3. Let X, Y be two normed spaces and let A be a non empty set and I, I 1, I be ideals on A. Then i) if I is not proper ideal, then every function f : A X is I convergent to each point of X. ii) If I 1 I, then for every function f : A X, we have I 1 lim f = x I lim f = x. Proof: i) Let x be arbitrary element of X, then ( z X)( ε > 0)A(ɛ) = a A : f (a) x, z ɛ} P(S) = I. ii) Let I 1 I, I 1 lim f = x. Then we have Hence I lim f = x. ( z X)( ε > 0)A(ɛ) = a A : f (a) x, z ɛ} I 1 I. Definition 3.4. Let (X,.,. ) be a normed space and I be an ideal on N. The sequence (x n ) n N in X is said to be I -convergent to a point l X, if there exists a set M F (I) such that x n.,. X l (on M). I lim x n = l. We introduce the definition of I K -convergence, we simply replace the ideal Fin by an arbitrary ideal on the set A. Definition 3.5. Let (X,.,. ) be normed space and let K and I be ideals on N, we say that a sequence (x n ) n N in X is I K - convergent to x X if: there exists a set M F (I) and the sequence (y n ) n N given by such that K lim x. I K lim x n = x.

4 M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), Remark 3.6. The definition of I K convergence can be reformulated in the form of decomposition. A sequence (x n ) n N is I K convergence if and only if (x n ) n N = (y n ) n N + (z n ) n N, where (y n ) n N is K- convergent and (z n ) n N is non-zero only on a set from I. Remark 3.7. Another definition of I K - convergence can be the form below: The sequence (x n ) n N is I K - convergent if there exists M F (I) such that the sequence x n M = (x n ) n M is K M -convergent to x. These two definitions are equivalent but definition 3.5 is simpler. We give some examples of ideals and corresponding I K convergence. Example 3.8. (i) Put I = K = }. I is the minimal ideal in N. A sequence (x n ) n N is I K - convergent if and only if it is constant. (ii) Let M N, M N. Put K = P(M), i.e. K is a proper ideal in N. Let I = }. A sequence (x n ) n N is I K - convergent if and only if it is constant on N \ M. (iii) Let K be an admissible ideal in N and I be a arbitrary ideal. A sequence (x n ) n N is I K convergent if there exists a set M F (I) and the sequence (y n ) n N given by definition such that the sequence y n is the usual converges. Corollary 3.9. Let (X,.,. ) be a normed space, and let (x n ) n N be a convergent sequence in X and l 1, l X. If I K lim x n = l 1 and I K lim x n = l then l 1 = l. Proof: Suppose l 1 l. Hence there exists z X such that l 1 l ( 0) and z are linearly independent. Put l 1 l, z = ε, with ε > 0. Since I K lim x n = l 1. By the definition, there exists M 1, M F (I) such that the sequences (y n ) n N, (z n ) n N given by 1 z l 1 if n M n = 1 l if n M have the following properties ( ε > 0)( z X) n N : y n l 1, z ɛ} K ( ε > 0)( z X) n N : z n l, z ɛ} K. Put M = M 1 M. We have ε = l 1 x n + x n l, z x n l 1, z + x n l, z = y n l 1, z + z n l, z Therefor n M : z n l, z < ε} n M : y n l 1, z ε} K. Hence n M : z n l, z < ε} K that is contradict with I φ. Corollary If (x n ) n N, (y n ) n N be sequences in -normed space (X,.,. ) and I K lim x n = a, I K lim b, then i)i K lim x n + a + b, ii)i K lim αx n = αa. Proof: (i) Let I K lim x n = a, I K lim b. By the definition, there exist M 1, M F (I) such that the sequences (z n ) n N, (t n ) n N given by z n = 1 a if n M 1 t n = yn if n M b if n M

5 M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), has the following properties K lim z n = a, and K lim t n = b. Now, we put M = M 1 M F (I) and define the sequence p n } by p n = xn + y n if n M a + b if n M, we have K lim z n + t n = K lim z n + K lim t n = a + b (see [1]). By the definition, I K lim x n + a + b. The proof of (ii) is similar to (i). In the next lemma, we show easily from the definitions that K- convergence implies the I K - convergence. Lemma Let K and I be ideals on a set N. If (X,.,. ) be a -normed space and (x n ) n N be a sequence in X such that K lim x n = x, then I K lim x n = x. Lemma 3.1. Let (X,.,. ) be a normed space and let I, I 1, I, K, K 1 and K be ideals on a set N such that I 1 I and K 1 K. Then for every sequence (x n ) n N in X, we have: i) I K 1 lim x n = x implies I K lim x n = x, ii) I K 1 lim x n = x implies I K lim x n = x. Proof: i) Suppose I K 1 given by lim x n = x, by definition, there exists M F (I 1 ) such that the sequence (y n ) n N has the following condition ( ε > 0)( z X)A(ɛ) = n N : y n x, z ɛ} K. On the other hand since I 1 I, then M F (I 1 ) F (I ) and by the definition (3.5), I K lim x n = x. ii) Let I K 1 lim x n = x. By definition, there exists M F (I) such that the sequence (y n ) n N given by has the following property Hence A(ɛ) K and the proof is complete. ( ε > 0)( z X)A(ɛ) = n N : y n x, z ɛ} K 1 K. In the next theorem, we prove the relationship between the I-convergence and I K - convergence. Theorem 3.1: Let (X,.,. ) be a -normed space and let K and let I be two ideals in N. Let (x n ) n N be a sequence in X. i) If I K lim x n = x implies I lim x n = x holds for some x X, which has at least one neighborhood different from X, then K I. ii) If K I then (I K lim x n = x implies I lim x n = x). Proof: i) Suppose that K is not subset of I. Then there exists a set A K such that A I. Let x X has a

6 M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), neighborhood U X such that U X and y X \ U. We define a sequence y n } on X by y if n A x if n A. Clearly, K lim x n = x. Thus by Lemma 3.11, we obtain I K lim x n = x. By the definition, n N : y n x, z ɛ} = A I. Hence the sequence (x n ) n N is not I-convergent to x. ii) Let (X,.,. ) be a -normed space and x X and let (x n ) n N be a sequence on X. Let K I, I K lim x n = x. By the definition of I K - convergence, there exists M F (I) such that the sequence (y n ) n N given by has the condition Hence A(ɛ) M K I. Consequently, and thus I lim x n = x. ( ε > 0)( z X)A(ɛ) = n N : y n x, z ɛ} = n M : x n x, z ɛ}. n N : x n x, z ɛ} (X \ M) (A(ɛ) M) I Example Let N = i=1 D i be a decomposition of N ( i.e. D i D j ). Assume that D i (i = 1,, ) are infinite sets (we can choose D i = 3 i 1 (t) : t N} for i = 1,, ). Denote by I the class of all A N such that A intersects only a finite numbers of D i. Let K be the family of all finite subsets of N. Define (x n ) n N as follows: For n D i, we put x n = 1 i (i = 1,, ). Obviously that I lim x n = 0. Now we show that I K lim x n 0. Assume that I K lim x n = 0. Then there exists M F (I) such that the sequence (y n ) n N given by 0 if n M, satisfying K lim 0, i.e.lim 0. Since M F (I), then there exists A I such that M = N \ A. By the definition of I, there exists an l N such that A D 1 D l. Then M contains the set D l+1 and x n = 1 l+1 for infinitely many n, s in M. This contradicts lim lim x n = 0. n n (n M) The concept of I and I -convergence of double sequences in -normed spaces was introduced in [19,1]. Now, we show that I K -convergence is a correct the generalization of I -convergence for double sequences in -normed spaces. Definition Let x = (x jk ) j,k N be a double sequence in -normed space (X,.,. ). A double sequence x=(x jk ) j,k N is said to be convergent to l X in Pringsheim s sense if ( z X)( ε > 0)( N N)( j, k N) x jk l, z < ε..,. X l. x jk

7 M. Eshaghi Gordji, S. Sarabadan, F. Amouei Arani / FAAC 4:1 (01), Definition A double sequence x=(x jk ) j,k N in -normed space (X,.,. ) is said to be I -convergent to l X, if for all ε > 0 and nonzero z X, A(ε) = (j, k) : x jk l, z ε} I. In this case we write it as I lim j,k x jk = l. The Pringsheim s ideal I on N N whose dual filter F (I ) is given by the filter base B = [n, ] [n, ]; n N}. Definition A double sequence x=(x jk ) j,k N in -normed space (X,.,. ) is said to be I -convergent to l X, if there exists a set M F (I) (i.e.n N\M I) such that lim m,n x mn =l (m, n) M and we write it as I lim j,k x jk = l. Remark The I -convergence of double sequences is the same as I I -convergence in N N. Hence the notion of the I K convergence is a correct generalization of I and I -convergence. Acknowledgements The authors would like to thank Professor Martin Sleziak for the valuable comments on this manuscript. References [1] V. Balá z, J. Cervenansky, P. Kostyrko, T. Salat, I-convergence and I-continuity of real functions, Acta Mathematica 5, Faculty of Natural Sciences, Constantine the Philosopher University, Nitra, (004) [] Y. J. Cho, P. C. S. Lin, S. S. Kim, A. Misiak, Theory of -inner product spaces, Nova Science, Huntington, NY, USA, (001). [3] J. S. Connor, The statistical and strong p-cesro convergence of sequences, Analysis, 8, (1988) [4] P. Das, P. Kostyrko, W. Wilczyncki, P. Malik, I and I -convergence of double sequence, Math. Slovaca, 58, (008), [5] H. Fast, Sur la convergence statistique, Colloq. Math,, (1951) [6] J. A. Fridy, On statistical convergence, Analysis, 105, (1985) [7] S. Gähler, normed spaces, Math. Nachr, 8, (1964) [8] S. Gähler, metrische Räumm und ihre topologische struktur, Math. Nachr, 6, (1963) [9] M. Gürdal, On ideal convergent sequences in normed Spaces, Thai. J. Math., 4(1), (006) [10] M. Gürdal and S. Pehlivan, The statistical convergence in -Banach Spaces, Thai. J. Math., (1), (004) [11] H. Gunawan and Mashadi, On finite dimensional -normed spaces, Soochow J. Math., 7(3), (001) [1] P. Kostyrko, T. Salat, W. Wilczyncki, I-convergence, Real Anal. Exchange,6, (000/001) [13] P. Kostyrko, M. Macaj, T. Salat, M. Sleziak, I-convergence and extremal I-limit points, Math. Slovaca, 55,(005) [14] B. Lahiri, P. Das, I and I -convergence in topological spaces,math. Bohem, 130, (005) [15] F. Moricz, Tauberian theorems for Cesro summable double sequences, Studia Math, 110, (1994) [16] M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 88, (003) [17] A. Pringsheim, Zur Ttheorie der zweifach unendlichen Zahlenfolgen, Math. Ann, 53, (1900) [18] W. Raymond, Y. Freese, J. Cho, Geometry of linear -normed spaces, N.Y. Nova Science Publishers, Huntington, 001. [19] S. Sarabadan, S. Talebi, Statistical convergence of double sequences in normed spaces, Int. J. Contemp. Math. Sciences, 6, (011) [0] S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in normed spaces, International Journal of Mathematics and Mathematical Sciences, Volume 011, Article ID , (011) 10 pages. [1] S. Sarabadan, S. Talebi, A condition for the equivalence of I- and I -convergence in - normed spaces, Int. J. Contemp. Math. Sciences., 6, (011) [] H. Steinhaus, Sur la convergence ordinarie et la convergence asymptotique, Colloq. Math,, (1953) [3] C. Tripathy, B. C. Tripathy, On I-convergent double series, Soochow J. Math, 31, (005)