Defined by Modulus. N. Subramanian [a],* Department of Mathematics, SASTRA University, Tanjore , India. *Corresponding author.

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1 Studies in Mathematical Sciences Vol.8, No. 204, pp DOI: /2954 On ISSN [Print] ISSN [Online] Defined by Modulus N. Subramanian [a],* [a] Department of Mathematics, SASTRA University, Tanjore-63 40, India. *Corresponding author. Address: Department of Mathematics. SASTRA University, Tanjore-63 40, India; Received: November 2, 203/ Accepted: January 5, 204/ Published online: February 26, 204 Abstract: In this paper we defined the defined by a modulus and exhibit some general properties of the space with an four dimensional infinite regular matrix. Key words: Gai sequence; Analytic sequence; Modulus function; Double sequences; Difference sequence; Lacunary sequence; Statistical convergence Subramanian, N. (204). On Λ 2q Defined by Modulus. Studies in Mathematical Sciences, 8(), Available from URL: DOI: dx.doi.org/0.3968/2954. INTRODUCTION Throughout ω χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w 2 for the set of all complex sequences x, where m n N, the set of positive integers. Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication. Some initial works on double sequence spaces is found in Bromwich (965). Later on, they were investigated by Hardy (97), Moricz (99), Moricz and Rhoades (988), Basarir and Solankan (999), Tripathy (2003), Turkmenoglu (999), and many others. Let us define the following sets of double sequences: M u t : = { x w 2 sup m n N x t < }, C p t { x w 2 : p lim m n x l t = for some l C }, C 0p t { x w 2 : p lim m n x t = }, 27

2 On Λ 2q Α φ μ Defined by Modulus L u t { x w 2 : Σ m= Σ n= x t < }, C bp t C p t M u t and C 0bp t = C 0p t M u t, Where t = t is the sequence of strictly positive reals t for all m n N and p lim m n denote the limit in the Pringsheim s sense. In the case t = for all m n N; M u t, C p t, C 0p t, L u t, C bp t and C 0bp t reduce to the sets M u, C p, C 0p, L u, C bp, and C 0bp, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Colak (2004, 2005) have proved that M u t, and C p t, C bp t are complete paranormed spaces of double sequences and gave the α-, β-, γ- duals of the spaces M u t and C bp t. Quite recently, in her Ph.D. thesis, Zelter (200) has essentially studied both the theory of topological double sequence spacesw and the theory of summability of double sequences. Mursaleen and Edely (2003) have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesὰro summable double sequences. Nextly, Mursaleen (2004) Mursaleen and Edely (2004) have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M - core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = x jk into one whose core is a subset of the M core of x. More recently, Altay and Basar (2005) have defined the spaces BS BS t ; CS p CS bp CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u t, C p, C bp, C r and L u, respectively, and also examined some properties of those sequence spaces and determined the α- duals of the spaces BS BV CS bp and the β v - duals of the spaces CS bp and CS r of double seties. Quite recently Basar and Sever (2009) have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and examined some properties of the space L q. Quite recently Subramanian and Misra (200) have studied the space 2 χ Μ p q u of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner (946), Maddox (979), and others. The class of sequences which are strongly Cesὰro summable withrespect to a modulus was introduced by Maddox (986) as an extension of the definition of strongly Cesὰro summable sequences. Connor (989) further extended this definition to a definition of strong A-summability with respect to a modulus where 28

3 Subramanian, N. / Studies in Mathematical Sciences, 8(), 204 A = a n k is a nonnegative regular matrix and established some connections between strong A-summability, strong A-summability with respect to a modulus, and A- statistical convergence. In (900) the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [35]-[38], and [39] the four dimensional matrix transformation Ax k l = Σ m= Σ n= a kl x was studied extensively by Robison and Hamilton. Let Φ denotes the set of all subsets of N, those do not contain more than elements. Further (Φ ) will denote a non-decreasing sequence of positive real numbers such that Φ m+ n+ m + n + Φ, for all m n N. Now, if u = u is any sequence such that u 0 for each m n and w 2 X denote the space of all sequences with elements in X, where (X, q) denotes a semi normed space, seminormed by q, and μ is any real number such that μ 0. This will be accomplished by presenting the following sequence space: Λ 2q Α φ μ = {x w 2 : sup m n σ Φ r σ s σ sr μ f q A rs u x m+n p rs < }; where f is a modulus function. Other implications, general properties and variations will also be presented. We need the following inequality in the sequel of the paper. For a, b 0 and 0<p< ; we have a + b p a p + b p. (.) The double series m n= x is called convergent if and only if the m double sequence s is convergent, where s = n i j= x ij m n N see []. A sequence x = x is said to be double analytic if sup x m+n <. The vector space of all double analytic sequences will be denoted by Λ 2. A sequence x = x is called double gai sequence if m + n! x m+n 0 as m n. The double gai sequences will be denoted byχ 2. LetΦ = {all finite sequences}. Consider a double sequence x = x ij. The m n th section x m n of the sequence is defined by x m n m = n i j=0 x ij I ij for all m n N; where I ij 29

4 On Λ 2q Α φ μ Defined by Modulus denotes the double sequence whose only non-zero term is a 30 i+j! in the i j th place for each i j N. An FK-space (or a metric space) X is said to have AK property if I is a Schauder basis for X. Or equivalently x m n x. An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = (x k x m n N are also continuous. Orlicz (936) used the idea of Orlicz function to construct the space (L M ). Lindenstrauss and Tzafriri (97) investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l M contains a subspace isomorphic to l P p. Subsequently, different classes of sequence spaces were defined by Parashar and Choudhary (994), Mursaleen et al. (999), Bektas and Altin (2003), Tripathy et al. (2003), Rao and Subramanian (2004), and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [6]. Recalling [3] and [6] an Orlicz function is a function M : [0, ) [0, ) which is continuous, non-decreasing, and convex with M (0) = 0, M (x)> 0, for x > 0 and M x as x. If convexity of Orlicz function M is replaced by subadditivity of M, then this function is called modulus function, defined by Nakano (973) and further discussed by Ruckle (986) and Maddox (986), and many others. An Orlicz function M is said to satisfy the 2 condition for all values of u if there exists a constant K > 0 such that M (2u) KM u u 0. The 2 condition is equivalent to M lu KlM u for all values of u and for l >. Lindenstrauss and Tzafriri (97) used the idea of Orlicz function to construct Orlicz sequence space l M = {x w k= The space l M with the norm M x k ρ < for some ρ > 0 }. x = inf {ρ > 0: k= M x k }, ρ becomes a Banach space which is called an Orlicz sequence space. For M(t)= t p p, the spaces l M coincide with the classical sequence space l P.

5 Subramanian, N. / Studies in Mathematical Sciences, 8(), 204 If X is a sequence space, we give the following definitions: i X = the continuous dual of X; (ii) X α ={a = a : m n= a x < for each x X}; iii X β {a = a : m n= a x is convegent for each x X}; (iv) X γ M = {a = a : sup N m n= a x < for each x X }; (v) let X bean F K space ; then X f = {f I : f X }; vi X δ = {a = a : sup a x m+n < for each x X}; X α, X β X γ are called α or Köothe Toeplitz dual of X, β - (or generalized - Köothe Toeplitz) dual of X, γ- dual of X, δ - dual of X respectively. X α is defined by Gupta and Kamptan (98). It is clear that x α X β and X α X γ but X β X γ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz (98) as follows Z Δ = {x = x k w: xk Z}, for Z = c, c 0 and l, where xk = x k -x k+ for all k N. Here c, c 0 and l denote the classes of convergent, null and bounded sclar valued single sequences respectively. The difference space bv p of the classical space l P is introduced and studied in the case p, BaȘar and Altay in [42] and in the case 0 p, by BaȘar and Altay in [43]. The spaces c (Δ), c 0 (Δ), l Δ and bv p are Banach spaces normed by x = x + sup k x k and x bvp = k= 3 x k p p p <. Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by Z Δ = {x = x w 2 : x Z}, where Z = Λ 2, X 2 and x = x x + x m+n x m+n+ = x x + x m+n + x m+n+ for all m, n N 2. DEFINITIONS AND PRELIMINATIES Λ 2q denote the Pringscheims sense of double analytic sequence space of modulus. 2. Definition. A modulus function was introduced by Nakano (953). We recall that a modulus f is a function from [0, ) [0, ), such that

6 On Λ 2q Α φ μ Defined by Modulus () f (x) = 0 if and only if x =0, (2) f x + y f x + f y for all x 0 y 0, (3) f is increasing, (4) f is continuous from the right at 0. Since f x f y f x y, it follows from here that f is continuous on [0, ) Definition. Let A = (a k l ) denote a four dimensional summability method that maps the complex double sequences x into the double sequence Ax where the k, l t term to Ax is as follows: Ax kl = m= n= a kl such transformation is said to be nonnegative if a kl is nonnegative. The notion of regularity for two dimensional matrix transformations was presented by Silverman (20) and Toeplitz (9). Following Silverman and Toeplitz, Robison and Hamilton presented the following four dimensional analog of regularity for double sequences in which they both added an adiditional assumption of boundedness. This assumption was made because a double sequence which is P- convergent is not necessarily bounded. 32 x 2.3 Definition. For a subspace ψ of a linear space is said to be sequence algebra if x y ψ, implies that x y = x y ψ. 2.4 Definition. A sequence E is said to be solid (Or normal) if (λ x E, wherener (x E) for all sequences of scalars λ = k with λ. 2.5 Definition. A double sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces. 2.6 Remark. From the above, it is clear that a sequence space E is solid implies that E is monotone. 2.7 Definition. Let X be a real or complex linear space, g be a function from X to the set R of real numbers. Then, the pair (X, g) is called a paranormed space and g is a paranorm for X, if the following axioms are satisfied for all elements x y E and for all scalars α (PN.) g (x) = 0 if x = θ. (PN.2) g (-x) =g (x). (PN.3) g (x + y) g (x) + g (y). (PN.4) If (α n ) is a sequence of scalars with α n α as n and x n, x

7 Subramanian, N. / Studies in Mathematical Sciences, 8(), 204 X for all n N with x n x as n then α n x n αx as n, in the sense that g(αx n - αx) 0 as n. 3. MAIN RESULTS 3. Theorem. Λ 2q Α φ μ is linear space over the complex field C. Proof: It is easy. Therefore omit the proof. 3.2 Theorem. Λ 2q Α φ μ is a paranormed space with g x = sup m n σε r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs M, (3.) if and only if = in rs > 0, where M = max (, H) and H = supp. ii Λ 2q is a complete paranormed linear metric space if the condition p in (3.) is satisfied. Proof(i): suffciency: Let h > 0. It is trivial that g θ = 0 and g x = g x. The inequality g x + y g x + g y follows from the inequality (3.), since p rs M for all positive integers r, s. We also may write g λx max λ λ h M g x, since λ p max λ h λ M for all positive integers r, s and for any λ C, the set of complex numbers. Using this inequality, it can be proved that λx θ, when x is fixed and λ 0, or λ 0 and x θ, or λ is fixed and x θ. Necessity: Λ 2q Α φ μ Let be a paranormed space with the paranorm. g x = sup m n σε r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs M and suppose that h = 0. Since λ p rs M and λ C such that 0 < λ <, we have g x = sup m n σε λ h M = for all positive integers r, s r σ s σ rs μ f q λ p rs 33 M =. Hence it follows that g λx = sup m n σε r σ s σ rs μ f q λ p rs M = for x= (α) Λ 2q as λ 0. But this contradicts the assumption Λ 2q is a paranormed space with g (x). (ii) The proof is clear.

8 On Λ 2q Α φ μ Defined by Modulus 3.3 Corollary. Λ 2q is a complete paranormed space with the natural paranorm if and only if Λ 2q =Λ 2q f. 3.4 Theorem. Λ 2q φ sup φ2 <. Proof: Let x Λ 2q sup m n σ Α φ μ φ sup φ2 sup m n σ T sup m n σ Λ 2q Α φ φ μ and T= sup φ2. Then Α φ 2 μ if and only if ) p rs φ2 r σ s σ rs μ f q ( A rs Δ u x m+n φ r σ s σ rs μ f q ( A rs x m+n φ r σ s σ rs μ f q ( A rs Δ u x m+n ) prs. ) p rs = Therefore x Λ 2q Δ u Α μ. Conversely, let Λ 2q x Λ 2q Α φ μ. We have Α φ μ 34 Λ 2q Α φ 2 μ sup m n σ φ r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs <. φ Suppose that sup φ2 <. Then there exists a sequence of positive natural numbers (m i n j ) such that lim i j φ m i n j φ2 =. Hence we can write mi n j sup m n σ φ2 r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs sup m n σ Therefore x Λ 2q r σ s σ rs μ f q ( A rs x m+n φ Α φ 2 μ ) p rs and =. which is a contradiction. Hence φ sup φ2 <. 3.5 Theorem. Let f be an modulus function which satisfies the 2 condition. Then Λ 2q Α φ μ = Λ 2q Α φ 2 μ if and only if φ sup < and sup φ 2 φ 2 φ <. 3.6 Theorem. Let f and f be modulus functions which satisfies the 2 condition. Then Λ 2q 2q Λ f 0 f p.

9 Subramanian, N. / Studies in Mathematical Sciences, 8(), 204 Proof: Let x Λ 2q and ϵ > 0 be given and choose δ with 0 < δ < such that f (t)< ϵ for 0 t δ. Therefore we have writ sup m n σ sup m n σ sup m n σ φ r σ s σ rs μ f f q ( A rs Δ u x m+n ) p rs φ rs μ f f q ( A rs Δ u x m+n ) p rs φ 2 rs μ f f q ( A rs Δ u x m+n ) p rs Where the summation is over f q ( A rs Δ u x m+n ) δ and the summation 2 is over f (q ( A rs Δ u x m+n )) > δ. Since f is continuous, we have sup m n σ φ rs μ f f q ( A rs Δ u x m+n ) p rs max{ f H }sup m n σ φ rs μ f q ( A rs Δ u x m+n ) p rs max{ f H }sup m n σ φ r σ s σ rs μ f q ( A rs Δ u x m+n ) prs. For f (q ( A rs Δ u x m+n )) > δ we use the fact that f (q ( A rs Δ u x m+n )) < f (q ( A rs Δ u x m+n )) δ + f (q ( A rs Δ u x m+n )) δ Since f satisfies the 2 condition, then there exists L > such that f (f (q ( A rs Δ u x m+n ))) f + f (q ( A rs Δ u x m+n )) δ f (2)+ f 2 f 2 L (q ( A rs Δ u x m+n )) δ + Lf 2 f 2 (q ( A rs Δ u x m+n )) δ = Lf (2) δ f (q ( A rs Δ u x m+n )). sup m n σ φ r σ s σ rs μ f f q ( A rs Δ u x m+n ) p rs max{ f H } sup m n σ r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs φ + max{ Lf 2 δ H } sup m n σ Therefore x Λ 2q. φ r σ s σ rs μ f q ( A rs x m+n ) p rs. +. = 35

10 On Λ 2q Α φ μ Defined by Modulus 3.7 Theorem. Λ 2q is not separable. Proof: f q ( A rs Δ u x m+n ) prs 0 as m n, so it may so happen that first row or colu may not be convergent, even may not be bounded. Let S be the set that has double sequences such that the first row is built up of sequences of zeros and ones. Then S will be uncountable. Consider open balls of radius 3 - units. Then these open balls will not cover Λ 2q. Hence is not separable. 3.8 Remark. Let f = (f) be a modulus function q and q 2 be two seminorms on X, we have 2q i Λ 2q Δu Λ 2 2 q Δu Λ +q 2, 2q (ii) If q is stronger than q 2 then Λ 2q Δu Λ 2 Δu, 2q (iii) If q is equivalent to q 2 then Λ Δu 3.9 Proposition. For every {Λ 2q {x = x : p = p rs {Λ 2q } γ = { 2q φ 2q = Λ 2 Δu. β } = 2q α {Λ } = } where 2q r σ s σ rs μ f q ( A rs Δ u x m+n ) p rs = N N {}, < }. Proof(): First we show that { 2q } 2q β {Λ }. Let { 2q } and y {Λ 2q } β. Then we can find a positive integer N such that ( y m+n ) < max ( sup m n φ r σ s σ rs μ f q ( A rs Δ u x m+n ) prs ) < < N, for all m, n. Hence we may write m n x y m n x y (f x y ) m n (f u x N m+n ). Since x { 2q }. The series on the right side of the above inequality is convergent, whence x { 2q }. Hence { 2q } 2q β {Λ }. Now we show that{λ 2q } β { 2q }. 36

11 Subramanian, N. / Studies in Mathematical Sciences, 8(), 204 {Λ 2q For this, let x {Λ 2q β } and suppose that x }. Then there exists a positive integer N > such that m n (f u x N m+n ) =. If we define y = N m+n Sgn u m, n =, 2, then y {Λ 2q }. But, since m n x y = (f x y ) = m n (f u x N m+n ) =, we get x {Λ 2q β } which contradicts to the assumption x {Λ 2q β }. Therefore x { 2q }. Therefore {Λ 2q } β = { 2q }. (ii) and (iii) can be shown in a similar way of (i). Therefore we omit it. 4. RESULT 4.. Proposition. The space Λ 2q are not solid. Proof: The space Λ 2q examples. Since the space Λ 2q is not monotone and such is not monotone follows from following is not monotone, is not solid is clear from the remark 2.6. Example: Let X = C and consider the sequence (x ) defined by if (x ) =.. for all m n N.. ( ) Consider the sequence (y) in the preimage space defined as Then (x ) Λ 2q monotone. y ={ i2 if m = n i N 0 otherwise. but (y Λ 2q are not 37

12 On Λ 2q Α φ μ Defined by Modulus 4.2 Proposition. The space Λ 2q are not convergence free in general. Proof: The proof follows from the following example: Example: Consider the sequences (x ), (y Λ 2q. Defined by Δ u x = m+n and Δ m+n u y = m n m+n. Hence m+n <. which implies sup m n x m+n <. Also sup m n m+n m+n sup m n the space Λ 2q 4.3. Proposition. Λ 2q m n m+n = 0. Hence sup m+n m n y m+n =0. Therefore are not convergence free. is not sequence algebra. Proof: This result is clear the following example: Example: Let (x ) = m n N. Then we have x y Λ x y Λ 2q. REFERENCES m m+n and (y m+n = 2q Δu but n m+n for all m+n [0] Apostol, T. (978). Mathematical analysis. London: Addison-Wesley. [02] Basarir, M., & Solancan, O. (999). On some double sequence spaces. J. Indian Acad. Math., 2(2), [03] Bektas, C., & Altin, Y. (2003). The sequence space l M (p, q, s) on seminormed spaces. Indian J. Pure Appl. Math., 34(4), [04] Bromwich, T. J. I'A. (965). An introduction to the theory of infinite series. New York: Macmillan and Co. Ltd. [05] Hardy, G. H. (97). On the convergence of certain multiple series (pp.86-95). Proc. Camb. Phil. Soc. [06] Krasnoselskii, M. A., & Rutickii, Y. B. (96). Convex functions and orlicz spaces. Netherlands: Gorningen. [07] Lindenstrauss, J., & Tzafriri, L. (97). On orlicz sequence spaces. Israel J. Math., 0, [08] Maddox, I. J. (986). Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc, 00(), [09] Moricz, F. (99). Extentions of the spaces c and c0 from single to double sequences. Acta. Math. Hung., 57(-2),

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