### Journal Menu >> Applied Mathematics, 2011, 2, 398-402 doi:10.4236/am.2011.24048 Published Online April 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function Vakeel A. Khan, Sabiha Tabassum Department of Mat hematics, Aligarh Muslim University, Aligarh, India E-mail: vakhan@math.com, sabihatabassum@math.com Received November 26, 2010; revised January 15, 2011; accepted Ja nu ary 18, 2011 Abstract The concept of statistical convergence was introduced by Stinhauss  in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2- normed spaces to be statistically Cauchy sequence in 2-normed spaces.1 Keywords: Double Sequence Spaces, Natural Density, Statistical Convergence, 2-Norm, Orlicz Function 1. Introduction In order to extend the notion of convergence of sequen- ces, statistical convergence was introduced by Fast  and Schoenberg  independently. Later on it was fur-ther investigated by Fridy and Orhan . The idea de-pends on the notion of density of subset of . The concept of 2-normed spaces was initially intro-duced by G ahler [5-7] in the mid of 1960’s. Since then, many researchers have studied this concept and ob- tained various results, see for instance . Let X be a real vector space of dimension d, where 2.d A 2-norm on X is a function .,. :XXR which satisfies the following four conditions: 1) 12,=0xx if and only if 12,xx are linearly de-pendent; 2) 12 21,=,xxxx: 3) 12 12,= ,xxxx, for any R: 4) 22 2,,,xxxxx xx The pair ,.,.X is then called a 2-normed space (see ). Example 1.1. A standard example of a 2-normed space is 2R equipped wit h t he f oll owing 2-norm ,:=xy the area of the triangle having vertices 0,,.xy Example 1.2. Let Y be a space of all bounded real-valued functions on R. For ,fg in Y, define ,=0fg , if f, g are linearly dependent, , =,suptRfgftgt if f, g are linearly independent. Then .,. is a 2-norm on Y. We recall some facts connecting with statistical con-vergence. If K is subset of positive integers , then nK denotes the set {:}.kKkn The natural density of K is given by ||=,lim nnKKn where nK deno- tes the number of elements in nK, provided this limit exists. Finite subsets have natural density zero and =1cKK where =\cKK, that is the comp- lement of K. If 12KK and 1K and 2K have natu-ral densities then 12.KK Moreover, if 12==1,KK then 12=1KK (see ). A real number sequence =jxx is statistically con- vergent to L provided that for every >0 the set :| |jnxL has natural density zero. The sequ- ence =jxx is statustically Cauchy sequence if for each >0 there is positive integer =NN such that :|= 0jNnxx  (see ). If =jxx is a sequence that satisfies some property P for all n except a set of natural density zero, then we say that jx satisfies some property P for “almost all n”. An Orlicz Function is a function :0, 0,M  which is continuous, nondecreasing and convex with 0=0M, >0Mx for >0x and Mx, as x. If convexity of M is replaced by Mxy Mx My, then it is called a Modulus funtion (see Mad- dox ). An Orlicz function may be bounded or un- 12000 Mathematics Subject Classification. 46E30, 46E40, 46B20. V. A. KHAN ET AL. Copyright © 2011 SciRes. AM 399bounded. For example,  =0<1pMx xp is un- bounded and =1xMx x is bounded. Lindesstrauss and Tzafriri  used the idea of Orlicz sequence space; =1:=:<, for some >0kMkxlxwM which is Banach space with the norm =1||=inf>0:1 .kMkxxM The space Ml is closely related to the space pl, which is an Orlicz sequence space with =pMxx for 1<.p An Orlicz function M satisfies the 2condition 2( f )Mor short if there exist constant 2K and 0>0u such that  2MuKMu whenever 0||uu. Note that an Orlicz function satisfies the inequality   for all with 0<<1.Mx Mx  Orlicz function has been studied by V. A. Khan [14-17] and many others. Throughout a double sequence =klxx is a double infinite array of elements klx for ,.kl Double sequences have been studied by V. A. Khan [18-20], Moricz and Rhoades  and many others. A double sequence =jkxx called statistically con- vergent to L if ,1,:, ,=0lim jkmn jkxLj mk nmn   where the vertical bars indicate the number of elements in the set. (see ) In this case we write 2lim =jkstxL. 2. Definitions and Preliminaries Let jx be a sequence in 2-normed space ,.,.X. The sequence jx is said to be statistically convergent to L, if for every >0, the set :,jjxLz has natural density zero for each nonzero z in X, in other words jx statistically converges to L in 2-normed space ,.,.X if 1:,=0lim jnjx Lzn  for each nonzero z in X. It means that for every zX, ,< ...jxLz aan In this case we write ,:=,.lim jnstx LzLz Example 2.1 Let 2=XR be equiped with the 2- norm by the formula 12211 21 2,=,= ,,=,.xyxyxyxxxyyy Define the jx in 2-normed space ,.,.X by 21, if =,,= 11, otherwise.jnnkkNxnn and let =1,1L and 12=,zzz. If 1=0z then =:,=jKj xLz for each z in 21||,:=,zXj nkk is a finite set, so 1221:,=:=,1 finite set.jjxLzjjkkz Therefore, 12211:,1:= ,101jjxLznjjkkzn  for each z in X. Hence, :, =0njxLz  for every >0 and zX. V. A. Khan and Sabiha Tabassum  defined a double sequence jkx in 2-nor med space ,.,.X to be Cauchy with respect to the 2-norm if ,,=0for every and ,.lim jk pqjp xxzzX kq If every Cauchy sequence in X converges to some ,LX then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be 2-Banach space. Example 2.2 Define the xi in 2-normed space ,.,.X by 20, if =,,=0,0 otherwise.jjjkkNx V. A. KHAN ET AL. Copyright © 2011 SciRes. AM 400and let =0,0L and 12=,zzz. If 1=0z then 2:,1,4,9,16, , ;jjxLzj We have that :, =0jjxLz  for every >0 and zX. This implies that ,=lim jnstxz ,Lz. But the sequence jx is not convergent to .L A sequence which converges statistically need not be bounded. This fact can be seen from Example [2.1] and Example [2.2]. 3. Main Results In this paper we define a double sequence jkx in 2-normed space ,.,.X to be statistically Cauchy with respect to the 2-norm if for ever y >0 and every nonzero zX there exists a number =,pp z and =,qqz such that ,1,:,,,=0lim jk pqmn jkNNxxzjmknmn   In this case we write 2lim, :=,jkstx LzLz . Theorem 3.1. Let jkx be a double sequence in 2-normed space ,.,.X and ,LL X. If 2lim, = ,jkstxzLz and 2lim, =,jkstxzLz, then =.LL Proof. Assume =,LL. Then =0LL, so there exists a zX, such that LL and z are linearly in-dependent. Therefore ,=2, with >0.LLz Now 2= , ,,.jk jkjk jkLxxL zxLzxLz  So ,: ,<,: ,0 and zX. ,:,,:=.jkjk jkjkN NxLzjkNNxy Therefore ,:,,:,,:=.jkjkjk jkjkN NxLzjkN NyLzjkNNxy   (3.1) Since ,=,lim jknyzLz for every zX, the set ,:,jkjkN NyLz contains finite number of integers. Hence, ,:,jkjkN NyLz =0. Using inequality [3.1], we get ,:,=0jkjkN NxLz for every >0 and .zX Consequently, 2lim,=, .jkstx LzLz Theorem 3.3. Let the double sequence jkx and jky in 2-normed space ,.,.X and ,LL X and a. If 2lim, = ,jkstxzLz and 2lim,=, ,jkstyzLz for every nonzero zX, then 1) 2lim, =,jk jkstxyzLLz , for each nonzero zX and 2) 2lim, =,jkstaxzaLz, for each nonzero zX. Proof 1) Assume that 2lim,=, ,jkstxzLz and 2lim, =,jkstyzLz, for every nonzero zX. Then 1=0K and 2=0K where  11=:=,: ,2jkKKjkNNx Lz  22=:=,: ,2jkKK jkNNyLz  for every >0 and zX. Let  =:=,:(), .jk jkKKjk NNxyLLz To prove that =0K, it is sufficient to prove that 12KKK. Suppose 00,jk K. Then 000,jk jkoxy LLz  (3.2) V. A. KHAN ET AL. Copyright © 2011 SciRes. AM 401Suppose to the contrary that 00 12,jk K K. Then 00 1,jk K and 002,jkK. If 001,jk K and 00 2,jk K then 00 ,<2jkxLz and 00,<.2jkxLz Then, we get 000000,,, <=22jk jkojk jkoxy LLzxLzyLz  which contradicts [3.2]. Hence 0012,jk K K, that is, 12KKK. 2) Let 2lim,=, ,jkstxzL za and =0a. Then ,:,=0.jkjkN NxLza Then we have ,:,=,: ,=,: ,.jkjkjkjkN NaxaLzjkNNa xLzjkN NxLza    Hence, the right handside of above equality equals 0. Hence, 2lim,=,,jkstaxzaLz for every nonzero .zX From Theorem 1 of Fridy  we have Theorem 3.4. Let jkx be statistically Cauchy se-quence in a finite dimensional 2-normed space ,.,.X. Then there exists a convergent double sequence jky in ,.,.X such that =jkjkxy for almost all n. Proof. See proof of Theorem 2.9 . Theorem 3.5. Let jkx be a double sequence in 2- normed space ,.,.X The double sequence ()jkx is statistically convergent if and only if ()jkx is a statisti-cally Cauchy sequence. Proof. Assume that 2lim, = ,jkstxzLz for every nonzero zX and >0. Then, for every zX, ,< almost all , 2jkxLz n and if =,pp z and =,qq z is chosen so that ,<,2pqxLz then, we have ,,,< almost all . 22= almost all .jk pqjkpqxxzxLz Lxznn Hence, jkx is statistically Cauchy sequence. Conversely, assume that jkx is a statistically Cauchy sequence. By Theorem 3.4, we have 2lim, =jkstxz ,Lz for each zX. 4. References  H. Stinhaus, “Sur la Convergence Ordinarie et la Conver- gence Asymptotique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 73-74.  H. Fast, “Sur la Convergence Statistique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244.  I. J. Schoenberg, “The Integrability of Certain Functions and Related Summability Methods,” American Mathema- tical Monthly, Vol. 66, No. 5, 1959, pp. 361-375. doi:10.2307/2308747  J. A. Fridy and C. 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