ON I-CONVERGENCE OF DOUBLE SEQUENCES IN THE TOPOLOGY INDUCED BY RANDOM 2-NORMS. Mehmet Gürdal and Mualla Birgül Huban. 1.

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1 MATEMATIQKI VESNIK 66, 1 (2014), March 2014 originalni nauqni rad research paper ON I-CONVERGENCE OF DOUBLE SEQUENCES IN THE TOPOLOGY INDUCED BY RANDOM 2-NORMS Mehme Gürdal and Mualla Birgül Huban Absrac. In his aricle we inroduce he noion of I-convergence and I-Cauchyness of double sequences in he opology induced by random 2-normed spaces and prove some imporan resuls. 1. Inroducion Probabilisic meric (PM) spaces were firs inroduced by Menger [19] as a generalizaion of ordinary meric spaces and furher sudied by Schweizer and Sklar [26, 27]. The idea of Menger was o use disribuion funcion insead of non-negaive real numbers as values of he meric, which was furher developed by several oher auhors. In his heory, he noion of disance has a probabilisic naure. Namely, he disance beween wo poins x and y is represened by a disribuion funcion F xy ; and for > 0, he value F xy () is inerpreed as he probabiliy ha he disance from x o y is less han. Using his concep, Sersnev [29] inroduced he concep of probabilisic normed space, which provides an imporan mehod of generalizing he deerminisic resuls of linear normed spaces, also having very useful applicaions in various fields, among which are coninuiy properies [1], opological spaces [3], linear operaors [7], sudy of boundedness [8], convergence of random variables [9], saisical and ideal convergence of probabilisic normed space or 2-normed space [14, 21 23, 25, 32] as well as many ohers. The concep of 2-normed spaces was iniially inroduced by Gähler [5, 6] in he 1960 s. Since hen, many researchers have sudied hese subjecs and obained various resuls [10 13, 28, 31]. P. Kosyrko e al. (cf. [17]; a similar concep was invened in [15]) inroduced he concep of I-convergence of sequences in a meric space and sudied some properies of such convergence. Noe ha I-convergence is an ineresing generalizaion 2010 AMS Subjec Classificaion: 40A35, 46A70, 54E70 Keywords and phrases: -norm; random 2-normed space; ideal convergence; ideal Cauchy sequences; F -opology. This work is suppored by Süleyman Demirel Universiy wih Projec 2947-YL

2 74 M. Gürdal, M. B. Huban of saisical convergence. The noion of saisical convergence of sequences of real numbers was inroduced by H. Fas in [2] and H. Seinhaus in [30]. There are many pioneering works in he heory of I-convergence. The aim of his work is o inroduce and invesigae he idea of I-convergence and I-Cauchy of double sequences in a more general seing, i.e., in random 2-normed spaces. 2. Definiions and noaions Firs we recall some of he basic conceps, which will be used in his paper. Definiion 1. [2, 4] A subse E of N is said o have densiy δ (E) if δ (E) = lim n n 1 n k=1 χ E (k) exiss. A number sequence (x n ) n N is said o be saisically convergen o L if for every ε > 0, δ ({n N : x n L ε}) = 0. If (x n ) n N is saisically convergen o L we wrie s-lim x n = L, which is necessarily unique. Definiion 2. [16, 17] A family I 2 Y of subses of a nonempy se Y is said o be an ideal in Y if: (i) I; (ii) A, B I imply A B I; (iii) A I, B A imply B I. A non-rivial ideal I in Y is called an admissible ideal if i is differen from P (N) and i conains all singleons, i.e., {x} I for each x Y. Le I P (Y ) be a non-rivial ideal. A class F (I) = {M Y : A I: M = Y \ A}, called he filer associaed wih he ideal I, is a filer on Y. Definiion 3. [17, 18] Le I 2 N be a nonrivial ideal in N. Then a sequence (x n ) n N in X is said o be I-convergen o ξ X, if for each ε > 0 he se A (ε) = {n N : x n ξ ε} belongs o I. Definiion 4. [5, 6] Le X be a real vecor space of dimension d, where 2 d <. A 2-norm on X is a funcion, : X X R which saisfies: (i) x, y = 0 if and only if x and y are linearly dependen; (ii) x, y = y, x ; (iii) αx, y = α x, y, α R; (iv) x, y + z x, y + x, z. The pair (X,, ) is hen called a 2-normed space. As an example of a 2-normed space we may ake X = R 2 being equipped wih he 2-norm x, y := he area of he parallelogram spanned by he vecors x and y, which may be given explicily by he formula x, y = x 1 y 2 x 2 y 1, x = (x 1, x 2 ), y = (y 1, y 2 ). Observe ha in any 2-normed space (X,, ) we have x, y 0 and x, y + αx = x, y for all x, y X and α R. Also, if x, y and z are linearly dependen, hen x, y + z = x, y + x, z or x, y z = x, y + x, z. Given a 2-normed space (X,, ), one can derive a opology for i via he following definiion of he limi of a sequence: a sequence (x n ) in X is said o be convergen o x in X if lim n x n x, y = 0 for every y X. All he conceps lised below are sudied in deph in he fundamenal book by Schweizer and Sklar [27].

3 On I-convergence of double sequences 75 Definiion 5. Le R denoe he se of real numbers, R + = {x R : x 0} and S = [0, 1] he closed uni inerval. A mapping f : R S is called a disribuion funcion if i is nondecreasing and lef coninuous wih inf R f () = 0 and sup R f () = 1. We denoe he se of all disribuion funcions by D + such ha f (0) = 0. If a R +, hen H a D +, where { 1 if > a, H a () = 0 if a. I is obvious ha H 0 f for all f D +. Definiion 6. A riangular norm (-norm) is a coninuous mapping : S S S such ha (S, ) is an abelian monoid wih uni one and c d a b if c a and d b for all a, b, c, d S. A riangle funcion τ is a binary operaion on D + which is commuaive, associaive and τ (f, H 0 ) = f for every f D +. Definiion 7. Le X be a linear space of dimension greaer han one, τ is a riangle funcion, and F : X X D +. Then F is called a probabilisic 2-norm and (X, F, τ) a probabilisic 2-normed space if he following condiions are saisfied: (i) F (x, y; ) = H 0 () if x and y are linearly dependen, where F (x, y; ) denoes he value of F (x, y) a R, (ii) F (x, y; ) H 0 () if x and y are linearly independen, (iii) F (x, y; ) = F (y, x; ) for all x, y X, (iv) F (αx, y; ) = F (x, y; α ) for every > 0, α 0 and x, y X, (v) F (x + y, z; ) τ(f (x, z; ), F (y, z; )) whenever x, y, z X, and > 0. If (v) is replaced by (vi) F (x + y, z; ) F (x, z; 1 ) F (y, z; 2 ) for all x, y, z X and 1, 2 R + ; hen (X, F, ) is called a random 2-normed space (for shor, RTN space). Remark 1. Noe ha every 2-norm space (X,, ) can be made a random 2-normed space in a naural way, by seing (i) F (x, y; ) = H 0 ( x, y ), for every x, y X, > 0 and a b = min {a, b}, a, b S; or (ii) F (x, y; ) = + x,y for every x, y X, > 0 and a b = ab for a, b S. Le (X, F, ) be an RTN space. Since is a coninuous -norm, he sysem of (ε, λ)-neighborhoods of θ (he null vecor in X) {N θ (ε, λ) : ε > 0, λ (0, 1)}, where N θ (ε, λ) = {x X : F x (ε) > 1 λ}, deermines a firs counable Hausdorff opology on X, called he F -opology. Thus, he F -opology can be compleely specified by means of F -convergence of sequences. I is clear ha x y N θ means y N x and vice-versa.

4 76 M. Gürdal, M. B. Huban A double sequence x = (x jk ) in X is said o be F -convergence o L X if for every ε > 0, λ (0, 1) and for each nonzero z X here exiss a posiive ineger N such ha x jk, z L N θ (ε, λ) for each j, k N or, equivalenly, In his case we wrie F -lim x jk, z = L. x jk, z N L (ε, λ) for each j, k N. Lemma 1. Le (X,, ) be a real 2-normed space and (X, F, ) be an RTN space induced by he random norm F x,y () = + x,y, where x, y X and > 0. Then for every double sequence x = (x jk ) and nonzero y in X lim x L, y = 0 F -lim (x L), y = 0. Proof. Suppose ha lim x L, y = 0. Then for every > 0 and for every y X here exiss a posiive ineger N = N () such ha We observe ha for any given ε > 0, which is equivalen o Therefore, by leing λ = x jk L, y < for each j, k N. ε + x jk L, y ε ε ε + x jk L, y > ε+ (0, 1) we have < ε + ε ε ε + = 1 ε +. F xjk L,y (ε) > 1 λ for each j, k N. This implies ha x jk, y N L (ε, λ) for each j, k N as desired. 2. I2 F and I F 2 -convergence for double sequences in RTN spaces In his secion we sudy he concep of I and I -convergence of a double sequence in (X, F, ) and prove some imporan resuls. Throughou he paper we ake I F 2 as a nonrivial admissible ideal in N N. Definiion 8. Le (X, F, ) be an RTN space and I be a proper ideal in N N. A double sequence x = (x jk ) in X is said o be I2 F -convergen o L X (I2 F -convergen o L X wih respec o F -opology) if for each ε > 0, λ (0, 1) and each nonzero z X, {(j, k) N N : x jk, z / N L (ε, λ)} I 2. In his case he vecor L is called he I F 2 -limi of he double sequence x = (x jk ) and we wrie I F 2 -lim x, z = L.

5 On I-convergence of double sequences 77 Lemma 2. Le (X, F, ) be an RTN space. If a double sequence x = (x jk ) is I F 2 -convergen wih respec o he random 2-norm F, hen I F 2 -limi is unique. Proof. Le us assume ha I F 2 -lim x, z = L 1 and I F 2 -lim x, z = L 2 where L 1 L 2. Since L 1 L 2, selec ε > 0, λ (0, 1) and each nonzero z X such ha N L1 (ε, λ) and N L2 (ε, λ) are disjoin neighborhoods of L 1 and L 2. Since L 1 and L 2 boh are I F 2 -limi of he sequence (x jk ), we have and A = {(j, k) N N : x jk, z / N L1 (ε, λ)} B = {(j, k) N N : x jk, z / N L2 (ε, λ)} boh belong o I F 2. This implies ha he ses A c = {(j, k) N N : x jk, z N L1 (ε, λ)} and B c = {(j, k) N N : x jk, z N L2 (ε, λ)} belong o F (I 2 ). In his way we obain a conradicion o he fac ha he neighborhoods N L1 (ε, λ) and N L2 (ε, λ) of L 1 and L 2 are disjoin. Hence we have L 1 = L 2. This complees he proof. Lemma 3. Le (X, F, ) be an RTN space. Then we have (i) F -lim x jk, z = L, hen I F 2 -lim x jk, z = L. (ii) If I F 2 -lim x jk, z = L 1 and I F 2 -lim y jk, z = L 2, hen I F 2 -lim (x jk + y jk ), z = L 1 + L 2. (iii) If I F 2 -lim x jk, z = L and α R, hen I F 2 -lim αx jk, z = αl. (iv) If I2 F -lim x jk, z = L 1 and I2 F -lim y jk, z = L 2, hen I2 F = L 1 L 2. -lim (x jk y jk ), z Proof. (i) Suppose ha F -lim x jk, z = L. Le ε > 0, λ (0, 1) and nonzero z X. Then here exiss a posiive ineger N such ha x jk, z N L (ε, λ) for each j, k > N. Since he se A = {(j, k) N N : x jk, z / N L (ε, λ)} {1, 2,..., N 1} {1, 2,..., N 1} and he ideal a I2 F = L. is admissible, we have A I F 2. This shows ha I F 2 -lim x jk, z (ii) Le ε > 0, λ (0, 1) and nonzero z X. Choose η (0, 1) such ha (1 η) (1 η) > (1 λ). Since I2 F -lim x jk, z = L 1 and I2 F -lim y jk, z = L 2, he ses )} A = {(j, k) N N : x jk, z / N L1 2, λ and )} B = {(j, k) N N : y jk, z / N L2 2, λ belong o I F 2. Le C = {(j, k) N N : (x jk + y jk ), z / N L1+L 2 (ε, λ)}. Since I F 2 is an ideal, i is sufficien o show ha C A B. This is equivalen o show

6 78 M. Gürdal, M. B. Huban ha C c A c B c where A c and B c belong o F (I 2 ). Le (j, k) A c B c, i.e., (j, k) A c and (j, k) B c, and we have F (xjk +y jk ) (L 1+L 2),z (ε) F xjk L 1,z F yjk L 2) 2,z 2) > (1 η) (1 η) > (1 λ). Since (j, k) C c A c B c F (I 2 ), we have C A B I F 2. (iii) I is rivial for α = 0. Now le α 0, ε > 0, λ (0, 1) and nonzero z X. Since I F 2 -lim x jk, z = L, we have This implies ha Le (j, k) A c. Then we have A = {(j, k) N N : x jk, z / N L (ε, λ)} I 2 A c = {(j, k) : x jk, z N L (ε, λ)} F (I 2 ). F αxjk αl,z (ε) = F xjk L,z ( ) ε α F xjk L,z (ε) F 0 α ε > (1 λ) 1 = (1 λ). So {(j, k) N N : αx jk, z / N αl (ε, λ)} I 2. Hence I2 F (iv) The resul follows from (ii) and (iii). ) -lim αx jk, z = αl. We inroduce he concep of I F 2 -convergence closely relaed o IF 2 -convergence of double sequences in random 2-normed space and show ha I2 F -convergence implies I2 F -convergence bu no conversely. Definiion 9. Le (X, F, ) be an RTN space. We say ha a sequence x = (x jk ) in X is I2 F -convergen o L X wih respec o he random 2-norm F if here exiss a subse K = {(j m, k m ) : j 1 < j 2 < ; k 1 < k 2 < } N N such ha K F (I 2 ) (i.e., N N \ K I 2 ) and F -lim m x jm,k m, z = L for each nonzero z X. In his case we wrie I F 2 -lim x, z = L and L is called he I F 2 -limi of he double sequence x = (x jk ). Theorem 1. Le (X, F, ) be an RTN space and I 2 be an admissible ideal. If 2 -lim x, z = L, hen I2 F -lim x, z = L. I F Proof. Suppose ha I F 2 -lim x, z = L. Then by definiion, here exiss K = {(j m, k m ) : j 1 < j 2 < ; k 1 < k 2 < } F (I 2 )

7 On I-convergence of double sequences 79 such ha F -lim m x jm,k m, z = L. Le ε > 0, λ (0, 1) and nonzero z X be given. Since F -lim m x jm k m, z = L, here exiss N N such ha x jm k m, z N L (ε, λ) for every m N. Since is conained in A = {(j m, k m ) K : x jmk m, z / N L (ε, λ)} B = {j 1, j 2,..., j N 1 ; k 1, k 2,..., k N 1 } and he ideal I 2 is admissible, we have A I 2. Hence {(j, k) N N : x jk, z / N L (ε, λ)} K B I 2 for ε > 0, λ (0, 1) and nonzero z X. Therefore, we conclude ha I F 2 -lim x, z = L. The following example shows ha he converse of Theorem 1 need no be rue. Example 1. Consider X = R 2 wih x, y := x 1 y 2 x 2 y 1 where x = (x 1, x 2 ), y = (y 1, y 2 ) R 2 and le a b = ab for all a, b S. For all (x, y) R 2 and > 0, consider F x,y () = + x, y. Then ( R 2, F, ) is an RTN space. Consider a decomposiion of N N as N N = i,j ij such ha for any (m, n) N N each ij conains infiniely many (i, j) s where i m, j n and ij mn = for (i, j) (m, n). Le I 2 be he class of all subses of N N which inersec a mos a finie number of ij s. Then ( I ) 2 is an 1 admissible ideal. We define a double sequence (x mn ) as follows: x mn = ij, 0 R 2 if (m, n) ij. Then for nonzero z X, we have F xmn,z () = as m, n. Hence I F 2 -lim m,n x mn, z = 0. + x mn, z 1 Now, we show ha I F 2 -lim m,n x mn, z 0. Suppose ha I F 2 -lim m,n x mn, z = 0. Then by definiion, here exiss a subse K = {(m j, n j ) : m 1 < m 2 < ; n 1 < n 2 < } N N such ha K F (I 2 ) and F -lim j x mj n j, z = 0. Since K F (I 2 ), here exiss H I 2 such ha K = N N \ H. Then here exiss posiive inegers p and q such ha ( p ( )) ( q ( H mn mn )). m=1 n=1 1 Thus p+1,q+1 K and so x mjn j = (p+1)(q+1) > 0 for infiniely many values (m j, n j ) s in K. This conradics he assumpion ha F -lim j x mj n j, z = 0. Hence I F 2 -lim m,n x mn, z 0. Hence he converse of Theorem 1 need no be rue. n=1 m=1 The following heorem shows ha he converse holds if he ideal I 2 saisfies condiion (AP).

8 80 M. Gürdal, M. B. Huban Definiion 10. [23] An admissible ideal I 2 P (N N) is said o saisfy he condiion (AP) if for every sequence (A n ) n N of pairwise disjoin ses from I 2 here are ses B n N, n N, such ha he symmeric difference A n B n is a finie se for every n and n N B n I 2. Theorem 2. Le (X, F, ) be an RTN space and he ideal I 2 saisfy he condiion (AP). If x = (x jk ) is a double sequence in X such ha I2 F -lim x, z = L, hen -lim x, z = L. I F 2 Proof. Since I F 2 -lim x, z = L, so for every ε > 0, λ (0, 1) and nonzero z X, he se {(j, k) N N : x jk, z / N L (ε, λ)} I 2. We define he se A p for p N as { A p = (j, k) N N : 1 1 p F x jk,z L < 1 1 }. p + 1 Then i is clear ha {A 1, A 2,... } is a counable family of muually disjoin ses belonging o I 2 and so by he condiion (AP) here is a counable family of ses {B 1, B 2,... } I 2 such ha he symmeric difference A i B i is a finie se for each i N and B = i=1 B i I 2. Since B I 2, here is a se K F (I 2 ) such ha K = N N \ B. Now we prove ha he subsequence (x jk ) (j,k) K is convergen o L wih respec o he random 2-norm F. Le η (0, 1), ε > 0 and nonzero z X. Choose a posiive q such ha q 1 < η. Then {(j, k) N N : x jk, z / N L (ε, η)} ( {(j, k) N N : x jk, z / N L ε, 1 )} q Since A i B i is a finie se for each i = 1, 2,..., q 1, here exiss (j 0, k 0 ) N N such ha ( q 1 ) B i {(j, k) N N : j j 0 and k k 0 } i=1 = ( q 1 i=1 q 1 i=1 A i. A i ) {(j, k) N N : j j 0 and k k 0 }. If j j 0, k k 0 and (j, k) K, hen (j, k) / q 1 i=1 B i and (j, k) / q 1 i=1 A i. Hence for every j j 0, k k 0 and (j, k) K we have x jk, z / N L (ε, η). Since his holds for every ε > 0, η (0, 1) and nonzero z X, so we have I F 2 - lim x, z = L. This complees he proof of he heorem.

9 On I-convergence of double sequences I2 F and I F 2 -double Cauchy sequences in RTN spaces In his secion we sudy he conceps of I 2 -Cauchy and I 2 -Cauchy double sequences in (X, F, ). Also, we will sudy he relaions beween hese conceps. Definiion 11. Le (X, F, ) be an RTN space and I be an admissible ideal of N N. Then a double sequence x = (x jk ) of elemens in X is called a I F 2 - Cauchy sequence in X if for every ε > 0, λ (0, 1) and nonzero z X, here exiss s = s (ε), = (ε) such ha {(j, k) N N : x jk x s, z / N θ (ε, λ)} I 2. Definiion 12. Le (X, F, ) be a RTN space and I be an admissible ideal of N N. We say ha a double sequence x = (x jk ) of elemens in X is a I F 2 -Cauchy sequence in X if for every ε > 0, λ (0, 1) and nonzero z X, here exiss a se K = {(j m, k m ) : j 1 < j 2 < ; k 1 < k 2 < } N N such ha K F (I 2 ) and (x jm,k m ) is an ordinary F -Cauchy in X. The nex heorem gives ha each I F 2 -double Cauchy sequence is a IF 2 -double Cauchy sequence. Theorem 3. Le (X, F, ) be an RTN space and I be a nonrivial ideal of N N. If x = (x jk ) is a I2 F -double Cauchy sequence, hen x = (x jk ) is a I2 F - double Cauchy sequence, oo. Proof. Le (x jk ) be a I F 2 -Cauchy sequence. Then for ε > 0, λ (0, 1) and nonzero z X, here exiss K = {(j m, k m ) : j 1 < j 2 < ; k 1 < k 2 < } F (I 2 ) and a number N N such ha x jm k m x jp k p, z N θ (ε, λ) for every m, p N. Now, fix p = j N+1, r = k N+1. Then for every ε > 0, λ (0, 1) and nonzero z X, we have x jm k m x pr, z N θ (ε, λ) for every m N. Le H = N N \ K. I is obvious ha H I 2 and A (ε, λ) = {(j, k) N N : x jk x pr, z / N θ (ε, λ)} H {j 1 < j 2 < < j N ; k 1 < k 2 < < k N } I 2. Therefore, for every ε > 0, λ (0, 1) and nonzero z X, we can find (p, r) N N such ha A (ε, λ) I 2, i.e., (x jk ) is a I F 2 -double Cauchy sequence. Now we will prove ha I F 2 -convergence implies IF 2 -Cauchy condiion in a 2-normed space. Theorem 4. Le (X, F, ) be an RTN space and I be an admissible ideal of N N. If a sequence x = (x jk ) is I2 F -convergen, hen i is a I2 F -double Cauchy sequence.

10 82 M. Gürdal, M. B. Huban Proof. By assumpion here exiss a se K = {(j m, k m ) : j 1 < j 2 < ; k 1 < k 2 < } N N such ha K F (I 2 ) and F -lim m x jm,k m, z = L for each nonzero z in X, i.e., here exiss N N such ha x jm k m, z N L (ε, λ) for every ε > 0, λ (0, 1), each nonzero z in X and m > N. Choose η (0, 1) such ha (1 η) (1 η) > (1 λ). Since ) ) F xj mkm x jpkp,z (ε) F xj L, z F xj mkm 2 L, z pkp 2 > (1 η) (1 η) > 1 λ for every ε > 0, λ (0, 1), each nonzero z in X and m > N, p > N, we have x jmk m x jpk p, z / N L (ε, λ) for every m, p > N and each nonzero z X, i.e., (x jk ) in X is an I2 F -double Cauchy sequence in X. Then by Theorem 3 (x jk ) is a I2 F -double Cauchy sequence in he RTN space. Theorem 5. Le (X, F, ) be an RTN space and I be an admissible ideal of N N. If a sequence x = (x jk ) of elemens in X is I F 2 -convergen, hen i is a I F 2 -double Cauchy sequence. Proof. Suppose ha (x jk ) is I2 F -convergen o L X. Le ε > 0, λ (0, 1) and nonzero z X be given. Then we have )} A = {(j, k) N N : x jk, z / N L 2, λ I 2 This implies ha )} A c = {(j, k) N N : x jk, z N L 2, λ F (I 2 ) Choose η (0, 1) such ha (1 η) (1 η) > (1 λ). Then for every (j, k), (s, ) A c, F xjk x s,z (ε) F xjk L,z F xs L,z > (1 η) (1 η) > (1 λ). 2) 2) Hence {(j, k) N N : x jk x s, z N θ (ε, λ)} F (I 2 ) for nonzero z X. This implies ha {(j, k) N N : x jk x s, z / N θ (ε, λ)} I 2, i.e., (x jk ) is a I F 2 -double Cauchy sequence. Acknowledgemen. The auhors would like o hank anonymous referees on suggesions o improve his ex. REFERENCES [1] C. Alsina, B. Schweizer, A. Sklar, Coninuiy properies of probabilisic norms, J. Mah. Anal. Appl. 208 (1997), [2] H. Fas, Sur la convergence saisique, Colloq. Mah. 2 (1951), [3] M.J. Frank, Probabilisic opological spaces, J. Mah. Anal. Appl. 34 (1971), [4] A.R. Freedman, J.J. Sember, Densiies and summabiliy, Pacific J. Mah. 95 (1981),

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