Selçuk J. Appl. Math. Selçuk Journal of
Vol. 10. No. 2. pp. 27-41, 2009 Applied Mathematics
On Ideal Convergence of Sequences in Intuitionistic Fuzzy Normed Spaces
Vijay Kumar1, Kuldeep Kumar2
1Department of Mathematics, Haryana College of Technology and Management,
Kaithal-136027, Haryana, India e-mail: vjy_ kaushik@ yaho o.com
2Department of Mathematics, National Institute of Technology, A Deemed University,
Kurukshetra-136119, Haryana, India e-mail:kuldeepnitk@ yaho o.com
Received Date: April 11, 2008 Accepted Date: July 2, 2009
Abstract. The notion of ideal convergence of single and double sequences were introduced in [16] and [17], respectively. This notion includes the notion of statistical convergence which has been intensively investigated in last twenty years. In this paper we de…ne and study ideal analogue of convergence and Cauchy sequences on intuitionistic fuzzy normed spaces(IFNS for short) and establish some properties related to these notions.
Key words: Intuitionistic fuzzy metric spaces, intuitionistic fuzzy normed spaces, statistical convergence, ideal convergence.
2000 Mathematics Subject Classi…cation: 40A05, 60B99. 1. Introduction
After the introduction of the concept of a fuzzy set by Zadeh [25], there has been a great e¤ort to obtain fuzzy analogues of classical theories. Among other …elds, a progressive developments is made in the …eld of fuzzy topology. The concept of fuzzy topology may have very important applications in quantum particle physics particularly in connections with both string and "1 theory which were given and studied by Naschie [19]. Atanassov [1 to 4] introduced the concept of intuitionistic fuzzy sets (IFS for short) whereas Deschrijver and Kerre [11]discussed some of their properties. Using this type of generalized fuzzy set, Coker [5 to 9] de…ned intuitionistic fuzzy topological spaces. These spaces and its generalizations are later studied by several authors. One of the most important problems in fuzzy topology is to obtain an appropriate concept of intuitionistic fuzzy normed space. This problem has been investigated by Saadati and Park
[20, 21]. They also de…ned the ordinary convergence and Cauchy sequence on IFNS and proved that every …nite dimensional intuitionistic fuzzy normed space is complete.
On the other side, the concept of statistical convergence for real number se-quences was …rst introduced by Fast [12] and Schonenberg [23] independently. Later on it was further investigated from sequence space point of view and linked with summability theory by Fridy [13], Salat [22] and many others. The idea is based on the notion of natural density of subsets of N , the set of positive integers. For A N; the natural density (A) is de…ned by
(A) = lim n!1
1
njfk < n : k 2 Agj;
where the vertical bar denotes the cardinality of the respective set.
Kostyrko et al [16] presented a very interesting generalization of statistical con-vergence and called it I-concon-vergence. They used the notion of an ideal I of subsets of N to de…ne such a concept. Dems [10] continued with this study and introduced the concept of I-Cauchy sequence in a metric space. He proved that a sequence in a metric space is I-convergent, if and only if, it is I-Cauchy. Na-biev et al [18] proved some more properties regarding I-Cauchy and I -Cauchy sequences. Gürgal [14] extended the idea of I-convergence in 2-normed spaces whereas Kumar [17] discussed the concept for double sequences.
Recently, Karakus et al [15] extended the idea of statistical convergence in IFNS and gave its useful characterization. In present paper we introduce and study the notions of I-convergence, I-Cauchy sequences, I - convergence and I -Cauchy sequences on IFNS. We also present example which shows that, for sequences in IFNS, I-convergence does not implies I -convergence.
2. Background and Preliminaries
Throughout the paper N, R respectively, will denote the sets of positive integers and real numbers whereas AC, P (A) respectively, will denote the complement and power set of any set A: We begin with some basic de…nitions used in this paper.
De…nition 2.1. [24] A triangular norm or brie‡y a t-norm is a binary operation on the closed interval [0; 1] which is continuous, commutative, associative, non-decreasing and has 1 as a netural element, i.e., it is the continuous mapping
: [0; 1] [0; 1] ! [0; 1] such that for all a; b; c and d 2 [0; 1], we have (i) a 1 = a ;
(ii) a b = b a;
(iii) c d a b if c a and d b; (iv) (a b) c = a (b c).
De…nition 2.2. [24] A triangular conorm or brie‡y a t-conorm is a binary operation on the closed interval [0; 1] which is continuous, commutative, asso-ciative, non-decreasing and has 0 as a netural element, i.e., it is the continuous mapping } : [0; 1] [0; 1] ! [0; 1] such that for all a; b; c and d 2 [0; 1], we have
(i) a}0 = a; (ii) a}b = b}a;
(iii) c}d a}b if c a and d b ; (iv) (a}b)}c = a}(b}c).
With the help of the De…nition 2.1 and De…nition 2.2, Saadati and Park [20] have recently introduced the concept of intuitionistic fuzzy normed space as follows:
De…nition 2.3. [20] The …ve tuple (X; ; ; ; }) is said to be an intuitionistic fuzzy normed space (IFNS in short) if X is a vector space, is a continuous t-norm, } is a continuous t-conorm, and , are fuzzy sets on X (0; 1) satisfying the following conditions.
For every x; y 2 X and s; t > 0, we have (i) (x; t) + (x; t) 1 ;
(ii) (x; t) > 0 ;
(iii) (x; t) = 1 if and only if x = 0 ; (iv) ( x; t) = (x;j jt ) for each 6= 0 ; (v) (x; t) (y; s) (x + y; t + s) ; (vi) (x; :) : (0; 1) ! [0; 1] is continuous ; (vii) limt!1 (x; t) =1 and limt!0 (x; t) = 0; (viii) (x; t) < 1 ;
(ix) (x; t) = 0, if and only if x = 0 ; (x) ( x; t) = (x;j jt ) for each 6= 0 ; (xi) (x; t)} (y; s) (x + y; t + s) ;
(xii) (x; :) : (0; 1) ! [0; 1] is continuous and (xiii) limt!1 (x; t) = 0 and limt!0 (x; t) = 1. In this case ( ; ) is called intuitionistic fuzzy norm.
Example 2.1. Suppose that ( X; k : k ) is a normed space and let a b = ab and a}b = minfa + b; 1g for all a; b 2 [0; 1]. For all x 2 X and every t > 0, consider
(x; t) = t
t + k x k and (x; t) =
k x k t + k x k:
Then it is easy to see that (X; ; ; ; }) is an intuitionistic fuzzy normed space. Saadati and Park [20] de…ned the concepts of ordinary convergence and Cauchy sequences in an intuitionistic fuzzy normed space as follows.
De…nition 2.4. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) of elements of X is said to be convergent to 2 X with respect to the intuitionistic fuzzy norm ( ; ) if for each " > 0 and t > 0 there exists a positive integer m such that (xn ; t) > 1 " and (xn ; t) < " whenever n m. The element is called the ordinary limit of the sequence (xn) with respect to the intuitionistic fuzzy norm ( ; ) and we shall write ( ; ) lim xn= .
De…nition 2.5. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) of elements of X is said to be Cauchy with respect to the intuitionistic fuzzy norm ( ; ) if for each " > 0 and t > 0 there exists a positive integer m0such that (xn xm; t) > 1 " and (xn xm; t) < " whenever n; m m0.
Karakus et al [15] have generalized these notions and introduced the concepts of statistical convergence and statistical Cauchy sequences as follows.
De…nition 2.6. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) of ele-ments in X is said to be statistically convergent to 2 X with respect to the intuitionistic fuzzy norm ( ; ) if for each " > 0 and t > 0 we have
(fn 2 N : (xn ; t) 1 " or (xn ; t) "g) = 0:
The element is called the statistical limit of the sequence (xn) with respect to the intuitionistic fuzzy norm ( ; ) and we write st( ; ) lim xn= .
De…nition 2.7. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) of elements in X is said to be statistically Cauchy with respect to the intuitionistic fuzzy norm ( ; ) if for each " > 0 and t > 0 there exist a positive integer m such that
(fn 2 N : (xn xm; t) 1 " or (xn xm; t) "g) = 0:
An interesting generalization of statistical convergence was introduced by Kostyrko et al [16] with the help of an admissible ideal I of subsets of N , the set of positive integers. Next we recall the basic terminology used by the authors to de…ne this new type of convergence.
De…nition 2.8. If X be a non-empty set then a family of sets I P (X)is called an ideal in X if and only if
(i) ; 2 I;
(ii) For each A; B 2 I we have A [ B 2 I ; (iii) For each A 2 I and B A we have B 2 I.
De…nition 2.9. Let X be a non-empty set. A non empty family of sets F P (X) is called a …lter on X if and only if
(i) ; =2 F ;
(ii) For each A; B 2 F we have A \ B 2 F ; (iii) For each A 2 F and B A we have B 2 F . An ideal I is called non-trivial if I 6= ; and X =2 I .
A non-trivial ideal I P (X) is called an admissible ideal in X if and only if it contains all singletons i.e., if it contains ffxg : x 2 Xg.
The following proposition express a relation between the notions of an ideal and a …lter.
Proposition 2.1. Let I P (X) be a non-trivial ideal. Then the class F = F (I) = fM N : M = X A, for some A 2 Ig is a …lter on X.
We shall call F = F (I) the …lter associated with the ideal I.
De…nition 2.10. An admissible ideal I P (N ) is said to be satisfy the condition (AP ) if for every countable family of mutually disjoint sets fA1; A2:::g belonging to I there exists a countable family fB1; B2:::g in I such that Ai Bi is a …nite set for each i 2 N and B = [1i=1Bi2 I.
For further study we shall denote by I, the admissible ideal of subsets of N unless otherwise stated.
3. I-convergence in an IFNS
In present section we introduce the notion of I-convergence in an intuitionistic fuzzy normed space and study some of its properties.
De…nition 3.1. Let I P (N ) be a nontrival ideal and (X; ; ; ; }) be an IFNS. A sequence x = (xn) of elements in X is said to I-convergent to 2 X with respect to the intuitionistic fuzzy norm ( ; ) if for each " > 0 and t > 0, the set
fn 2 N : (xn ; t) 1 " or (xn ; t) "g 2 I:
In this case the element is called the I-limit of the sequence (xn) with respect to the intuitionistic fuzzy norm ( ; ) and we write I( ; ) lim xn = .
Example 3.1. (i) If we take I = fA N : A is a …nite set g, then it is clear that I is an admissible ideal in N and the corresponding I-convergence coincides to the usual convergence with respect to the intuitionistic fuzzy norm ( ; ).
(ii) If we take I = fA N : (A) = 0g, then it is clear that I is an admis-sible ideal in N and the corresponding I-convergence coincides with statistical convergence in an IFNS.
With the help of the De…nition 3.1, we have the following theorem.
Theorem 3.1. Let (X; ; ; ; }) be an IFNS and x = (xn) be a sequence in X. Then for each " > 0 and t > 0, the following statements are equivalent:
(i) I( ; ) limn!1xn = ;
(ii) fn 2 N : (xn ; t) 1 "g 2 I and fn 2 N : (xn ; t) "g 2 I; (iii) fn 2 N : (xn ; t) > 1 " and (xn ; t) < "g 2 F (I);
(iv) fn 2 N : (xn ; t) > 1 "g 2 F (I)and fn 2 N : (xn ; t) < "g 2 F (I) and
(v) I limn!1 (xn ; t)= 1 and I limn!1 (xn ; t) =0.
Proof. It is not hard to prove the equivalence of (i), (ii), (iii) and (iv). So we only prove the equivalence of (ii) and (v). Suppose that (ii) holds. Since for every " > 0 and t > 0, we have
fn 2 N : (xn ; t) 1 "g
and for " > 0 the set fn 2 N : (xn ; t) 1 + "g = ; 2 I, it follows together with (ii) that fn 2 N : j (xn ; t) 1j "g 2 I: Hence we have I limn!1 (xn ; t)= 1.
Similarly, the fact that for every " > 0 and t > 0
fn 2 N : j (xn ; t) 0j "g = fn 2 N : (xn ; t) "g[
fn 2 N : (xn ; t) "g
and fn 2 N : (xn ; t) "g = ; 2 I, implies that I limn!1 (xn ; t) = 0. Also it is clear that (v) implies (ii).
Theorem 3.2. Let (X; ; ; ; }) be an IFNS and x = (xn) be a sequence in X. If (xn) is I-convergent with respect to the intuitionistic fuzzy norm ( ; ), then its I( ; )-limit is unique.
Proof. Suppose that there exist two distinct elements , 2 X such that I( ; ) limn!1xn= and I( ; ) limn!1xn= . Let " > 0 be given. Choose r > 0 such that (1) (1 r) (1 r) > 1 " and r}r < ": For t > 0, de…ne K1= fn 2 N : (xn ; t) 1 rg; K2= fn 2 N : (xn ; t) rg; K3= fn 2 N : (xn ; t) 1 rg; K4= fn 2 N : (xn ; t) rgand K = (K1[ K3) \ (K2[ K4):
Since I( ; ) limn!1xn= and I( ; ) limn!1xn= , therefore all the sets K1, K2, K3, K4 and K belongs to I. This implies that KC is a nonempty set in F (I). Let m 2 KC. Then we have m 2 K1C\ K3C or m 2 K2C\ K4C. Case (i) Suppose that m 2 KC
1 \ K3C. Then we have (xm ;2t) > 1 r, (xm ;2t) > 1 r and therefore ( ; t) (xm ; t 2) (xm ; t 2) > (1 r) (1 r) > 1 " (by(1)): Since " > 0 was selected arbitrary therefore we have ( ; t) = 1 for every t > 0. This implies that = 0. Hence we have = .
Case (ii) Suppose that m 2 K2C\ K4C. Then we have (xm ;t2) < r, (xm ;2t) < r and therefore ( ; t) < (xm ; t 2)} (xm ; t 2) < r } r < " (by(1)):
As " > 0 was selected arbitrary therefore we have ( ; t) = 0 for every t > 0. This implies that = ,
Hence in both the cases we have = . This shows that I( ; )-limit of the sequence (xn) is unique.
Theorem 3.3. Let (X; ; ; ; }) be an IFNS and x = (xn), y = (yn) be two sequences in X.
If ( ; ) limn!1xn= , then I( ; ) limn!1xn= ; If I( ; ) limn!1xn= and I( ; ) limn!1yn= , then
I( ; ) lim
n!1(xn+ yn) = ( + );
If I( ; ) limn!1xn= and be any real number, then I( ; ) limn!1 xn = .
Proof. (i) As ( ; ) limn!1xn = , so for each " > 0 and t > 0 there exists a positive integer m (say) such that (xn ; t) > 1 " and (xn ; t) < " for every n m. Since A = fn 2 N : (xn ; t) 1 " or (xn ; t) "g is contained in f1; 2; : : : m 1g and the ideal I is admissible so A 2 I. This shows that I( ; ) limn!1xn= .
(ii) Let " > 0 be given. Choose r > 0 such that
(1 r) (1 r) > 1 " and r}r < ": For t > 0, de…ne
K1= fn 2 N : (xn ; t) 1 rg; K2= fn 2 N : (xn ; t) rg;
K3= fn 2 N : (yn ; t) 1 rg; K4= fn 2 N : (yn ; t) rg and
K = (K1[ K3) [ (K2[ K4):
Since I( ; ) limn!1xn= and I( ; ) limn!1yn = , so for t > 0, all the sets K1, K2, K3, K4 and K belongs to I. It follows that KC is a non empty set in F (I). Next we shall show that
KC fn 2 N : ((xn+yn) ( + ); t) > 1 " and ((xn+yn) ( + ); t) < "g: For this let m 2 KC. Then we have,
(xm ; t 2) > 1 r; (ym ; t 2) > 1 r; (xm ; t 2) < r and (ym ; t 2) < r: Now we have, ((xm+ ym) ( + ); t) (xm ; t 2) (ym ; t 2) > (1 r) (1 r) > 1 " and ((xm+ ym) ( + ); t) (xm ; t 2)} (ym ; t 2) < r}r < ": This shows that
KC fn 2 N : ((xn+yn) ( + ); t) > 1 " and ((xn+yn) ( + ); t) < "g: Since KC 2 F (I), it follows that the later set belongs to F (I). HenceI
( ; ) limn!1(xn+ yn) = ( + ).
(iii) Case-(i) If = 0, then for each " > 0 and t > 0, (0xn 0 ; t) = (0; t) = 1 > 1 " and (0xn 0 ; t) = (0; t) = 0 < ". This implies that ( ; ) limn!10xn= and so by part (i), we have I( ; ) limn!10xn= . Case-(ii) Take 6= 0. As I( ; ) limn!1xn = so for each " > 0 and t > 0,
(2) A = fn 2 N : (xn ; t) > 1 " and (xn ; t) < "g 2 F (I):
Next to prove the result it is su¢ cient to prove that for each " > 0 and t > 0, A fn 2 N : ( xn ; t) > 1 " and ( xn ; t) < "g. For this let m 2 A. Then we have (xm ; t) > 1 " and (xm ; t) < ".
Now, ( xm ; t) = ((xm );j jt ) ((xm ); t) (0;j jt t) = ((xm ); t) 1 = ((xm ); t) > 1 " and ( xm ; t) = ((xm );j jt ) ((xm ); t) } (0;j jt t) = ((xm ); t) } 0 = ((xm ); t) < " :
Hence we have A fn 2 N : ( xn ; t) > 1 " and ( xn ; t) < "g. But then (2) shows that I( ; ) limn!1 xn = .
Before giving the next theorem we recall the following: Let (X; ; ; ; }) be an IFNS. The open ball B(x; "; t) with center x and radius 0 < " < 1 is given as
B(x; "; t) = fy 2 X : (x y; t) > 1 " and (x y; t) < "g where t > 0: A subset A of X is said to be IF -bounded if there exists t > 0 and 0 < " < 1 such that (x; t) > 1 " and (x; t) < " for each x 2 A.
Let `1
( ; )(X) denotes the space of all I F -bounded sequences whereas by I( ; )1 (X) we shall denote the space of all I F -bounded and I-convergent sequences in an intuitionistic fuzzy normed space (X; ; ; ; }). Now we have the following theorem.
Theorem 3.4. Let (X; ; ; ; }) be an IFNS. Then I( ; )1 (X) is a closed linear space of `1( ; )(X).
Proof. It is clear by Theorem 3.3 thatI1
( ; )(X) is a subspace of `1( ; )(X). Next we prove the closedness of I1
( ; )(X). As I( ; )1 (X) I( ; )1 (X) always (here bar denotes the closure of the respective set), so we prove that I1
( ; )(X) I( ; )1 (X). For this, let x 2 I1
( ; )(X). ThenB(x; r; t) \I( ; )1 (X) 6= ;, for each open ball B(x; r; t) centered at x. Let y 2 B(x; r; t) \ I1
( ; )(X). Let t > 0 and " 2 (0; 1). Choose r 2 (0; 1) such that
(1 r) (1 r) > 1 " and r}r < ": As y 2 B(x; r; t) \ I1
( ; )(X) so there exists a subset K of N such that K 2 F (I) and for alln 2 K we have (xn yn;2t) > 1 r, (xn yn;2t) < r, (yn;2t) > 1 r and (yn;t2) < r. But then for every n 2 K, we have
(xn; t) = (xn yn+ yn; t) (xn yn;2t) (yn;t2) > (1 r) (1 r) > 1 "
and
(xn; t) = (xn yn+ yn; t) (xn yn;2t)} (yn;2t) < r}r < ":
This implies that K fn 2 N : (xn; t) > 1 " and (xn; t) < "g. Since K 2 F (I), it follows that fn 2 N : (xn; t) > 1 " and (xn; t) < "g 2 F (I). Hence we have x 2 I1
( ; )(X). 4. I -convergence in an IFNS
Karakus et al [15] proved that a sequence x = (xn) in an IFNS (X; ; ; ; }) is statistically convergent to 2 X if and only if there exists an increasing index
sequence K = fk1; k2; : : : : k1< k2< : : :g of natural numbers such that (K) = 1 and ( ; ) limn!1xkn = . We use this result to introduce the concept of
I - convergence in an intuitionistic fuzzy normed space (X; ; ; ; }) as follows. De…nition 4.1. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) in X is said to be I -convergent to 2 X with respect to the intuitionistic fuzzy norm ( ; ) if there exists a subset K = fk1; k2; : : : : k1 < k2 < : : :g of N such that K 2 F (I) and ( ; ) limn!1xkn = . The element is called the I -limit
of the sequence (xn) with respect to the intuitionistic fuzzy norm ( ; ) and we write I( ; ) limn!1xn= .
Theorem 4.1Let (X; ; ; ; }) be an IFNS and x = (xn) be a sequence in X. If I( ; ) limn!1xn= , then I( ; ) limn!1xn = .
Proof. As I( ; ) limn!1xn = so there exists a subset K = fk1; k2; : : : : k1 < k2 < : : :g of N such that K 2 F (I) and ( ; ) limn!1xkn = . But
then for each " > 0 and t > 0 there exists a positive integer m (say) such that (xkn ; t) > 1 " and (xkn ; t) < " for every n m. Since the set fkn2
K : (xkn ; t) 1 " or (xkn ; t) "gis contained infk1; k2; :::km 1gand
the ideal I is admissible therefore the set fkn 2 K : (xkn ; t) 1 " or
(xkn ; t) "g 2 I. Also K 2 F (I), therefore by de…nition of F (I) there is
a set H 2 I such that K = N H. Hence fn 2 N : (xn ; t) 1 " or (xn ; t) "g H [ fk1; k2; :::km 1g. As the set on the right side belongs to I, therefore the set fn 2 N : (xn ; t) 1 " or (xn ; t) "g 2 I. Since this holds for each " > 0 and t > 0 therefore we have I( ; ) limn!1xn= . We next give example which shows that the converse of Theorem 4.1 is not true in general.
Example 4.1. Suppose that (R; j:j) denotes the space of real numbers with the usual norm, and let a b = ab, a}b = minfa + b; 1g for all a; b 2 [0; 1]. If for x 2 R and every t > 0, we consider
0(x; t) = t t + jxj and 0(x; t) = jxj t + jxj; then (R; 0; 0; ; }) is an IFNS.
Let N =S[1i = 1 be a decomposition of N such that each Niis an in…nite set and Ni\ Nj = ;for i 6= j. If we take I = fA N : A [pi=1Ni;for some …nite positive integer pg, then it is easy to see that Iis a non-trivial admissible ideal in N . Now we de…ne a sequence x = (xn)as follows. For n 2 Ni, we put xn = 1i (i = 1; 2; : : :). Now 0(xn; t) = t+jxtnj ! 1as n ! 1and 0(xn; t) =
jxnj
t+jxnj! 0as n ! 1. Hence by Theorem 3.1, I( 0; 0) limn!1xn = 0.
Next we shows that the I( ; ) limn!1xn= 0 does not holds. We suppose that I( ; ) limn!1xn = 0. By de…nition there exists a subset K = fk1; k2; : : : : k1< k2< : : :g of N such that K 2 F (I) and ( 0; 0) limn!1xkn= 0. Since
But then there is a positive integer p (say) such that H S[p i = 1. This implies that the set Np+1 Kand therefore xki =
1
p+1for in…nitely many k ;
ifrom K. This contradicts ( 0; 0) limn!1xkn= 0. Hence, I( ; ) limn!1xn= 0does not holds.
Theorem 4.2. Let (X; ; ; ; })be an IFNS and the ideal Isatisfy the condition (AP ). If x = (xn)be a sequence in Xsuch that I( ; ) limn!1xn = , then I( ; ) limn!1xn= .
Proof. Since I( ; ) limn!1xn = so for each " > 0and t > 0, the set fn 2 N : (xn ; t) 1 " or (xn ; t) "g 2 I: For k 2 Nand t > 0, we de…ne
Ak= fn 2 N : 1 1 k (xn ; t) < 1 1 k + 1 or t 1 k + 1 < (xn ; t) 1 kg: Now it is clear that fA1; A2; : : :g is a countable family of mutually disjoint sets belonging to I and therefore by the condition (AP ) there is a countable family of sets fB1; B2; : : :g in I such that Ai Bi is a …nite set for each i 2 N and B =S[1i = 1. Since B 2 Iso by de…nition of associate …lter F (I)there is set Kin F (I)such that K = N B. Now to prove the result it is su¢ cient to prove that the subsequence (xn)n2Kis ordinary convergent to with respect to the intuitionistic fuzzy norm ( ; ).
For this, let > 0and t > 0. Choose a positive integer qsuch that 1q < . Then we have fn 2 N : (xn ; t) 1 or (xn ; t) g fn 2 N : (xn ; t) 1 1 q or (xn ; t) 1 qg q+1S i=1 Ai
Since Ai Bi be a …nite set for each i = = 1 ,2, 3,...q+1, therefore there exist a positive integer n0 such that
([q+1i=1Bi) \ fn 2 N : n n0g = ([q+1i=1Ai) \ fn 2 N : n n0g:
If n > n0 and n 2 K, then n =2 B. This implies that n =2 [q+1i=1Bi and therefore n =2 [q+1i=1Ai. Hence for every n n0, n 2 K, we have (xn ; t)1 and (xn ; t) < . As this holds for every > 0 and t > 0, so I( ; ) limn!1xn = .
Theorem 4.3. Let (X; ; ; ; }) be an IFNS. Then for any sequence x = (xn) in X the following conditions are equivalent.
(i) I( ; ) limn!1xn = .
(ii) There exists two sequences y = (yn) and z = (zn) in X such that x = y + z, ( ; ) limn!1yn = and the set fn 2 N : zn 6= g 2 I where denotes the zero elements of X.
Proof. Suppose that condition (i) holds. Then there exists a subset K = fk1; k2; : : : : k1< k2< : : :g of N such that
(3) K 2 F (I) and t( ; ) lim
n!1xkn= : We de…ne the sequences y = (yn) and z = (zn) as follow:
yn =
xn; if n 2 K if n 2 KC
and (zn) = (xn) (yn) for n 2 N. Now for n 2 KC, for each " > 0 and t > 0, we have (yn ; t) = 1 > 1 " and (yn ; t) = 0 < ". This shows together with (3) that ( ; ) limn!1yn= . Since fn 2 N : zn6= g KC, it follows that fn 2 N : zn6= g 2 I.
Assume that (ii) holds. Let K = fn 2 N : zn = g, clearly K 2 F (I) and therefore is an in…nite set (as otherwise it belongs to I). Let K = fk1; k2; : : : : k1 < k2 < : : :g . Since xkn = ykn and ( ; ) limn!1yn = , therefore
( ; ) limn!1xkn= . This shows that I( ; ) limn!1xn= . Theorem 4.4. Let (X; ; ; ; }) be an IFNS. Then I( ; );1(X) = I1
( ; )(X), where I( ; );1(X) denotes the space of all bounded and I -convergent sequences in an IFNS and bar the closure of the respective set.
Proof. Since I( ; );1(X) I1
( ; )(X) and I( ; )1 (X) is a closed subspace of `1( ; )(X) by Theorem 3.4, so we have I( ; );1(X) I1
( ; )(X). We next prove that, I( ; )1 (X) I( ; );1(X). To prove this it su¢ ces to prove that for each y 2 I1
( ; )(X) and 0 < " < 1 we have I( ; );1\ B(y; "; t) 6= ;. As y 2 I1
( ; )(X) so, let I( ; ) limn!1yn = . Choose r 2 (0; ") arbitrary. Since I( ; ) limn!1yn = so the set
A = fn 2 N : (yn ; t) 1 r or (yn ; t) rg 2 I and
AC= fn 2 N : (yn ; t) > 1 r and (yn ; t) < rg 2 F (I): Now de…ne a sequence x = (xn) as follows.
xn=
; if n 2 AC yn if n 2 A:
Then it is clear that x 2 `1( ; )(X), I( ; ) limn!1xn = and x 2 B(y; "; t). Hence we have, I( ; );1 \ B(y; "; t) 6= ;.
5. I and I -Cauchy Sequences in an IFNS
In this section we introduce and study the concepts of I-Cauchy and I -Cauchy sequences in an IFNS (X; ; ; ; }).
De…nition 5.1. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) of elements in X is said to I-Cauchy with respect to the intuitionistic fuzzy norm ( ; ) or I( ; )-Cauchy sequence if for each " > 0 and t > 0, there exists a positive integer m (say) such that the set fn 2 N : (xn xm; t) 1 " or (xn xm; t) "g 2 I:
De…nition 5.2. Let (X; ; ; ; }) be an IFNS. A sequence x = (xn) in X is said to be I -Cauchy with respect to the intuitionistic fuzzy norm ( ; ) or I( ; )-Cauchy sequence if there exists a subset K = fk1; k2; : : : : k1 < k2< : : :g of N such that K 2 F (I) and the subsequence (xkn) is an ordinary Cauchy
sequence with respect to the intuitionistic fuzzy norm ( ; ).
Analogous to the previous section one easily can have the following theorems. Theorem 5.1. Let (X; ; ; ; }) be an IFNS and x = (xn) be a sequence in X. If (xn) is I( ; )-Cauchy sequence then it is also I( ; )-Cauchy.
Theorem 5.2. Let (X; ; ; ; }) be an IFNS and the ideal I satisfy the con-dition (AP ). If x = (xn) is any I( ; )-Cauchy sequence in X then it is also I( ; )-Cauchy.
Next Theorem shows that in an intutionistic fuzzy normed space (X; ; ; ; }), I( ; )-convergent sequences are I( ; )-Cauchy.
Theorem 5.3. Let (X; ; ; ; }) be an IFNS. If x = (xn) is any I( ; ) -convergent sequence in X, then (xn) is I( ; )-Cauchy.
Proof.Assume that there exists 2 X such that I( ; ) limn!1xn = . Let " > 0 be given. Choose r > 0 such that
(1 r) (1 r) > 1 " and r}r < ":
Since I( ; ) limn!1xn= , it follows that for all t > 0, we have
(4) A = fn 2 N : (xn ; t) 1 r or (xn ; t) rg 2 I: This implies that ; 6= AC = fn 2 N : (x
n ; t) > 1 r and (xn ; t) < rg 2 F (I).Let m 2 AC. But then for t > 0 we have, (x
m ; t) > 1 r and (xm ; t) < r.
If we take B = fn 2 N : (xn xm; t) 1 " or (xn xm; t) "g; t 0, then to prove the result it is su¢ cient to prove B is contained in A.
Let n 2 B, then we have (xn xm; t) 1 " or (xn xm; t) " for all t 0. Case (i) If (xn xm; t) 1 ", then we have (xn ;2t) 1 r and therefore n 2 A.
As otherwise i.e., if (xn ;t2) > 1 r, then we have 1 " (xn xm; t) (xn ; t 2) (xm ; t 2) > (1 r) (1 r) > 1 ";
which is not possible. Hence B A.
Case (ii) If (xn xm; t) "; then we have (xn ;2t) r and therefore n 2 A. As otherwise i.e., if (xn ;2t) < r, then we have
" (xn xm; t) (xn ; t 2)} (xm ; t 2) < r}r < ";
which is not possible. Hence B A.
Thus in all the cases we B A. By (4), B 2 I. This shows that (xn) is I( ; )-Cauchy.
Theorem 5.4Let (X; ; ; ; }) be an IFNS and x = (xn) be any sequence in X. If (xn) is I( ; )-convergent then it is I( ; )-Cauchy.
6. Acknowledgment
The authors are grateful to the referees and the editor in chief for their cor-rections and suggestions, which have greatly improved the readability of the paper. We also wish to thank Prof. N. Singh, formerly in the department of mathematics,Kurukshetra University, Kurukshetra, for his valuable suggestions and guidelines that improved the presentation of the paper.
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