On rough I^* and I^K-convergence of sequences in normed linear spaces
aa r X i v : . [ m a t h . GN ] F e b ON ROUGH I ∗ AND I K -CONVERGENCE OF SEQUENCES IN NORMED LINEARSPACES AMAR KUMAR BANERJEE AND ANIRBAN PAULA
BSTRACT . In this paper, we have introduced first the notion of rough I ∗ -convergence in a normedlinear space as an extension work of rough I -convergence and then rough I K -convergence in moregeneral way. Then we have studied some properties on these two newly introduced ideas. We alsoexamined the relationship between rough I -convergence with both of rough I ∗ -convergence and rough I K -convergence. Key words and phrases:
Rough convergence, Rough I -convergence, Rough I ∗ -convergence,Rough I K -convergence, (AP) condition. (2010) AMS subject classification : NTRODUCTION
The concepts of convergence of a sequence of real numbers has been generalised to statisticalconvergence independently by Fast [7] and Steinhaus [20]. Then a lot of developments were carriedout in this area by many authors. The concepts of statistical convergence of sequence has been ex-tended to I -convergence by Kostyrko et al. [10, 12] using the structure of the ideal I of subsets ofthe set of natural numbers. An another type of convergence which is closely related to the ideas of I -convergence is the idea of I ∗ -convergence given by Kostyrko et al. [9]. It is seen in [10] that thesenotions are equivalent if and only if the ideal satisfies the property (AP). Several works have beendone in recent years on I -convergence (see [2, 3, 4, 5, 14, 15]).The idea of rough convergence in a finite dimensional space was introduced by Phu [18] in 2001.In 2014, D ¨undar et al. [6] introduced the notion rough I -convergence using the concepts of I -convergence and rough convergence. For given two arbitrary ideals I and K on a set S , the idea of I K -convergence in topological space was given by M. Maˇcaj and M. Sleziak in [16] as a generaliza-tion of the notion of I ∗ -convergence. In their paper they modified the condition (AP) and have showedthat when such condition termed as AP( I , K ) holds then I -convergence implies I K -convergence andthe converse of this result also holds for the first countable space which is not finitely generated( orAlexandroff space). Indeed, they worked with functions instead of sequences. One of the reasons isthat using functions sometimes helps to simplify notations.In our work we have introduced the notion of rough I ∗ -convergence and rough I K -convergence in anormed linear space. Rough I K -convergence is a common generalization of rough I ∗ -convergence.Here we have studied the ideas of rough I ∗ -convergence and rough I K -convergence in terms ofsequences instead of functions. We then intend to find the relation between rough I -convergenceand rough I ∗ -convergence. We also tried to find the relation of rough I -convergence with rough I K -convergence and we have observed that the condition AP( I , K ) is an necessary and sufficient con-dition for the rough I -limit set to be a subset of rough I K -limit set. We have tried to verify whethersome property of I K -convergence as in [16] also holds for rough I K -convergence.We now recall some definitions and notions which will be needed in sequel.
2. P
RELIMINARIES
Throughout the paper, N denotes the set of all natural numbers, R the set of all real numbers unlessotherwise stated. Definition 2.1. [7] Let K be a subset of the set of natural numbers N and let us denote the set K i = { k ∈ K : k ≤ i } . Then the natural density of K is given by d ( K ) = lim i →∞ | K i | i , where | K i | denotes the number of elements in K i . Definition 2.2. [7] A sequence { x n } n ∈ N of real numbers is said to be statistically convergent to x iffor any ε > , d ( A ( ε )) = 0 , where A ( ε ) = { n ∈ N : | x n − x | ≥ ε } .Let I be a collection of subset of a set S . Then I is called an ideal on S if ( i ) A, B ∈ I ⇒ A ∪ B ∈ I and ( ii ) A ∈ I and B ⊂ A ⇒ B ∈ I [11].An ideal I on S is called admissible if it contains all singletons, that is, { s } ∈ I for each s ∈ S . I iscalled nontrivial if S / ∈ I [11]. From the definition it is noted that φ ∈ I .If S = N , the set of all positive integers then I is called an ideal on N . We will denote by Fin theideal of all finite subsets of a given set S . Definition 2.3. [13] Let S = φ . A non empty class F is called a filer in S provided: ( i ) φ / ∈ F , ( ii ) A, B ∈ F ⇒ A ∩ B ∈ F , ( iii ) A ∈ F, A ⊂ B ⇒ B ∈ F . Lemma 2.1. [10] If I is a non trivial ideal on N , then the class F ( I ) = { M ⊂ N : there exists A ∈ I such that M = N \ A } is a filter on N , called the filter associated with I . Definition 2.4. [10] An admissible ideal I ⊂ N is said to satisfy the condition (AP) if for everycountable family of mutual disjoint sets { A , A , · · · } belonging to I there exists a countable familyof sets { B , B , · · · } such that the symmetric difference A j ∆ B j is a finite set for each j ∈ N and B = ∞ [ j =1 B j ∈ I . Several example of countable family satisfying (AP) are seen in [10]. Definition 2.5. [10, 12] Let ( X, || · || ) be a normed linear space and I ⊂ N be a non-trivial ideal.A sequence { x n } n ∈ N of elements of X is said to be I -convergent to x ∈ X if for each ε > theset A ( ε ) = { n ∈ N : || x n − x || ≥ ε } belongs to I . The element x is here called the I -limit of thesequence { x n } n ∈ N .It should be noted here that if I is an admissible ideal then usual convergence in X implies I -convergence in X . Example 2.1. [10] If I d denotes the class of all A ⊂ N with d ( A ) = 0 . Then I d is non trivialadmissible ideal and I d convergence coincides with the statistical convergence. Definition 2.6. [9, 10] Let ( X, || · || ) be a normed linear space and I ⊂ N be a non-trivial ideal. Asequence { x n } n ∈ N in X is said to be I ∗ -convergent to x if there exists a set M = { m < m < · · ·
A sequence { x n } n ∈ N in a normed linear space X is bounded if and only ifthere exists an r ≥ such that LIM r x n = φ . For all r > , a bounded sequence { x n } n ∈ N alwayscontains a subsequence { x m k } k ∈ N with LIM r x m k = φ . Proposition 2.3. [18] If { x ′ n } n ∈ N is a sub sequence of { x n } n ∈ N in a normed linear space ( X, || · || ) ,then LIM r x n ⊂ LIM r x ′ n . Proposition 2.4. [18]
For all r ≥ , the r -limit set LIM r x n of an arbitrary sequence { x n } n ∈ N in anormed linear space ( X, || · || ) is closed set. Proposition 2.5. [18]
For an arbitrary sequence { x n } n ∈ N in a normed linear space ( X, || · || ) the r -limit set LIM r x n is convex. Definition 2.8. [6] A sequence { x n } n ∈ N in a normed linear space ( X, || · || ) is said to be I -boundedif there exists a positive real number M such that the set { n ∈ N : || x n || ≥ M } ∈ I Definition 2.9. [6] Let I be an admissible ideal and r be a non-negative real number. A sequence { x n } n ∈ N in a normed linear space ( X, || · || ) is said to be rough I -convergent of roughness degree r to x , denoted by x n r − I −−→ x provided that { n ∈ N : || x n − x || ≥ r + ε } ∈ I for every ε > and x iscalled rough I -limit of { x n } n ∈ N of roughness degree r . Remark 1. If I is an admissible ideal, then the usual rough convergence implies rough I -convergence. Note 1.
If we take r = 0 , then we obtain the definition of ordinary I -convergence. In general, therough I -limit of a sequence may not be unique for the roughness degree r > . So we have to considerthe so-called rough I -limit set of a sequence { x n } n ∈ N which is defined by I − LIM r x n := { x ∈ X : x n r − I −−→ x } . A sequence { x n } n ∈ N is said to be rough I -convergent if I − LIM r x n = φ . Theorem 2.6. [6]
Let I ⊂ N be an admissible ideal. A sequence { x n } n ∈ N in ( X, || · || ) is I -boundedif and only if there exists a non negative real number r such that I − LIM r x n = φ . Theorem 2.7. [6]
Let I ⊂ N be an admissible ideal. If { x m k } k ∈ N is a sub sequence of { x n } n ∈ N in ( X, || · || ) then I − LIM r x n ⊂ I − LIM r x m k Theorem 2.8. [6]
Let I ⊂ N be an admissible ideal. The rough I -limit set of a sequence { x n } n ∈ N in ( X, || · || ) is closed. Theorem 2.9. [6]
Let I ⊂ N be an admissible ideal. The rough I -limit set of a sequence { x n } n ∈ N in ( X, || · || ) is convex. We now give some basic ideas on I K -convergence in a topological space studied by M. Maˇcaj andM. Sleziak [16]. Definition 2.10. [16] Let I be an ideal on a set S and X be a topological space. A function f : S X is said to be I -convergent to x if f − ( U ) = { s ∈ S : f ( s ) ∈ U } ∈ F ( I ) holds for everyneighbourhood U of the point x and we write I − limf = x If S = N , then we obtain the usual definition of I -convergence of sequences. In this case thenotation I − limx n = x is used. AMAR KUMAR BANERJEE AND ANIRBAN PAUL
Definition 2.11. [16] Let I be an ideal on a set S and let f : S X be a function to a topologicalspace X . The function f is called I ∗ -convergent to the point x if there exists a set M ∈ F ( I ) suchthat the function g : S X defined by g ( s ) = ( f ( s ) , if s ∈ Mx, if s / ∈ M is Fin-convergent to x . If f is I ∗ -convergent to x , then we write I ∗ − limf = x .The usual notion of I ∗ -convergence of sequences is a special case when S = N . Definition 2.12. [16] Let K and I be ideals on a set S and X be a topological space. The function f : S X is said to be I K -convergent to x ∈ X if there exists a set M ∈ F ( I ) such that the function g : S X given by g ( s ) = ( f ( s ) , if s ∈ Mx, if s / ∈ M is K -convergent to x . If f is I K -convergent to x ,then we write I K − limf = x .When S = N , then we speak about I K -convergence of sequences. Remark 2.
The definition of I K -convergence may also be treated from [9] as follows: there exists M ∈ F ( I ) such that the function f | M is K | M -convergent to x , where K | M = { A ∩ M : A ∈ K } isthe trace of K on M . The two definitions are equivalent but the definition given in 2.12 is somewhatsimpler [16]. Lemma 2.10. [16] If I and K are ideals on a set S and f : S X is a function such that K − limf = x then I K − limf = x . Definition 2.13. [16] Let K be an ideal on a set S . we write A ⊂ K B whenever A \ B ∈ K . if A ⊂ K B and B ⊂ K A then we write A ∼ K B .Clearly A ∼ K B ⇔ A ∆ B ∈ K .Now we recall a lemma from [16] where several equivalent formulations of a condition for ideals I and K have been described. Lemma 2.11. [16]
Let I and K be ideals on the same set S . Then the following conditions areequivalent: ( i ) For every sequence { A n } n ∈ N of sets from I there is A ∈ I such that A n ∼ K A for all n ’s. ( ii ) Any sequence { F n } n ∈ N of sets from F ( I ) has a K -pseudo intersection in F ( I ) . ( iii ) For every sequence { A n } n ∈ N of sets belonging to I there exists a sequence { B n } n ∈ N of sets from I such that A j ∼ K B j for j ∈ N and B = S j ∈ N B j ∈ I . ( iv ) For every sequence of mutually disjoint sets { A n } n ∈ N belonging to I there exists a sequence { B n } n ∈ N of sets belonging to I such that A j ∼ K B j for j ∈ N and B = S j ∈ N B j ∈ I . ( v ) For every non-decreasing sequence A ⊂ A ⊂ · · · ⊂ A n ⊂ · · · of sets from I there exists asequence { B n } n ∈ N of sets belonging to I such that A j ∼ K B j for j ∈ N and B = S j ∈ N B j ∈ I . Definition 2.14. [16] Let I and K be two ideals on a same set S . We say that I has the additiveproperty with respect to K , more briefly that AP ( I, K ) holds, if any one of the equivalent conditionsof Lemma 2.11 holds.The condition (AP) from [10], is equivalent to the condition AP( I , Fin). Now we recall the follow-ing two theorems from [16]. Theorem 2.12. [16]
Let I and K be ideals on a set S and X be a first countable topological space.If I has the additive property with respect to K , then for any function f : S X I convergenceimplies the I K -convergence or in other words, if the condition AP ( I, K ) holds then the I -convergenceimplies the I K -convergence. N ROUGH I ∗ AND I K -CONVERGENCE OF SEQUENCES IN NORMED LINEAR SPACES 5 Let us recall that a topological space X is called finitely generated space or Alexandroff space ifintersection of any number of open sets of X is again an open set (see [1]). Equivalently, X is finitelygenerated if and only if each point x has a smallest neighbourhood. Theorem 2.13. [16]
Let I , K be ideals on a set S and X be a first countable topological spacewhich is not finitely generated. If the I -convergence implies the I K -convergence for any function f : S X , then the ideal I has the additive property with respect to K or briefly the conditionAP ( I, K ) holds.
3. M
AIN RESULTS
Through out our discussion ( X, || · || ) or simply X will always denote a normed linear space overthe field C or R and I , K always assumed to be non trivial admissible ideals on N unless otherwisestated. Definition 3.1.
Let r be a non-negative real number and I be a non trivial admissible ideal on N .Then a sequence { x n } n ∈ N in ( X, || · || ) is said to be rough I ∗ -convergent of roughness degree r to x if there exists a set M = { m < m < m < · · · < m k < · · · } in F ( I ) such that the sub sequence { x m k } k ∈ N is rough convergent of roughness degree r to x . Thus for any ε > there exists a N ∈ N such that || x m k − x || < r + ε for all k ≥ N . we denote this by x n r − I ∗ −−−→ x .Here x is called the rough I ∗ -limit of the sequence { x n } n ∈ N of roughness degree r . For r = 0 we have the definition of I ∗ -convergence of sequences in normed linear spaces. Obviously rough I ∗ -limit of a sequence in normed linear spaces is not unique. Therefore we have to consider the rough I ∗ -limit set of the sequence { x n } n ∈ N defined as follows: I ∗ − LIM r x n = { x ∈ X : x n r − I ∗ −−−→ x } . Definition 3.2.
Let r be a non-negative real number. Also let I and K be two non trivial admissibleideals on N . Then a sequence { x n } n ∈ N in a normed linear space ( X, || · || ) is said to be rough I K -convergent of roughness degree r to x if there exists a set M = { m < m < · · · < m k < · · · } in F ( I ) such that the sub sequence { x m k } k ∈ N is rough K | M -convergent of roughness degree r to x , where K | M = { A ∩ M : A ∈ K } is the trace of K on M . That is for any ε > , the set { k ∈ N : || x m k − x || ≥ r + ε } ∈ K | M . we denote this by x n r − I K −−−→ x .Here x is called the rough I K -limit of the sequence { x n } n ∈ N of roughness degree r . For r = 0 wehave the definition of I K -convergent of sequences in normed linear spaces. It should be noted thatfor M ∈ F ( I ) , the trace K | M = { A ∩ M : A ∈ K } of K on M also forms an ideal on N . Clearlyrough I K -limit of a sequence in normed linear spaces is not unique. Therefore we will consider therough I K -limit set of the sequence { x n } n ∈ N defined by I K − LIM r x n = { x ∈ X : x n r − I K −−−→ x } .If the ideal K is such that it is the class of all finite subsets of N then definition 3.1 and definition 3.2coincides. Obviously if x is a rough I ∗ -limit of a sequence { x n } n ∈ N then x is also a rough I K -limit of { x n } n ∈ N . But it may happen that x is rough I K -limit of a sequence { x n } n ∈ N in normed linear spacewithout being rough I ∗ -limit of the the sequence { x n } , which is seen from the next example. So, ingeneral, for a sequence { x n } n ∈ N in a normed linear space and for any non-negative real number r ,we have I ∗ − LIM r x n ⊂ I K − LIM r x n . Example 3.1.
Let us consider a decomposition of N by N = A ∪ ∞ [ i =1 A i , where A = { , , , · · · } and A i = { n (2 i −
1) : n ∈ N } . Then each of A i ’s are disjoint from each other and each of A i ’sare disjoint from A also. Let I be the collections of all those subsets of N such that the sets whichbelongs to I can intersects with A and with only a finite numbers of A i ’s. Then I is an non trivial AMAR KUMAR BANERJEE AND ANIRBAN PAUL admissible ideal on N . Let N = ∞ [ j =1 D j be another decomposition of N such that D j = { j − (2 s −
1) : s = 1 , , · · · } . Then each of D j is infinite and D j ∩ D k = φ for j = k . Let K be the ideal of allthose subsets of N which intersects with only a finite numbers of D j ’s. Then K is a non trivialadmissible ideal on N . Let us consider the sequence in real number space with usual norm define by x n = j if n ∈ D j . Let us take M = N ∈ F ( I ) . Then K | M = K . Now let r > be arbitrary.Since by Archimedean property for any arbitrary ε > there exists a l ∈ N such that ε > l . So { k ∈ N : | x k − ( − r ) | = | x k + r | ≥ r + ε } ⊂ D ∪ D ∪ · · · ∪ D l ∈ K = K | M . Therefore − r ∈ I K − LIM r x n .If possible let − r ∈ I ∗ − LIM r x n . So there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) for which the sub sequence { x n } n ∈ M of the sequence { x n } n ∈ N is rough convergent to x of roughnessdegree r . Now as M ∈ F ( I ) therefore we have N \ M = H (say) ∈ I . Therefore exists a p ∈ N suchthat H ⊂ A ∪ A ∪ A ∪ · · · ∪ A p and so A k ⊂ M for all k ≥ p + 1 . Now as each of the set A k ’scontains an element from each of the set D i ’s for i ≥ , so there exists a s ∈ N such that x m k = s for infinitely k ’s when m k ∈ D s . As − r ∈ I ∗ − LIM r x n , so for ε = s +1 there exists a N ∈ N such that | x m k − ( − r ) | = | x m k + r | < r + ε for all k ≥ N → ( i ) . Since x m k = s for infinitelymany k ’s, therefore the condition in ( i ) does not holds. Thus we arrived at a contradiction. Hence − r / ∈ I ∗ − LIM r x n . Theorem 3.1.
Let { x n } n ∈ N be a sequence in a normed linear space ( X, || · || ) and r be a non-negative real number. Then for an non trivial admissible ideal I if { x n } n ∈ N is rough I ∗ -convergentof roughness degree r to x then it is also rough I -convergent of roughness degree r to x .Proof. If possible let { x n } n ∈ N be rough I ∗ -convergent of roughness r to x . Therefore there existsa set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such that { x m k } k ∈ N is rough convergent ofroughness degree r to x . Thus for any ε > there exists N ∈ N such that || x m k − x || < r + ε forall k ≥ N . So { k ∈ N : || x k − x || ≥ r + ε } ⊂ N \ M ∪ { m , m , · · · , m N − } → ( i ) . Now sincethe right hand side of ( i ) belongs to I , so { k ∈ N : || x k − x || ≥ r + ε } ∈ I . Therefore the sequence { x n } n ∈ N is rough I -convergent of roughness degree r to x . (cid:3) In view of Theorem 3.1, it follows that rough I ∗ -limit set of roughness degree r is a subset ofrough I -limit set of same roughness degree r . Converse of the theorem 3.1 not necessarily true. Thatis if a sequence { x n } n ∈ N is rough I -convergent of some roughness degree r to x then the sequence { x n } n ∈ N may not be rough I ∗ -convergent of same roughness degree r to x . This fact can be seenfrom the next example. Example 3.2.
Let N = ∞ [ j =1 D j be a decomposition of N such that D j = { ( j − (2 s −
1) : s =1 , , · · · } . Then each of D j is infinite and disjoint from each others. Let I be the class of all thosesubsets of N which intersects with only a finite numbers of D j ’s. Then I is an admissible ideal on N .Let us define a sequence in real numbers space with usual norm by x n = j j if n ∈ D j . Let r be anarbitrary non-negative real number. Let ε > be arbitrarily chosen, then there exists a l ∈ N suchthat ε > l l . Then [ − r, r ] ⊂ I − LIM r x n , as { n ∈ N : | x n − x | ≥ r + ε } ⊂ D ∪ D ∪ · · · D l ∈ I for any x ∈ [ − r, r ] .If possible suppose that the sequence defined above is rough I ∗ -convergent to − r of same roughnessdegree r . Therefore, there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such thatthe sub sequence { x m k } k ∈ N is rough convergent to − r of roughness degree r . Now as M ∈ F ( I ) ,so N \ M = H ( say ) ∈ I . Hence there exists a p ∈ N such that H ⊂ D ∪ D ∪ · · · ∪ D p and so N ROUGH I ∗ AND I K -CONVERGENCE OF SEQUENCES IN NORMED LINEAR SPACES 7 D p +1 ⊂ M . Therefore x m k = p +1) p +1 for m k ∈ D p +1 . Now for ε = p +2) p +1 and m k ∈ D p +1 we see that | x m k + r | ≥ r + ε for infinitely many k ’s. Therefore the sequence { x n } n ∈ N is not rough I ∗ -convergent of roughness degree r to − r although − r ∈ I − LIM r x n .Let r be a non-negative real number. Then for a sequence { x n } n ∈ N in a normed linear space rough I -limit of the sequence { x n } n ∈ N of roughness degree r is also a rough I ∗ -limit of same roughnessdegree r if the ideal I satisfies the condition (AP). To prove this we need the following lemma. Lemma 3.2. [17]
Let { A n } n ∈ N be a countable family of subsets of N such that each A n belongs to F ( I ) , the filter associated with an admissible ideal I which has the property (AP). Then there existsa set B ⊂ N such that B ∈ F ( I ) and the set B \ A n is finite for all n ∈ N . Theorem 3.3.
Let I be an ideal which has the property (AP) and { x n } n ∈ N be a sequence in a normedlinear space ( X, ||·|| ) . Then if x is a rough I -limit of the sequence { x n } n ∈ N of some roughness degree r then x is also a rough I ∗ -limit of the sequence { x n } n ∈ N of same roughness degree r .Proof. Let I be an ideal on N which satisfies the condition (AP) and { x n } n ∈ N be a sequence in anormed linear space ( X, || · || ) . Also let us suppose that x be a rough I -limit of the sequence { x n } n ∈ N of roughness degree r for some r ≥ . Therefore for any ε > the set { n ∈ N : || x n − x || ≥ r + ε } ∈ I . Let l be any arbitrary positive real number, so li is also positive real number for each i ∈ N . Define A i = { n ∈ N : || x n − x || < r + li } for each i ∈ N . Then A i ∈ F ( I ) for each i ∈ N . Also bythe lemma 3.2 there exists a set B ⊂ N such that B ∈ F ( I ) and B \ A i is finite for all i ∈ N . Nowfor any arbitrary ε > there exists a j ∈ N such that ε > lj . Since B \ A j is finite, so there exists k = k ( j ) ∈ N such that n ∈ B ∩ A j for all n ∈ B with n ≥ k . Now || x n − x || < r + lj < r + ε forall n ∈ B and n ≥ k . Thus the sub sequence { x n } n ∈ B is rough convergent of roughness degree r to x . Therefore x is also a rough I ∗ -limit of roughness degree r . Hence the result follows. (cid:3) Corollary 1.
Let { x n } n ∈ N be a sequence in a normed linear space ( X, || · || ) and r be a non-negativereal number. Let I be an ideal on N such that it satisfies the condition (AP). Then both the rough I -limit set of roughness degree r and rough I ∗ -limit set of roughness degree r of a sequence { x n } n ∈ N are equal.Proof. In view of theorem 3.1 and theorem 3.3 the result follows. (cid:3)
Rough I -limit set of a sequence { x n } n ∈ N in a normed linear space is a subset of rough I -limit setof a sub sequence { x n k } k ∈ N . But rough I ∗ -limit set of a sequence { x n } n ∈ N in a normed linear spacemay not be a subset of rough I ∗ -limit set of a sub sequence { x n k } k ∈ N . This fact can be justified bythe following example. Example 3.3.
Let I be the ideal of all subsets of N whose natural density is zero. Let us considera sequence { x n } n ∈ N in real number space with usual norm as follows: x n = ( − n = k n n = k , where k ∈ N . Now as the natural density of the set A = { n ∈ N : n = k , k ∈ N } is zero, therefore A ∈ I .So N \ A = M ( say ) ∈ F ( I ) . Put M = { m < m < · · · < m k < · · · } . Then for any arbitrary ε > we can see that | x m k − | < ε holds for all k ∈ N . Hence is a rough I ∗ -limit of roughnessdegree r = 1 of the sequence { x n } n ∈ N . Let A be enumerated as A = { n < n < · · · < n k < · · · } and consider the sub sequence { x n k } k ∈ N of { x n } n ∈ N . Since for any sub sequence { x n km } m ∈ N of { x n k } k ∈ N , we can see that for < ε < we have | x n km − | > ε for all m . Hence forthis choice of ε , there does not exists any N ( ε ) ∈ N for which | x n km − | < ε holds for all m ≥ N ( ε ) . Therefore there does not exists any M ′ = { m ′ < m ′ < · · · < m ′ k < · · · } ∈ F ( I ) forwhich { x n m ′ k } k ∈ N is rough convergent to of roughness degree r = 1 . So is not a rough I ∗ -limitof roughness degree r = 1 of the sub sequence considered above. AMAR KUMAR BANERJEE AND ANIRBAN PAUL
Theorem 3.4. If I is an ideal which satisfies the condition (AP), then the rough I ∗ -limit set of asequence { x n } n ∈ N of some roughness degree r is a subset of rough I ∗ -limit set of a sub sequence { x n k } k ∈ N of same roughness degree r .Proof. Let x be a rough I ∗ -limit of a sequence { x n } n ∈ N of some roughness degree r . As a rough I ∗ -limit is also a rough I -limit of { x n } n ∈ N , so x is a rough I -limit of { x n } n ∈ N . Since rough I -limitof a sequence { x n } n ∈ N is a subset of rough I -limit of a sub sequence { x n k } k ∈ N , hence x is rough I -limit of the sub sequence { x n k } k ∈ N . Now as I satisfies the condition (AP), so x is also a rough I ∗ -limit of the sub sequence { x n k } k ∈ N . (cid:3) Theorem 3.5.
Let I and K be two admissible ideals on N and r be a non-negative real number.Suppose that a sequence { x n } n ∈ N in a normed linear space ( X, || · || ) is rough I K -convergent to x of roughness degree r then { x n } n ∈ N is also rough I -convergent to x of same roughness degree r if K ⊂ I .Proof. Let I and K be two admissible ideals on N such that K ⊂ I . Also let r be a non-negativereal number. Suppose that a sequence { x n } n ∈ N is rough I K -convergent to x of roughness degree r .Then there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such that for any ε > theset A ( ε ) = { k ∈ N : || x m k − x || ≥ r + ε } ∈ K | M . Suppose that A ( ε ) = { k ∈ N : || x m k − x || ≥ r + ε } = K ∩ M for some K ∈ K . Now as K is an ideal and K ∩ M ⊂ K , so K ∩ M ∈ K .Again { n ∈ N : || x n − x || ≥ r + ε } ⊂ ( K ∩ M ) ∪ N \ M . Since N \ M ∈ I and K ⊂ I , therefore ( K ∩ M ) ∪ N \ M ∈ I . So { n ∈ N : || x n − x || ≥ r + ε } ∈ I . Hence the result follows. (cid:3) Converse part of the theorem 3.5 is also valid, i.e., if a rough I K -limit x of a sequence { x n } n ∈ N of some roughness degree r implies that x is also a rough I -limit of same roughness degree r then K ⊂ I . To prove this we need the following lemma. Lemma 3.6. If I and K are ideals on N . Then a rough K -limit of a sequence { x n } n ∈ N of someroughness degree r is also a rough I K -limit of { x n } n ∈ N of same roughness degree r .Proof. Let I and K be two ideals on N and r be a non-negative real number. Let x be a rough K -limitof { x n } n ∈ N of roughness degree r i.e., x ∈ K − LIM r x n . Then for any ε > , { n ∈ N : || x n − x || ≥ r + ε } ∈ K . Now as φ ∈ I , so N ∈ F ( I ) . Let M = { m < m < · · · < m k < · · · } = N ∈ F ( I ) ,then { x m k } = { x n } and K | M = K . Therefore { k ∈ N : || x m k − x || ≥ r + ε } = { n ∈ N : || x n − x || ≥ r + ε } ∈ K = K | M . So x ∈ I K − LIM r x n . Hence the result follows. (cid:3) Theorem 3.7.
If rough I K -limit of a sequence { x n } n ∈ N of some roughness degree r is x implies that x is also a rough I -limit of { x n } n ∈ N of same roughness degree r then K ⊂ I .Proof. Suppose that K I and r be a non-negative real number. Then there exists a set A ∈ K \ I .Let us choose x, y ∈ X such that || x || = 1 and y = ( r + 2) x , then we have || x − y || ≥ r + ε for < ε ≤ and || x − y || < r + ε for ε > . Now define a sequence { x n } n ∈ N as follows x n = ( x + r, n ∈ N \ Ay, n ∈ A . Then for any ε > the set { n ∈ N : || x n − x || ≥ r + ε } is either theset A (when < ε ≤ ) or φ (when ε > ). Since K is an admissible ideal and A ∈ K , therefore { n ∈ N : || x n − x || ≥ r + ε } ∈ K . Thus x is a rough K -limit of { x n } n ∈ N of roughness degree r .Now by lemma 3.6, x is a rough I K -limit of { x n } n ∈ N of roughness degree r . Since for < ε ≤ , { n ∈ N : || x n − x || ≥ r + ε } = A and A / ∈ I . So { n ∈ N : || x n − x || ≥ r + ε } / ∈ I and hence x is not a rough I -limit of roughness degree r . But by our assumption, x is also a rough I -limit of { x n } n ∈ N . Thus we arrive at a contradiction and so, K ⊂ I . (cid:3) Corollary 2.
Let I and K be two ideals on N . Then rough I K -limit set is a subset of I -limit set of asequence { x n } n ∈ N of some roughness degree r if and only if K ⊂ I . N ROUGH I ∗ AND I K -CONVERGENCE OF SEQUENCES IN NORMED LINEAR SPACES 9 Proof.
In view of Theorem 3.5 and Theorem 3.7 the result follows. (cid:3)
In general, if x is a rough I -limit of a sequence { x n } n ∈ N in a normed linear space then it doesnot necessarily implies that x is also a rough I K -limit of { x n } n ∈ N . The following is an example insupport of this assertion. Example 3.4.
Let I be ideal as in example 3.2. Also let K be the ideal on N such that it is thecollection of all subsets of N whose natural density is zero. Now let us define a sequence in realnumbers space with usual norm by x n = j if n ∈ D j . Let r be a arbitrary non-negative realnumber. Let ε > be arbitrarily chosen, then there exists a l ∈ N such that ε > l . Clearly [ − r, r ] ⊂ I − LIM r x n , as { n ∈ N : | x n − x | ≥ r + ε } ⊂ D ∪ D ∪ · · · D l ∈ I for any x ∈ [ − r, r ] .If possible let − r be a rough I K -limit of { x n } of roughness degree r . Then there exists a M = { m 1) : s = 1 , , · · · } andnatural density of D p +1 = p +1 , therefore natural density of the set { k ∈ N : || x m k + r || ≥ r + p +1 } is not zero. Hence { k ∈ N : || x m k + r || ≥ r + p +1 } / ∈ K | M , since natural density of each setbelongs to K | M is also zero. Therefore − r is not a rough I K -limit of { x n } of roughness degree r .Rough I -limit x of a sequence { x n } n ∈ N is also a rough I K -limit of { x n } n ∈ N if the ideal I satisfiesthe condition (AP). Thus we have the following Theorem. Theorem 3.8. Let I and K be two admissible ideals on N such that the ideal I satisfies the condition(AP). Also let r be a non-negative real number and { x n } n ∈ N be a sequence in a normed linear space ( X, || · || ) . Then x ∈ I − LIM r x n implies x ∈ I K − LIM r x n .Proof. Suppose that I and K be two ideals on N such that the ideal I satisfies the condition (AP). Let { x n } n ∈ N be sequence such that x ∈ I − LIM r x n . Now since I satisfies the condition (AP), hence x ∈ I ∗ − LIM r x n . Now as I ∗ − LIM r x n ⊂ I K − LIM r x n , therefore x ∈ I K − LIM r x n . (cid:3) Theorem 3.9. Let I and K be two ideals on N . If for any sequence { x n } n ∈ N in a normed linearspace ( X, || · || ) the implication I − LIM r x n ⊂ I K − LIM r x n holds, then the the ideal I has theadditive property with respect to K , i.e., AP( I , K ) holds.Proof. Let { A n } n ∈ N be a sequence of mutually disjoint sets belonging to I and r > be a realnumber. Define a sequence { x n } n ∈ N in real number space with usual norm as follows x n = ( r + i , n ∈ A i , n ∈ N \ ∪ i A i . Then for any ε > , there exists p ∈ N such that { n ∈ N : || x n − || ≥ r + ε } ⊂ A ∪ A ∪ · · · ∪ A p ∈ I . So ∈ I − LIM r x n . Consequently, by our assumption, ∈ I K − LIM r x n . Thus there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such thatthe sub sequence { x m k } k ∈ N is rough K | M -convergent to of roughness degree r . Now if ∪ i A i ∈ I then by taking A i = B i for i ∈ N the results follows directly by using ( iv ) of the lemma 2.11. Solet ∪ i A i / ∈ I . Since M ∈ F ( I ) , so the set M contains a infinite numbers of A i ’s. Now for arbitrary ε > the set { k ∈ N : | x m k − | ≥ r + ε } ∈ K | M . For each i ∈ N either A i ∩ M = φ or A i ∩ M = φ .By the construction of the sequence and by the fact that ∈ I K − LIM r x n in both cases we have A i ∩ M ∈ K | M and since ε > is arbitrary so, A i ∩ M ∈ K for each i ∈ N . Now as M ∈ F ( I ) ,so N \ M = B ( say ) ∈ I . Let us put B i = A i ∩ B for each i . Then each of B i belongs to I . Also as ∞ [ i =1 B i = ∞ [ i =1 ( A i ∩ B ) = B ∩ ∞ [ i =1 A i ⊂ B , so ∞ [ i =1 B i ∈ I . Now as B i ⊂ A i , so A i \ B i = A i ∩ M . Thus A i \ B i ∈ K . Therefore A i ∼ K B i for i ∈ N , since A i ∼ K B i ⇔ A i ∆ B i ∈ K and A i ∆ B i = A i \ B i in this case. Thus by the virtue ( iv ) of the lemma 2.11 the result follows. (cid:3) In the next example we will see that, as in the case rough I ∗ -limit, rough I K -limit set of a sequence { x n } n ∈ N in a normed linear space may not be a subset of rough I K -limit set of a sub sequence { x n k } k ∈ N . Example 3.5. Let I be the collection of all subsets of N whose natural density is zero. Then I is a non trivial admissible ideal on N . Also let N = ∞ [ j =1 D j be a decomposition of N such that D j = { j − (2 s − 1) : s = 1 , , · · · } . Then each D j is infinite and D j ∩ D k = φ for j = k . Now K be the ideal such that it is the class of all subsets of N which intersects with only a finite numbers of D j ’s. Let us consider the sequence in real number space with usual norm, where x n = j if n ∈ D j .Now as φ ∈ I , so N ∈ F ( I ) . Let us take N = M . Let us enumerate M as, M = { m < m Let I and K be two admissible ideal on N such that K ⊂ I and AP( I , K ) holds.Then the rough I K -limit set of a sequence { x n } n ∈ N of some roughness degree r is a subset of rough I K -limit set of a sub sequence { x n k } k ∈ N of same roughness degree r .Proof. Let x be a rough I K -limit of a sequence { x n } n ∈ N of some roughness degree r . Also let I and K be two ideals on N such that AP( I , K ) holds and K ⊂ I . Now since K ⊂ I , therefore x is alsoa rough I -limit of the sequence { x n } n ∈ N . So x is also a rough I -limit of a sub sequence { x n k } k ∈ N of the sequence { x n } n ∈ N . Again since AP( I , K ) holds, so x is a rough I K -limit of the sub sequence { x n k } k ∈ N . Hence the results follows. (cid:3) Remark 3. Rough I K -limit of a sequence { x n } n ∈ N of some roughness degree r is also a rough I ∗ -limit of { x n } n ∈ N of same roughness degree r if some additional condition holds as given in thefollowing Theorem. Theorem 3.11. Let I and K be two admissible ideal on N such that K ⊂ I and the ideal I satisfiesthe condition (AP). Then for a sequence { x n } n ∈ N in X rough I K -limit of some roughness degree r isa rough I ∗ -limit of same roughness degree r .Proof. Let x be a rough I K -limit of a sequence { x n } n ∈ N of some roughness degree r . Also let I and K be two ideals on N such that AP( I , K ) holds and K ⊂ I . Again since K ⊂ I , so x is also a rough I -limit of the sequence { x n } n ∈ N . Again since the ideal I satisfies the condition (AP), so x is a rough I ∗ -limit of the sequence { x n } n ∈ N by Theorem 3.3. (cid:3) Definition 3.3 (c.f. [6]) . Let I and K be two admissible ideals on N . A sequence { x n } n ∈ N in anormed linear space is said to be K | M -bounded if there exists a set M = { m < m < · · · Let I be the ideal of the class of all those subsets of N whose natural density is zero.Then I is an admissible ideal on N . Also let K be any admissible ideal on N . Let us consider thesequence { x n } n ∈ N in real number space with usual norm as follows: x n = ( , n = k n, n = k , for some k ∈ N . Now as A ( say ) = { n ∈ N : n = k for some k ∈ N } ∈ I , so N \ A = M ( say ) ∈ F ( I ) . Letus enumerate M as, M = { m < m < · · · < m k < · · · } . Now we see that for r = 1 and for any ε > , the set { k ∈ N : | x m k − x | ≥ r + ε } = φ ∈ K | M for any x ∈ [0 , . So [0 , ⊂ I K − LIM r x n .But the sequence considered here is unbounded. Theorem 3.12. Let I and K be two admissible ideals on N . Then a sequence { x n } n ∈ N is K | M -bounded if and only if there exists a non-negative real number r such that I K − LIM r x n = φ .Proof. Suppose that the sequence { x n } n ∈ N is K | M -bounded. Then there exists a set M = { m Let I and K be two ideals on N and r be a non-negative real number. Then for asequence { x n } n ∈ N , the rough I K -limit set I K − LIM r x n is convex.Proof. Let us assume that x , x ∈ I K − LIM r x n . Then there exists M ′ = { m ′ < m ′ < · · · Let I be an ideal on N . Since for any non-negative real number r and for a sequence { x n } n ∈ N in X rough I ∗ -limit of roughness degree r is also a rough I K -limit of same roughnessdegree r , therefore rough I ∗ -limit set of roughness degree r of { x n } n ∈ N is also a convex set. Theorem 3.14. Let I , I , I , K , K and K be ideals on N such that I ⊂ I and K ⊂ K .Also let { x n } n ∈ N be a sequence and r be a non-negative real number. Then ( i ) I K − LIM r x n ⊂ I K − LIM r x n . ( ii ) I K − LIM r x n ⊂ I K − LIM r x n . Proof. ( i ) Suppose x ∈ I K − LIM r x n . Thus there exists a set M = { m < m < · · · < m k · · · } ∈ F ( I ) such that the sub sequence { x m k } k ∈ N is rough K | M -convergent to x of roughness degree r .Now as I ⊂ I , therefore N \ M ∈ I ⊂ I . Hence M ∈ F ( I ) . So x ∈ I K − LIM r x n . ( ii ) Let x ∈ I K − LIM r x n . Therefore the exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such that the sub sequence { x m k } k ∈ N is rough K | M -convergent to x of roughness degree r . Now as K ⊂ K , therefore the sub sequence { x m k } k ∈ N is also rough K | M -convergent to x of roughnessdegree r . Hence x ∈ I K − LIM r x n . (cid:3) Theorem 3.15. Let I and K be two admissible ideals on N . Then rough I K -limit set of a sequence { x n } n ∈ N is closed.Proof. The proof is trivial when I K − LIM r x n = φ , where r ≥ . Let us assume that I K − LIM r x n = φ for some r ≥ . Also let { y n } n ∈ N be a sequence in I K − LIM r x n such that y n → y .Since y n → y , so for a given ε > there exists a N ∈ N such that || y n − y || < ε for all n > N . Let n > N , therefore || y n − y || < ε . Again since { y n } n ∈ N is a sequence in I K − LIM r x n , therefore y n ∈ I K − LIM r x n . Therefore there exists a set M = { m < m < · · · < m k < · · · } ∈ F ( I ) such that { k ∈ N : || x m k − y n || ≥ r + ε } = K ( say ) ∈ K | M . Now for k ∈ K ∁ we have, || x m k − y || = || x m k − y n + y n − y || ≤ || x m k − y n || + || y n − y || < r + ε + ε = r + ε . Thus { k ∈ N : || x m k − y || ≥ r + ε } ∈ K | M . Therefore y ∈ I K − LIM r x n and hence the resultfollows. (cid:3) R EFERENCES [1] F. G. Arenas, Alexandroff spaces , Acta Math. Univ. Comenian., 17-25, (1999), .[2] Amar Kumar Banerjee and Apurba Banerjee, I -convergence classes of sequences and nets in topological spaces ,Jordan Journal of Mathematics and Statistics (JJMS), pp 13-31, (2018), .[3] Amar Kumar Banerjee and Rahul Mondal, A note on convergence of double sequences in a topological space , Mat.Vesnik, 144-152, June (2017), .[4] Amar Kumar Banerjee and Rahul Mondal, Rough convergence of sequences in a cone metric space , The Journal ofAnalysis, (2019), .[5] Amar Kumar Banerjee and Apurba Banerjee, A study on I -Cauchy sequences and I -divergence in S -metric spaces ,Malaya Journal of Matematik(MJM), pp 326-330, (2018), .[6] E. D¨undar and C. C¸ akan, Rough I -convergence, Demonstratio Mathematica, No (2014), Vol. XLVII.[7] H. Fast, Sur la convergence statistique , Colloq. Math., 241-244, (1951), .[8] J. A. Friday, On statistical convergence , Analysis, 301-313,(1985), .[9] P. Kostyrko, M. Maˇcaj, and T. ˇSal´at, Statistical convergence and I -convergence , 1999, Unpublished; http://thales.doa.fmph.uniba.ak/macaj/ICON.pdf. [10] Pavel Kostyrko, Tibor ˇSal´at, Wladyslaw Wilczy´nski, I -convergence, Real Analysis Exchange, pp.- 669-686,(2000/2001), Vol. .[11] C. Kuratowski, Topologie I , PWN Warszawa, 1958.[12] P. Kostyrko, M. Maˇcaj, T. ˇS˜al´at and M. Sleziak, I -convergence and extremal I -limit points , Math Slovaca, 443-464,(2005), .[13] J. L. Kelley, General Topology , Springer-Verlag, New York, 1955.[14] B. K. Lahiri and Pratulananda Das, Further results on I -limit superior and I -limit inferior , Math. Commun., 151-156,(2003), .[15] B. K. Lahiri and Pratulananda Das, I and I ∗ -convergence in topological spaces , Math. Bohemica, 153-160, (2005), .[16] M. Maˇcaj and M. Sleziak, I K -convergence , Real Analysis Exchange, pp. 177- 194, (2010/2011), .[17] A. Nabiev, S. Pehlivan, M. Gurdal, On I -Cauchy sequences , Taiwanese J. Math., 569- 576, (2007), .[18] H. X. Phu, Rough convergence in normed linear space , Numer. Funct. Anal. Optim., 199-222, (2001), .[19] T. ˇSal´at, On statistically convergent sequences of real numbers , Math. Slovaca, 139-150, (1980), .[20] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique , Colloq. Math., 73-74, (1951), . N ROUGH I ∗ AND I K -CONVERGENCE OF SEQUENCES IN NORMED LINEAR SPACES 13 (Amar Kumar Banerjee) D EPARTMENT OF M ATHEMATICS , T HE U NIVERSITY OF B URDWAN , G OLAPBAG ,B URDWAN -713104, W EST B ENGAL , I NDIA . Email address : [email protected], [email protected] (Anirban Paul) D EPARTMENT OF M ATHEMATICS , T HE U NIVERSITY OF B URDWAN , G OLAPBAG , B URDWAN -713104, W EST B ENGAL , I NDIA . Email address ::