Generalized almost statistical convergence
aa r X i v : . [ m a t h . F A ] N ov GENERALIZED ALMOST STATISTICAL CONVERGENCE
ABSOS ALI SHAIKH AND BISWA RANJAN DATTA Abstract.
The objective of this paper is to introduce the notion of generalized almost statisti-cal (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almostconvergence as well as statistical convergence of bounded real sequences. As a special kind ofBanach limit functional, we also introduce the concept of Banach statistical limit functionaland the notion of GAS convergence mainly depends on the existence of Banach statistical limitfunctional. We prove the existence of Banach statistical limit functional. Then we have shownthe existence of a GAS convergent sequence, which is neither statistical convergent nor almostconvergent. Also, some topological properties of the space of all GAS convergent sequences areinvestigated. Introduction
The theory of convergence arises from the study of the situation when the indices of asequence become larger and larger. A sequence ( ξ n ) of real numbers is said to be convergentto a real number l if for any ǫ > n ∈ N such that | ξ k − l | < ǫ for all k > n . But this usual notion of convergence can not always capture the detailed characteristicsof a non-convergent sequence. Because there are various non-convergent sequences which are“nearly to be” convergent. There are several generalizations of usual convergence, viz. almostconvergence [1, 2], statistical convergence [3, 4] etc. Nevertheless, it is always better to havelarger set of convergent sequences in more generalized sense under investigation.The existence of Banach limit functionals was proven by Banach [1] in 1932. Using Banachlimits, in 1948, Lorentz [2] introduced the notion of almost convergence, which is a generalizationof usual convergence of real sequences. Moricz and Rhoades [18] extended the idea of almostconvergence for double sequences and later it was studied in [22]. Again in 1951 Fast [3] andSteinhaus [4] introduced independently the notion of statistical convergence by a rigorous use ofnatural density of subsets of N , which is another generalization of usual convergence. Salat [5],Fridy [8, 13], Miller [14] and many others [9, 10, 23, 15] studied the concept of statisticalconvergence. In last few decades, several types of generalizations of usual convergence have been Mathematics Subject Classification.
Primary 46B45; Secondary 40A35, 40G15, 40H05 .
Key words and phrases.
Banach limit functional, almost convergence, statistical convergence. introduced and lots of research papers can be found in literature. Mursaleen [16] introducedthe idea of λ -statistical convergence in 2000. If the sequence λ is choosen, particularly, by λ = (1 , , , , . . . ), then λ -statistical convergence coincides with the statistical convergence. In2001 Kostyrko et al. [17] introduced the notion of ideal convergence of real sequences. If weconsider the ideal of all subsets of N having natural density zero, then the ideal convergencecoincides with statistical convergence. Later Lahiri and Das [19] extended the concept of idealconvergence for nets in topological spaces. In 2003, Mursaleen and Edely [7] extended theidea of statistical convergence for real double sequences. Further some generalizations of usualconvergence were introduced and studied in [20, 21].In this paper, the existence of a certain type of Banach limit functionals is shown whichare designated as Banach statistical limit functionals. With the help of Banach statisticallimit functionals we have introduced generalized almost statistical (briefly, GAS) convergenceof bounded real sequences. Then we have shown that GAS convergence is a generalization ofalmost convergence as well as statistical convergence. Hence a natural question arises: doesthere exist any GAS convergent sequence which is neither almost convergent nor statisticalconvergent? The answer to the above question is affirmative and the existence of such sequenceis shown in Example 3.2. At the end, we also investigated some topological properties ofthe space of all GAS convergent sequences. The outline of this paper is as follows: Section 2deals with some rudimentary facts about Banach limit functionals, almost convergence, naturaldensity, statistical convergence etc. as preliminaries. Section 3 is devoted to the investigationof main results of the paper (see Theorem 3.2) and the conclusion of the hole paper is drawnin the last section. 2.
Preliminaries
The limit functional f on the space c of all convergent real sequences defined by f ( x ) =lim n →∞ x n , x ∈ c , can be extended to the space l ∞ of all bounded real sequences by Hahn-Banach Extension theorem, where l ∞ is the normed linear space with sup-norm defined by || x || ∞ = sup n ∈ N | x n | for all x ∈ l ∞ . Banach [1] showed the existence of certain extensions whichare called Banach limits, defined as follows: Definition 2.1.
A functional B : l ∞ → R is called Banach limit if it satisfies the following :(i) || B || = 1,(ii) B | c = f , where f is the limit functional on c ,(iii) If x ∈ l ∞ with x n ≥ ∀ n ∈ N , then B ( x ) ≥ ENERALIZED ALMOST STATISTICAL CONVERGENCE 3 (iv) If x ∈ l ∞ , then B ( x ) = B ( Sx ), where S is the shift operator defined by S (( x n ) ∞ n =1 ) = S (( x n ) ∞ n =2 ).The concept of almost convergence, which is a generalization of usual convergence, wasintroduced by Lorentz [2] in 1948 by using Banach limit functionals. Definition 2.2.
For some x ∈ l ∞ , if B ( x ) is unique (i.e. invariant) for all Banach limitfunctionals B , then the sequence x is called almost convergent to B ( x ).Let A ⊂ l ∞ be the set of all almost convergent real sequences. Clearly c ( A . Definition 2.3.
Let P ⊂ N . If the limit δ ( P ) = lim n →∞ | P ∩ { , , . . . , n }| n exists, then δ ( P ) is called the natural density or asymptotic density of P in N . Note.
Any subset P of N may not have natural density as it depends totally on the existenceof the limit.By using the concept of natural density, in 1951, Fast [3], Steinhaus [4] introduced indepen-dently the notion of statistical convergence, which is another generalization of usual conver-gence. Definition 2.4.
A sequence ( p k ) k of real numbers is called convergent statistically to ℓ ∈ R iffor any ǫ > δ ( { k ∈ N : | p k − ℓ | > ǫ } ) = 0. We use the notation p k stat −−→ ℓ or stat lim n →∞ p n = ℓ . Example 2.1.
The sequence ( λ n ) n defined by λ j = j if j = k , k ∈ N st be the set of all bounded statistically convergent real sequences.Throughout this paper, we say that a condition C is satisfied by x k for almost all k (briefly, a.a.ka.a.ka.a.k ) if E = { k ∈ N : x k does not satisfy condition C } = ⇒ δ ( E ) = 0. We’ll now state somewell known results which will be used in the sequel. Lemma 2.1.
Let
C, D ⊂ N . If δ ( C ) , δ ( D ) , δ ( C ∪ D ) exist, then max { δ ( C ) , δ ( D ) } ≤ δ ( C ∪ D ) ≤ min { δ ( C ) + δ ( D ) , } . Furthermore, if δ ( C ) = δ ( D ) = 0 , then δ ( C ∪ D ) exists and equals to . A. A. SHAIKH AND B. R. DATTA
Lemma 2.2. [5]
Let ( p n ) n be a real sequence. Then, p n stat −−→ ℓ if and only if there exists some J ⊂ N with J = { α < α < · · · < α n < . . . } such that δ ( J ) = 1 and lim j →∞ p α j = ℓ . Lemma 2.3.
Let ( r n ) n be a statistically convergent real sequence and stat lim n →∞ r n = ℓ . If ( s n ) n is defined by s n = r n − ℓ for all n ∈ N , then ( s n ) n converges statistically to . Lemma 2.4.
Let ( p n ) n be a statistically convergent real sequence. Then, the telescoping se-quence of ( p n ) n is statistically convergent to i.e. stat lim n →∞ ( p n − p n +1 ) = 0 .Proof. Let stat lim n →∞ p n = l and ǫ > n ∈ N , | p n +1 − p n | ≤| p n +1 − l | + | p n − l | . Thus, | p n +1 − l | ≤ ǫ and | p n − l | ≤ ǫ = ⇒ | p n +1 − p n | ≤ ǫ . Contrapositively, | p n +1 − p n | > ǫ = ⇒ | p n +1 − l | > ǫ or | p n − l | > ǫ . Hence, { n ∈ N : | p n +1 − p n | > ǫ } ⊂ { n ∈ N : | p n +1 − l | > ǫ } ∪ { n ∈ N : | p n − l | > ǫ } . Therefore, δ ( { n ∈ N : | p n +1 − p n | > ǫ } ) = 0. Thus stat lim n →∞ ( p n − p n +1 ) = 0. (cid:3) Lemma 2.5.
Let ( x n ) n and ( y n ) n be two real sequences such that x n stat −−→ ℓ and y k = x k a.a.k .Then, y n stat −−→ ℓ .Proof. Let E = { k ∈ N : y k = x k } and ǫ > δ ( E ) = 0. Now, { n ∈ N : | y n − ℓ | > ǫ } ⊂ { n ∈ N : | x n − ℓ | > ǫ } ∪ E, which implies that δ ( { n ∈ N : | y n − ℓ | > ǫ } ) = 0. Hence y n stat −−→ ℓ . (cid:3) Definition 2.5.
A family I of subsets of N is said to be an ideal of N if(i) A ∪ B ∈ I for each A, B ∈ I ,(ii) A ⊂ B with B ∈ I implies A ∈ I . Definition 2.6. [17] A real sequence ( z k ) k is said to be I -convergent to l if for any ǫ >
0, theset { k ∈ N : | z k − l | > ǫ } ∈ I .In [17] Kostyrko et al. showed that I δ = { A ⊂ N : δ ( A ) = 0 } is an ideal and I δ -convergence isactually statistical convergence. The following result is a consequence of Hahn-Banach theorem. Proposition 2.6.
Let X be a normed linear space and Y be a subspace space of X . If α ∈ X − Y and µ = d ( α, Y ) := inf { d ( α, y ) : y ∈ Y } , then there exists a bounded linear functional f : X → R such that f ( α ) = 1 , f ( y ) = 0 , ∀ y ∈ Y with k f k = µ − . Lorentz [2] proved that A is a closed linear non-separable subspace of l ∞ and A is dense initself but nowhere dense in l ∞ . Also, Lorentz [2] characterized the almost convergence given asfollows: ENERALIZED ALMOST STATISTICAL CONVERGENCE 5
Proposition 2.7. [2]
Let x = ( x n ) n ∈ l ∞ . Then ( x n ) n is almost convergent to some ℓ if andonly if lim k →∞ x p + x p +1 + · · · + x p + k − k = ℓ holds for each p ∈ N . Example 2.2. [14] The divergent sequencce (1 , , , , . . . ) ∈ l ∞ is almost convergent to . Butit is not statistically convergent. Example 2.3. [14] Consider a sequence x = ( x n ) n in { , } constructed as follows: x = ( 0 , , . . . , | {z }
100 copies ,
10 copies z }| { , , . . . , , , , . . . , | {z } copies , copies z }| { , , . . . , , , , . . . , | {z } copies , copies z }| { , , . . . , , . . . )Then, ( x n ) n is statistically convergent to 0. But it is not almost convergent.In [14] Miller and Orhan showed that almost convergence and statistical convergence areincomparable, i.e., there exists a statistically convergent (resp. almost convergent) sequencewhich is not almost convergent (resp. statistically convergent). Thus st
6⊂ A (see, Example 2.3)and
A 6⊂ st (see, Example 2.2). In this paper, we have defined a new type of convergence whichis a generalization and comparable with both almost and statistical convergence. To this endwe have to introduce the concept of Banach limits in the context of statistical convergence.3. Main Results
Salat [5] showed that st is a closed subspace of l ∞ . Since, φ n stat −−→ a , ξ n stat −−→ b imply φ n + ξ n stat −−→ a + b and λξ n stat −−→ λb for any λ ∈ R and φ, ξ ∈ st , the map g : st → R definedby g ( x ) = stat lim n →∞ x n is a linear functional on st . We call this functional g by statistical limitfunctional on st . Lemma 3.1.
The linear functional g : st → R defined by g ( x ) = stat lim n →∞ x n is a bounded with || g || = 1 .Proof. Let x ∈ st . Then | g ( x ) | = | stat lim n →∞ x n | ≤ | sup n ∈ N x n | ≤ sup n ∈ N | x n | = || x || ∞ , which implies | g ( x ) ||| x || ∞ ≤
1. Thus || g || ≤
1. Again, consider y = ( λ, λ, λ, . . . ) ∈ st . Then g ( y ) = λ = || y || ∞ . Thus ∃ y ∈ st such that | g ( y ) ||| y || ∞ = 1 = ⇒ || g || ≥
1. Hence || g || = 1. (cid:3) A. A. SHAIKH AND B. R. DATTA
Thus by Hahn-Banach theorem, g can be extended to l ∞ preserving norm i.e., ∃ L ∈ ( l ∞ ) ∗ such that L | st = g and || L || = || g || , where ( l ∞ ) ∗ is the continuous dual (dual space) of l ∞ . Wenow state and prove the main result of the paper. Theorem 3.2.
There exists a functional F : l ∞ → R satisfying the following:(i) ||F || = 1 ,(ii) F | st = g , where g is the statistical limit functional on st ,(iii) If s ∈ l ∞ with s n ≥ a.a.n , then F ( s ) ≥ ,(iv) If x ∈ l ∞ , then F ( x ) = F ( T x ) , where T : l ∞ → l ∞ is a map with ( T x ) k = ( Sx ) k a.a.k for each x ∈ l ∞ and S is the shift operator defined by S (( x n ) ∞ n =1 ) = ( x n ) ∞ n =2 .Proof. Let G = { x − Sx : x ∈ l ∞ } and N = { y ∈ l ∞ : y k = x k a.a.k, x ∈ G } . So G ⊂ N ⊂ l ∞ .Clearly G is a subspace of l ∞ . Let p, q ∈ N and any µ, λ ∈ R . Then ∃ x, y ∈ l ∞ such that p n = ( x − Sx ) n a.a.n and q n = ( y − Sy ) n a.a.n . Since, G is a subspace, µ ( x − Sx )+ λ ( y − Sy ) ∈ G .So by Lemma 2.1, we have ( µp + λq ) n = ( µ ( x − Sx )+ λ ( y − Sy )) n a.a.n which implies µp + λq ∈ N .Therefore N is a subspace of l ∞ .Now, we claim that d (1 , N ) = 1, where 1 = (1 , , , . . . ). For, 0 ∈ G = ⇒ d (1 , G ) ≤ G ⊂ N , d (1 , N ) ≤ d (1 , G ) ≤
1. Let p ∈ N . Then p n = ( b − Sb ) n a.a.n for some b ∈ l ∞ .If p n ≤ n ∈ N , then || − p || ∞ = sup n ∈ N | − p n | ≥
1. Again, if p n ≥ , ∀ n ∈ N ,then ( b − Sb ) n ≥ a.a.n which implies b n ≥ b n +1 a.a.n . Thus ( b n ) n has a subsequence ( b n k ) k such that b n k ≥ b n k +1 ∀ k ∈ N with δ ( { n k : k ∈ N } ) = 1. Since ( b n k ) k is a monotonicallydecreasing bounded sequence, lim n →∞ b n k = l (say) exists. Thus by Lemma 2.2 stat lim n →∞ b n = l .Therefore using Lemma 2.4 we have stat lim n →∞ ( b n − b n +1 ) = 0, i.e. stat lim n →∞ ( b − Sb ) n = 0. So,by Lemma 2.5 stat lim n →∞ p n = 0. Hence using Lemma 2.2, there is a subsequence ( p n j ) j of ( p n ) n such that δ ( { n j : j ∈ N } ) = 1 with lim j →∞ p n j = 0. So, || − p || ∞ = sup n ∈ N | − p n | ≥ sup j ∈ N | − p n j | = 1.Thus, for any p ∈ N we have || − p || ∞ ≥
1, which implies d (1 , N ) ≥
1. Hence d (1 , N ) = 1.Clearly 1 / ∈ N . By the Proposition 2.6 there exists a functional F : l ∞ → R such that F (1) = 1, F ( y ) = 0 , ∀ y ∈ N with ||F || = d (1 , N ) − = 1. Let st be the collection of allbounded sequences which converge stasistically to 0.We claim that st ⊂ ker F . For, let ξ ∈ st and ǫ > ξ ∈ l ∞ and ξ n stat −−→ ⇒ if U ǫ = { k ∈ N : | ξ k | > ǫ } , then δ ( U ǫ ) = 0. Again, F ( y ) = 0 , ∀ y ∈ N implies that F ( y ) = 0 , y k = ( z − Sz ) k a.a.k, ∀ z ∈ G . So by the linearity of F we can write F ( x ) = F ( T x ), where T : l ∞ → l ∞ is any map such that σ T ( z ) = { j ∈ N : ( T z ) j = ( Sz ) j } and δ [ σ T ( z )] = 0 , ∀ z ∈ l ∞ . ENERALIZED ALMOST STATISTICAL CONVERGENCE 7
Choosing σ T ( x ) = U ǫ for all x ∈ l ∞ , we consider the map T : l ∞ → l ∞ defined by T x = r ,for all x ∈ l ∞ , where r k = k + 1 ∈ U ǫ x k +1 otherwise.Then F ( ξ ) = F ( T ξ ) which implies that |F ( ξ ) | = |F ( T ξ ) | ≤ ||F || || T ξ || ∞ = sup {| ( T ξ ) k | : k ∈ N } ≤ ǫ . Since ǫ > F ( ξ ) = 0. Thus st ⊂ ker F .Next we claim that F | st = g . For, let x ∈ st , i.e. x n stat −−→ Ω (say). Then the sequence e defined by e j = x j − Ω , ∀ j ∈ N . Now by Lemma 2.3, x − Ω1 = e ∈ st which implies e ∈ ker F .Now F ( x ) = F ( x − Ω1) + F (Ω1) = F ( e ) + Ω F (1) = Ω F (1) = Ω = stat lim n →∞ x n = g ( x ) . Thus
F | st = g .Next we show that if u, v ∈ l ∞ with u k = v k a.a.k , then F ( u ) = F ( v ). For, let w n = u n − v n , ∀ n ∈ N . Let K = { k : w k = 0 } . Then δ ( K ) = 0. Now consider the map T : l ∞ → l ∞ with σ T ( w ) = K defined by T a = b , for all a ∈ l ∞ , where b k = k + 1 ∈ Ka k +1 otherwise.Therefore T w = 0. Now, F ( u ) − F ( v ) = F ( u − v ) = F ( w ) = F ( T w ) = F (0) = 0. Thus F ( u ) = F ( v ).If possible, suppose that there exists z ∈ l ∞ with z n ≥ a.a.n such that F ( z ) <
0. Consider y ∈ l ∞ defined by y n = z n || z || ∞ . Clearly y n ≥ a.a.n and F ( y ) <
0. Again consider x ∈ l ∞ defined by x k = y k < y k if y k ≥ . Then, x n = y n , a.a.n and 0 ≤ x n ≤ , ∀ n ∈ N . Therefore F ( x ) = F ( y ) < || − x || ∞ = sup n ∈ N | − x n | ≤ . Again F (1 − x ) = F (1) − F ( x ) = 1 − F ( x ) >
1. Thus we get 1 < |F (1 − x ) | ≤ ||F || || − x || ∞ = || − x || ∞ ≤
1, which is a contradiction. Hence, if s ∈ l ∞ with s k ≥ a.a.k , then F ( s ) ≥ (cid:3) The functional F in the Theorem 3.2 is named as Banach statistical limit functional anddefined as follows:
A. A. SHAIKH AND B. R. DATTA
Definition 3.1.
A functional F : l ∞ → R is called a Banach statistical limit if it satisfies thefollowing :(i) ||F || = 1,(ii) F | st = g , where g is the statistical limit functional on st ,(iii) If s ∈ l ∞ with s n ≥ a.a.n , then F ( s ) ≥ x ∈ l ∞ , then F ( x ) = F ( T x ), where T : l ∞ → l ∞ is a map with ( T x ) k = ( Sx ) k a.a.k for each x ∈ l ∞ and S is the shift operator defined by S (( x n ) ∞ n =1 ) = ( x n ) ∞ n =2 . Corollary 3.2.1.
Every Banach statistical limit functional is a Banach limit functional on l ∞ . Corollary 3.2.2.
Let F be any Banach statistical limit functional on l ∞ . If u, v ∈ l ∞ with u k = v k a.a.k , then F ( u ) = F ( v ) . Now, by using Banach statistical limit functional, we introduce a new type of convergence,called, generalized almost statistical convergence (briefly, GAS convergence), defined as follows:
Definition 3.2.
Let x ∈ l ∞ . Then x is said to be generalized almost statistically conver-gent to λ , if F ( x ) = λ for all Banach statistical limit functionals F , i.e., if F ( x ) is invariant(unique) for each Banach statistical limit fnctionals F on l ∞ .Let S be the set of all GAS convergent real sequences. The following result is easily obtainedfrom the Definitions 3.1 and 3.2. Corollary 3.2.3.
Every statistically convergent sequence is GAS convergent with the samelimit, i.e. st ⊂ S . The converse is not true (see Example 3.1). Clearly st ( S .From the Definitions 3.1 and 3.2, it follows that every bounded statistically convergent se-quence is GAS convergent with the same limit. Lemma 3.3.
Every almost convergent real sequence is GAS convergent with the same limit,i.e.
A ⊂ S .Proof.
Suppose r = ( r i ) i ∈ l ∞ is an almost convergent real sequence with limit κ . Then for anyBanach limit functional B : l ∞ → R we have B ( r ) = κ . By Corollary 3.2.1, we can easily saythat F ( r ) = κ for any Banach statistical limit functional F . Thus ( r n ) n is generalized almoststatistically convergent to κ . (cid:3) ENERALIZED ALMOST STATISTICAL CONVERGENCE 9
The converse is not true. Because in Example 2.3 the sequence x = ( x n ) n is not almostconvergent. But it is statistically convergent, which implies x ∈ S (by Corollary 3.2.3). Clearly A ( S . Example 3.1.
Consider the real bounded sequence x = ( x n ) n defined by x k = k is a perfect square number0 if k is even and not a perfect square1 if k is odd and not a perfect squarei.e., ( x n ) n = (5 , , , , , , , , , , , , , , , , , , , , , , , , , , . . . )Clearly, x is not convergent in usual sense. Even x is not convergent statistically. But x isGAS convergent to . For, let S = { n : n ∈ N } . Now consider the map T : l ∞ → l ∞ (as inthe Theorem 3.2) with σ T ( x ) = S defined by T z = y , for all z ∈ l ∞ , where y k = z k +1 if k + 1 / ∈ S k + 1 ∈ S and k is odd1 if k + 1 ∈ S and k is even.Then T x = (0 , , , , , , . . . ). Let F be any Banach statistical limit functional. Since, u, v ∈ l ∞ with u k = v k a.a.k implies F ( u ) = F ( v ) (as shown in the proof of the Theorem 3.2),then F ( x ) = F (1 , , , , , , . . . ). Then by the Theorem 3.2 we have F ( x ) = F ( T x ) = F (0 , , , , , , . . . ) = F ((1 , , , , . . . ) − (1 , , , , . . . )) = F (1 , , , , . . . ) −F (1 , , , , . . . ) =1 − F ( x ). Thus F ( x ) = .The following example shows that there exists a GAS convergent sequence, which is neitheralmost convergent nor statistically convergent. i.e. st ∪ A ( S . Example 3.2.
Consider the sequence ξ = ( ξ n ) n defined as follows: ξ = ( 1 , , , , . . . | {z }
100 terms ,
10 terms z }| { , , . . . , , , , , , . . . | {z } terms , terms z }| { , , . . . , , , , , , . . . | {z } terms , terms z }| { , , . . . , , . . . ) . It is easy to check that ξ is neither a statistically convergent nor an almost convergent sequence.But ξ is GAS convergent. For, let E = ∞ S i =1 ( b i − i , b i ] ∩ N with b i = i P j =1 (100 j + 10 j ) for each i ∈ N . Let T : l ∞ → l ∞ defined by T x = z with z k = x k +1 , if k + 1 / ∈ E , otherwise. Then z = (0 , , , , , . . . ). Since δ ( E ) = 0, z k = ( Sx ) k a.a.k . Therefore, F ( x ) = F ( T x ) = F (0 , , , , , . . . ) = F (1 , , , , . . . ) = . Theorem 3.4.
GAS convergence can not be characterized by ideal convergence for proper idealsof N .Proof. If possible, suppose that GAS convergence coincides with ideal convergence for someproper ideal I of N . Since, ξ = (1 , , , , . . . ) is almost convergent to , ξ is GAS convergent to . Hence, for any ǫ > A ǫ := { k ∈ N : | ξ k − | ≥ ǫ } ∈ I . But A = N ∈ I , which contradictsthat I is a proper ideal of N . (cid:3) Proposition 3.5.
The following are some topological properties of S (i) S is closed(ii) S is non-separable in l ∞ (iii) S is first countable but not second countable.(iv) S is not Lindel¨of and not compact.Proof. (i) S is closed. For, let s ∈ ¯ S . Then, by the sequence lemma, there exists a sequence( s ( n ) ) n ∈ N in S such that lim n →∞ s ( n ) = s . Let ρ, τ be any two Banach statistical limit functional.Since ρ, τ are continuous, ρ ( s ) = ρ ( lim n →∞ s ( n ) ) = lim n →∞ ( ρ ( s ( n ) )) = lim n →∞ ( τ ( s ( n ) )) = τ ( lim n →∞ s ( n ) ) = τ ( s ). So s ∈ S . Hence S is closed.(ii) S is non-separable. For, consider Λ ⊂ l ∞ defined as follows:Λ = x ∈ l ∞ : x k = k is a perfect square0 otherwise. Clearly Λ is an uncountable subset of
A ⊂ S , which implies that S is uncountable. Now for anydistinct u, v ∈ Λ, d ( u, v ) = || u − v || ∞ = 1. Thus Λ is an uncountable discrete subset of S . Let D be any dense set in S i.e. ¯ D = S . Let us consider any s, t ∈ Λ with s = t . Then s, t ∈ S = ¯ D .Then the disjoint open balls B d ( s, ) and B d ( t, ) must have non-empty intersections with D ,which implies that there are two distinct elements of D . Since Λ is uncountable, D is alsouncountable. Thus S does not contain any countable dense subset, i.e. S is non-separable.(iii) S is not second countable due to its non-separability.(iv) Since S is metrizable and not second countable, S is non-Lindel¨of and non-compact. (cid:3) Conclusion
The notion of GAS convergence is introduced in this paper as a generalization of almostconvergence as well as statistical convergence of bounded real sequences. The existence of GAS
ENERALIZED ALMOST STATISTICAL CONVERGENCE 11 convergence is ensured by the existence of Banach statistical limit functional (see Theorem3.2). Some topological properties of the space of all GAS convergent sequences are obtained(see Proposition 3.5). The GAS convergence can not be characterized by ideal convergence forproper ideals of N . Open problem: In the analogy of Proposition 2.7, is there any necessaryand sufficient condition for GAS convergence? Acknowledgement:
The second author is grateful to The Council Of Scientific and Indus-trial Research (CSIR), Government of India, for the award of JRF (Junior Research Fellowship).
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Department of Mathematics, The University of Burdwan, Burdwan–713101, West Bengal,India.
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