Ratio Mathematica Volume 47, 2023 Weak and weak* IK-convergence in normed spaces Amar Kumar Banerjee* Mahendranath Paul† Abstract The main object of this paper is to study the concept of weak IK - convergence, a generalization of weak I∗-convergence of sequences in a normed space, introducing the idea of weak* IK -convergence of sequences of functionals where I, K are two ideals on N, the set of all positive integers. Also we study the ideas of weak IK and weak* IK -limit points to investigate the properties in the same space. Keywords: Weak IK -convergence, Weak* IK -convergence, Condi- tion AP(I, K), Weak IK -limit points, Weak* IK -limit points. 2020 AMS subject classifications: 40A35; 40H05. 1 *Department of Mathematics, The University of Burdwan, Purba Burdwan -713104, W.B., India; akbanerjee1971@gmail.com, akbanerjee@math.buruniv.ac.in. †Department of Mathematics, The University of Burdwan, Purba Burdwan -713104, W.B., India; mahendrabktpp@gmail.com. 1Received on October 19, 2022. Accepted on June 18, 2023. Published on June 30, 2023. DOI: 10.23755/rm.v39i0.882. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 309 A. K. Banerjee, M. Paul 1 Introduction The idea of statistical convergence, an extended form of ordinary convergence, based on the concept of natural density of subsets of N, was introduced indepen- dently by Steinhaus [27] and by Fast [15]. Over the years and under different forms of statistical convergence turned out to be one of the most active research areas in the summability theory after the remarkable works of Friday [16, 17] and Salat [21, 26]. In 2000, Cannor et al.[10] introduced the idea of weak statistical convergence which has been used to characterize Banach spaces with seperable duals. Last few years some basic properties of this concept were studied by many authors in [8, 24]. Recently the concept of weak* statistical convergence of se- quence of functionals has been given by Bala [1]. In 2001, Kostyrko et al. [18] extended the idea of statistical convergence into I and I∗-convergence which depends on the structure of the ideals I of N. The mutual relation between I and I∗-convergence was given in [19] using the condition AP of the ideals.(such ideals are often called P-ideals [2]). Later many works on ideal convergence have been done in [3, 4, 5, 11, 12, 13]. In 2010, Pehlivan et al.[25] introduced the idea of weak I and I∗-convergence in a normed space and using the condition AP, they established a relation between such types convergence. In 2012, Bhardwaj et al. [9] extended the idea of weak* statistical convergence to weak* ideal convergence of sequence of functionals and shew that the ideas of weak ideal convergence and weak* ideal convergence are identical in a reflexive Banach space. In 2010, Macaj et al. [22] introduced the idea of IK -convergence which is a gen- eralization of all types of I∗-convergence. They have shown that if the ideal I has additive property with respect to an another ideal K (i.e. if condition AP(I, K) holds) then I-convergence implies IK -convergence. Very recently more results and applications of IK -convergence have been carried out [23, 7, 10]. It seems therefore reasonable to think if we extend the idea of weak and weak* conver- gence using double ideals in a normed space and in that case we intend to investi- gate how far several basic properties are affected. In our paper, we have studied the idea of weak IK -convergence of sequences which is a generalization of weak I∗-convergent sequences as defined in [25] and we have presented an interrelation between weak I and weak IK -convergence using the condition AP(I, K). Next we have introduced the concept of weak* IK - convergence for a sequence of functionals and observed that the ideas of weak and weak* IK -convergence of sequences of functionals are same in a reflexive Banach space. In the last section of this paper we have discussed the notion of weak IK - limit point of sequences and weak* IK -limit points of sequences of functionals. Since the importance of the notion of weak and weak* convergence in functional analysis is very significant, we have realized that the ideas of weak and weak* IK - 310 Weak and weak* IK -convergence in normed spaces convergence in a normed space give more general frame for functional analysis to study summability theory as well. 2 Basic definitions and notations Throughout the paper, we use N to denote the set of all positive integers and X for a normed linear space and X∗ for dual of X. First recall that a subset A of N is said to have natural density d(A) if d(A) = lim n 1 n n∑ k=1 χA(k), provided the limit exists where χA is characteristic function of A ⊂ N. Definition 2.1. [15] A sequence {xn} in X is said to be statistically convergent to l if for every ϵ > 0 the set K(ϵ) = {k ∈ N : ||xk − l|| ≥ ϵ} has natural density zero. Definition 2.2. [10] Let X be a normed linear space then a sequence {xn}n∈N in X is said to be weak statistically convergent to x ∈ X provided that for any f ∈ X∗, the sequence {f(xn − x)}n∈N is statistically convergent to 0. In this case we write w-st- lim n→∞ xn = x. Definition 2.3. Let S be a non empty set and a class I ⊂ 2S of subset of S is said to an ideal if (i)A, B ∈ I implies A ∪ B ∈ I and (ii)A ∈ I, B ⊂ A implies B ∈ I. I is said nontrivial ideal if S /∈ I and I ̸= {ϕ}. In view of condition (ii) ϕ ∈ I. If I ⫋ 2S we say that I is proper ideal on S. A nontrivial ideal I is said admissible if it contains all the singletons of S. A nontrivial ideal I is said non-admissible if it is not admissible. Definition 2.4. Let F be a class of subsets of non-empty set S. Then F is said to be a filter in S if (i) ϕ /∈ F , (ii) A, B ∈ F implies A ∩ B ∈ F and (iii) A ∈ F, A ⊂ B implies B ∈ F . If I is a non-trivial on a non-void set S then F = F(I) = {A ⊂ S : S\A ∈ I} is clearly a filter on S and conversely. Again F(I) is said associated filter with respect to ideal I. Definition 2.5. [18] A sequence {xn}n∈N in X is said to be I-convergent to x if for any ϵ > 0 the set A(ϵ) = {n ∈ N : ||xn − x|| ≥ ϵ} ∈ I. In this case we write I − lim n→∞ xn = x. 311 A. K. Banerjee, M. Paul Definition 2.6. [25] A sequence {xn}n∈N in X is said to be weak I-convergent to x ∈ X if for any ϵ > 0 and for any f ∈ X∗ the set A(f, ϵ) = {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ∈ I. In this case we write w − I − lim n→∞ xn = x. Note 2.1. It is easy to observe that weak I-limit of a weak I-convergent sequence is unique and moreover for an admissible ideal I, weak convergence implies weak I-convergence with the same limit point but converse part is not true which has been shown in paper [25] by an interesting examples. Note 2.2. It is obvious that if two ideals I1, I2 on N such that I1 ⊆ I2 then for a sequence {xn} w-I1-lim xn = x implies w-I2-lim xn = x. 3 Weak IK-convergence We have already mentioned that our aim to generalize the notion of weak I∗- convergence of sequences. We need to modify this definition introduced in [25]. Definition 3.1. (cf.[25]) A sequence {xn}n∈N in X is said to be weak I∗-convergent to x ∈ X if there exists a set M ∈ F(I) such that the sequence {yn}n∈N ⊂ X defined by yn = { xn if n ∈ M x if n /∈ M is weak-convergent to x. we denote it by the notation w-I∗-lim xn = x. Definition 3.2. (cf.[22]) Let I, K be two ideals on the set N. A sequence {xn}n∈N in X is said to be weak IK -convergent to x ∈ X if there exists a set M ∈ F(I) such that the sequence {yn}n∈N ⊂ X defined by yn = { xn if n ∈ M x if n /∈ M is weak K-convergent to x. we denote it by the notation w-IK -lim xn = x. Remark 3.1. We can give an equivalent definition of weak-IK -convergence in the following way: if there exists an M ∈ F(I) such that the sequence {xn}n∈M is weak-K|M -convergent to x where K|M = {B ∩ M : B ∈ K}. Lemma 3.1. If I and K are ideals on N, the set of all positive integers and if {xn}n∈N is a sequence in X such that w-K-lim{xn} = x, then w-IK -lim{xn} = x. The proof follows from the definition of weak K-convergence taking M = N ∈ F(I) and yn = xn. 312 Weak and weak* IK -convergence in normed spaces Proposition 3.1. Let X be normed space and I be an ideal on N. A sequence {xn}n∈N ⊂ X is weak II -convergent to x if and only if it is weak I-convergent to x. Proof. Let {xn} be weak II -convergent to x then there exists an M ∈ F(I) such that the sequence {xn}n∈M is weak-I|M -convergent to x. So there exists G ∈ F(I) such that {n ∈ N : |f(xn) − f(x)| < ϵ} ∩ M = G ∩ M. Clearly G ∩ M ∈ F(I) and {n ∈ N : |f(xn) − f(x)| < ϵ} ⊇ G ∩ M. Therefore {n ∈ N : |f(xn) − f(x)| < ϵ} ∈ F(I) i.e. {xn} is weak I-convergent to x. Converse part follows from Lemma 3.1 taking K = I. Proposition 3.2. Let X be a normed space and I, I1, K and K1 be ideals on N such that I ⊆ I1 and K ⊆ K1. Then for any sequence {xn}n∈N, we have (i) w-IK -lim xn = x ⇒ w-IK1 -lim xn = x and (ii) w-IK -lim xn = x ⇒ w-IK1 -lim xn = x. Proof. (i) Now as w-IK -lim xn = x so there exists an M ∈ F(I) such that the sequence {xn}n∈M is weak-K|M -convergent to x where K|M = {B ∩ M : B ∈ K}. Here M ∈ F(I) ⊆ F(I1) as I ⊆ I1. So obviously w-IK1 -lim xn = x. (ii) Again w-IK -lim xn = x then there exists a set M ∈ F(I) such that the {yn} ∈ X given by yn = { xn if n ∈ M x if n /∈ M is weak K-convergent to x. Since K ⊆ K1 and from the Note 2.2 we get {yn} is weak K-convergent to x. Hence w-IK1 -lim xn = x. Theorem 3.1. Let I and K be ideals on N and {xn}n∈N be a sequence in X then (i) w-I-lim xn = x ⇒ w-IK -lim xn = x if I ⊆ K. (ii) w-IK -lim xn = x ⇒ w-I-lim xn = x if K ⊆ I. Proof. (i) Since {xn} is weak I-convergent to x ∈ X then for any ϵ > 0 and f ∈ X∗ the set A(f, ϵ) = {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ∈ I. Again I ⊆ K so A(f, ϵ) ∈ K. Therefore the sequence {xn} is weak K-convergent to x. So from the Lemma 3.1 we get {xn} is weak IK -convergent to x. (ii) Now w-IK -lim xn = x then there exists a set M ∈ F(I) such that the sequence {yn} given by yn = { xn if n ∈ M x if n /∈ M is weak K-convergent to x. So A(f, ϵ) = {n ∈ N : |f(yn) − f(x)| ≥ ϵ} = {n ∈ N : |f(xn)−f(x)| ≥ ϵ}∩M ∈ K ⊆ I. Consequently {n ∈ N : |f(xn)−f(x)| ≥ ϵ} ⊆ (N \ M) ∪ A(f, ϵ) ∈ I. So w-I-lim xn = x. 313 A. K. Banerjee, M. Paul Note 3.1. If K ̸⊂ I and I ̸⊂ K then none of these implications in Theorem3.1 may not be true. To support this we cite an example which is weak-IK -convergence but not weak-I-convergence. Example 3.1. Let I and K be two ideals on N such that K ̸⊂ I and I ̸⊂ K, but K ∩ I ̸= ϕ. Let x, y ∈ X,x ̸= y and consider a set M ∈ K \ I. Let us now consider a sequence {xn} with define by xn = { x if n /∈ M y if n ∈ M then for every ϵ > 0 and f ∈ X∗ we get {n ∈ N : |f(xn)−f(x)| ≥ ϵ} ⊂ M ∈ K. So w − K − lim xn = x. But, since x − y ̸= θ so from Hanh Banach theorem there exist a f ∈ X∗ such that f(x − y) = ||x − y||. Choose an ϵ = ||x−y|| 2 . then {n ∈ N : |f(xn) − f(x)| ≥ ϵ}={n ∈ M : |f(xn) − f(x)| ≥ ϵ} ∪ {n ∈ N \ M : |f(xn) − f(x)| ≥ ϵ} = {n ∈ M : |f(y) − f(x)| ≥ ||x−y|| 2 } = {n ∈ M : ||x − y|| ≥ ||x−y|| 2 }M /∈ I. So w − I − lim xn ̸= x. Note 3.2. Consider any two ideals I and K on N then we can construct a new ideal I ∨ K = {A ∪ B : A ∈ I, B ∈ K} containing both I, K.The dual filter of I ∨K is F(I ∨K) = {G∩H : G ∈ F(I), H ∈ F(K)}, when I ∨K is non-trivial. It should be noted that if I ∨ K is non-trivial ideal and I, K are proper subsets of I ∨ K then both I and K are non-trivial. But converse part may or may not be true always. To establish this, following examples are given. Example 3.2. Let the two sets P = {5p : p ∈ N} and S = {5s − 1 : s ∈ N} now it is clear that 2P , 2S and 2P ∨ 2S all ideals are non-trivial on N. Example 3.3. Now let P be set of all odd integers and S be set of all even integers. Then I = 2P , K = 2S both are non-trivial on the whole set N but I ∨ K is not a non-trivial ideal on N. Theorem 3.2. If I ∨K is non-trivial ideal on N and X is normed space then weak IK -limit of a sequence {xn}n∈N in X is unique. Proof. If possible let sequence {xn}n∈N has two distinct weak IK -limits say x and y. Since x ̸= y i.e. (x − y) ̸= θ then by a consequence of Hahn Banach theorem there exists f such that f(x − y) = ||x − y|| ̸= θ then f(x) ̸= f(y) and let ϵ = |f(x)−f(y)| 3 > 0. Since {xn}n∈N has weak IK -limit x then there exists a set A1 ∈ F(I) such that the {yn} ∈ X given by yn = { xn if n ∈ A1 x if n /∈ A1 314 Weak and weak* IK -convergence in normed spaces is weak K-convergent to x. So,{n ∈ N : |f(yn) − f(x)| ≥ ϵ} ∈ K i.e. {n ∈ N : |f(yn) − f(x)| < ϵ} ∈ F(K) which implies that {n ∈ A1 : |f(yn) − f(x)| < ϵ} ∪ {n ∈ N \ A1 : |f(yn) − f(x)| < ϵ} ∈ F(K) i.e. (N \ A1) ∪ {n ∈ A1 : |f(yn)−f(x)| < ϵ} ∈ F(K) i.e. N\(A1\{n ∈ A1 : |f(yn)−f(x)| < ϵ}) ∈ F(K) so A1 \ B1 ∈ K where B1 = {n ∈ A1 : |f(xn) − f(x)| < ϵ}. Similarly as {xn} has weak IK -limit y, so there exists a set A2 ∈ F(I) such that A2 \ B2 ∈ K where B2 = {n ∈ A2 : |f(xn) − f(y)| < ϵ}. So, (A1 \ B1) ∪ (A2 \ B2) ∈ K then (A1∩A2)∩(B1∩B2)c ⊂ (A1∩Bc1)∪(A2∩Bc2) ∈ K. Thus (A1∩A2)∩(B1∩B2)c ∈ K i.e. (A1 ∩ A2) \ (B1 ∩ B2) ∈ K. Now by our construction we get B1 ∩ B2 = ϕ. For if B1∩B2 ̸= ϕ, let n ∈ B1∩B2 then |f(xn)−f(x)| < ϵ and |f(xn)−f(y)| < ϵ. Therefore, 3ϵ = |f(x) − f(y)| ≤ |f(x) − f(xn)| + |f(xn) − f(y)| < 2ϵ, which is a contradiction. So A1 ∩ A2 ∈ K i.e. N \ (A1 ∩ A2) ∈ F(K) −→ (i). Since A1, A2 ∈ F(I) so A1 ∩ A2 ∈ F(I) −→ (ii). Since I ∨ K is non-trivial so the dual filter F(I ∨ K) exits. Now from (i) and (ii) we get ϕ ∈ F(I ∨ K), which is a contradiction. Hence the weak IK -limit is unique. Theorem 3.3. Let X be normed space and I, K be two ideals on N. A sequence {xn}n∈N ∈ X is weak IK -convergent to x if and only if it is weak (I ∨ K)K - convergent to x. Proof. Suppose that {xn} is weak IK -convergent to x then there exists an M ∈ F(I) such that the sequence {xn}n∈M is weak-K|M -convergent to x. Since M ∈ F(I) so it is clear that M ∈ F(I ∨ K). Therefore {xn} is also weak (I ∨ K)K - convergent to x. Conversely, let {xn} is weak (I ∨ K)K -convergent to x then there exists an M ∈ F(I ∨ K) such that the sequence {xn}n∈M is weak-K|M -convergent to x. So for any ϵ(> 0) and for every f ∈ X∗ there exists G ∈ F(K) such that A(f, ϵ) ∩ M = G ∩ M where A(f, ϵ) = n ∈ N : |f(xn) − f(x)| < ϵ. Since M ∈ F(I ∨ K) then M = M1 ∩ M2 for some M1 ∈ F(I) and M2 ∈ F(K). Now we have A(f, ϵ) ∩ M1 ⊇ A(f, ϵ) ∩ M = (G ∩ M2) ∩ M1. Since G ∩ M2 ∈ F(K), this shows that A(f, ϵ) ∩ M1 ∈ F(K|M1) i.e. {xn} is weak IK -convergent to x. In the rest of this section, using additive property of ideals we will investi- gate the relationship between weak-I and IK -convergence. Now we recall the definition of K-pseudo intersection and then AP(I, K)-condition. Definition 3.3. [21] Let K be an ideal on N. We denote A ⊂K B whenever A \ B ∈ K. If A ⊂K B and B ⊂K A then we denote A ∼K B. Clearly A ∼K B ⇔ A △ B ∈ K. If A ⊂K An holds for each n ∈ N then we say that a set A is K-pseudo intersec- tion of a system {An : n ∈ N}. 315 A. K. Banerjee, M. Paul Definition 3.4. [21] Let I, K be ideals on the set X. We say that I has additive property with respect to K or that the condition AP(I, K) holds if any one of the equivalent condition of following holds: (a) For every sequence (An)n∈N of sets from I there is A ∈ I such that An ⊂K A for every n′s. (b) Any sequence (Fn)n∈N of sets from F(I) has K-pseudo intersection in F(I). (c) For every sequence (An)n∈N of sets from the ideal I there exists a sequence (Bn)n∈N ⊂ I such that Aj ∼K Bj for j ∈ N and B = ∪j∈NBj ∈ I. (d) For every sequence of mutually disjoint sets (An)n∈N ⊂ I there exists a se- quence (Bn)n∈N ⊂ I such that Aj ∼K Bj for j ∈ N and B = ∪j∈NBj ∈ I. (e) For every non-decreasing sequence A1 ⊆ A2 ⊆ · · · ⊆ An · · · of sets from I ∃ a sequence (Bn)n∈N ⊂ I such that Aj ∼K Bj for j ∈ N and B = ∪j∈NBj ∈ I. (f) In the Boolean algebra 2S/K the ideal I corresponds to a σ-directed subset, i.e. every countable subset has an upper bound. Note that the proof of the conditions (a) to (f) in the definition 3.4 are equiva- lent has been given in [21][Lemma 3.9]. Above definition is reformulation of the definition given below: Definition 3.5. [14] Let I, K be ideals on the non-empty set S. We say that I has additive property with respect to K or that the condition AP(I, K) holds if for every sequence of pairwise disjoint sets An ∈ I, there exists a sequence Bn ∈ I such that An △ Bn ∈ K for each n and ∪n∈NBn ∈ I. Theorem 3.4. If the condition AP(I, K) holds then weak-I-convergence implies weak-IK -convergence, where I, K are two ideals on N. Proof. Let {xn} be weak I-convergent sequence to x ∈ X. Let f ∈ X∗ and choose a sequence of rationals {ϵi : i ∈ N} so that {(f(x)−ϵi, f(x)+ϵi) : i ∈ N} be a countable base for R at the point f(x). By weak I-convergence of {xn} we have Bi = {n : |f(xn) − f(x)| < ϵi} ∈ F(I) for each i, thus by definition 3.4(b) there exists a set A ∈ F(I) with A ⊂K Bi i.e. A \ Bi ∈ K for all i’s. Now it suffices to show that the sequence {yn} ∈ X given by yn = { xn if n ∈ A x if n /∈ A is weak K-convergent to x. Now {n ∈ N : |f(yn) − f(x)| < ϵi} = {n ∈ A : |f(yn) − f(x)| < ϵi} ∪ {n ∈ N \ A : |f(yn) − f(x)| < ϵi} = (N \ A) ∪ {n ∈ A : |f(xn) − f(x)| < ϵi} = (N \ A) ∪ (Bi ∩ A) = N \ (A \ Bi). As A \ Bi ∈ K then N \ (A \ Bi) ∈ F(K). Thus {n ∈ N : |f(yn) − f(x)| < ϵi} ∈ F(K) for each i and every f ∈ X∗. Thus {yn} is weak K-convergent to x. Hence {xn} is weak IK -convergent to x. 316 Weak and weak* IK -convergence in normed spaces Theorem 3.5. Let I, K be ideals on N. If for any sequence {xn}n∈N in X weak I-convergence implies weak IK -convergence then the condition AP(I, K) holds. Proof. Let {yn} be a sequence in X which is weak I-convergent to x, Since X is first countable and f(x) is not isolated point in R then there exists a sequence {zn} of points from X \ {x} which weak convergent to x. Let {An : n ∈ N} be a system of mutually disjoint sets from I. Let us define a sequence {xn} as xn = { zj if n ∈ Aj yn if n /∈ ∪Aj Let f ∈ X∗ be arbitrary. Now {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ⊂ {n ∈ N : |f(yn) − f(x)| ≥ ϵ} ∪ ∪nj=1Aj implies {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ∈ I. This shows that {xn} is weak I-convergent to x. By our assumption this implies {xn} is weak IK -convergent to x i.e. there exists a set M ∈ F(I) such that {xn}n∈M is weak K|M -convergent to x i.e. {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ∩ M = A ∩ M for some A ∈ K. This implies that {n ∈ N : |f(xn) − f(x)| ≥ ϵ} ∩ M ∈ K. Let us define Bi = Ai \ M we have ∪i∈NBi ⊆ N \ M ∈ I. At the same time, for the set Bi △Ai = Ai ∩M we have Ai ∩M ⊆ {n ∈ N : |f(xn)−f(x)| ≥ ϵ}∩M for any ϵ > 0. Consequently Bi △ Ai ∈ K. Hence the condition AP(I, K) holds. 4 Weak* IK-convergence In this section, Following Bala [1] and Bhardwaj et al. [9], now we intro- duce the concept of weak* IK -convergence of sequence of functionals and present some result. Definition 4.1. [9] A sequence {fn}n∈N in X∗ is said to be weak* I-convergent to f ∈ X∗ if for any ϵ > 0 and for each x ∈ X the set A(x, ϵ) = {n ∈ N : |fn(x) − f(x)| ≥ ϵ} ∈ I. In this case we write w∗-I- lim n→∞ fn = f. Definition 4.2. A sequence {fn}n∈N in X∗ is said to be weak* I∗-convergent to f ∈ X∗ if there exists a set M = {m1 < m2 < ... < mk < ...} ∈ F(I) such that lim k→∞ fmk(x) = f(x) for each x ∈ X. In this case we write w ∗-I∗- lim n→∞ fn = f. Theorem 4.1. Let X be a normed space and {fn}n∈N be a sequence in X∗. If {fn} is weak* I∗-convergent to f ∈ X∗ then it is weak* I-convergent to f. Proof. By assumption, there exists a set H ∈ I such that for M = N\H = {m1 < m2 < ... < mk < ...} we have lim k→∞ fmk(x) = f(x) for each x ∈ X. Now let ϵ > 0 and for this there exists an N(ϵ, x) ∈ N such that |fmk(x)−f(x)| < ϵ for each k > 317 A. K. Banerjee, M. Paul N(ϵ, x). Then we have {n ∈ N : |fn(x) − f(x)| ≥ ϵ} ⊂ H ∪ m1, m2, ..., mN(ϵ,x). Since I is an admissible ideal so right-hand side of the above relation belongs to I. Hence the result. Remark 4.1. We can reformulate the definition4.2 in the following way: if there exists a set M ∈ F(I) such that the sequence {gn} ∈ X∗ given by gn = { fn if n ∈ M f if n /∈ M is weak* convergent to f. Definition 4.3. Let X be a normed space with a separable dual X∗ and I, K be two ideals on N. A sequence {fn}n∈N in X∗ is said to be weak* IK -convergent to f ∈ X∗ if there exists a set M ∈ F(I) such that the sequence {gn} ∈ X∗ given by gn = { fn if n ∈ M f if n /∈ M is weak* K-convergent to f and we write w∗-IK - lim n→∞ fn = f Theorem 4.2. If I ∨ K is a non-trivial ideal on N and X is normed space with dual X∗ then weak* IK -limit of a sequence {fn}n∈N in X∗ is unique. The proof is parallel to proof of Theorem3.2 with slight modification. Theorem 4.3. Let X be a normed space. If a sequence {fk} in X∗ is weak IK - convergent to f ∈ X∗ then it is weak* IK -convergent. Proof. By our assumption, w −IK −lim fk = f then there exists a set M ∈ F(I) such that the sequence {gk} ∈ X∗ given by gk = { fk if k ∈ M f if k /∈ M is weak K-convergent to f. Then for every h ∈ X∗∗ and ϵ > 0, we have {k : |h(gk) − h(f)| ≥ ϵ} ∈ K. Let x ∈ X and Fx = C(x) where C : X → X∗∗ is the canonical mapping we have Fx(gk) = gk(x) and Fx(f) = f(x) for every x ∈ X. So in particular for each x ∈ X,{k : |Fx(gk) − Fx(f)| ≥ ϵ} ∈ K i.e. {k : |gk(x) − f(x)| ≥ ϵ} ∈ K. So the sequence {gk} is weak* K-convergent to f. Hence the result. Theorem 4.4. Let X be a reflexive normed space with dual X∗. If a sequence {fk} in X∗ is weak* IK -convergent to f ∈ X∗ then it is weak IK -convergent to f. 318 Weak and weak* IK -convergence in normed spaces Proof. By our assumption, w∗−IK −lim fk = f. So there exists a set M ∈ F(I) such that the sequence {gk} ∈ X∗ given by gk = { fk if k ∈ M f if k /∈ M is weak* K-convergent to f. Then for each x ∈ X and ϵ(> 0) the set {k ∈ N : |gk(x) − f(x)| ≥ ϵ} ∈ K. Let F ∈ X∗∗ then F = C(x0) for some x0 ∈ X where C : X → X∗∗ is the canonical mapping. We have in particular {k ∈ N : |gk(x0) − f(x0)| ≥ ϵ} ∈ K sinceF(gk) = gk(x0) and F(f) = f(x0). We have {k ∈ N : |F(gk) − F(f)| ≥ ϵ} ∈ K for each ϵ(> 0) and F ∈ X∗∗. So the sequence {gk} is weak K-convergent to f. Hence the result. 5 Weak and weak* IK-limit points In this last part, we introduce weak and weak* IK -limit points of sequences and sequence of functionals respectively. First we define weak I-limit point of a sequence. Definition 5.1. (cf. [17]) Let X be a normed space and a sequence {xn} be a sequence in X. Then y ∈ X is called an weak I-limit point of {xn} if there exists a set M /∈ I such that the sequence {yn}n∈N ∈ X defined by yn = { xn if n ∈ M y if n /∈ M is weak-convergent to y. Definition 5.2. Let X be a normed space and I, K be two ideals on N. Then y ∈ X is called an weak IK -limit point of a sequence {xn} if there exists a set M /∈ I, K such that the sequence {yn}n∈N ∈ X defined by yn = { xn if n ∈ M y if n /∈ M is weak K-convergent to y. We denote I(Lw) and IK(Lw) the collection of all weak I and weak IK -limit points of xn ∈ X. Theorem 5.1. If K is an admissible ideal and K ⊂ I then I(Lw) ⊂ IK(Lw). 319 A. K. Banerjee, M. Paul Proof. Let y ∈ I(Lw), so there exists a set M /∈ I such that the sequence {yn} give by yn = { xn if n ∈ M y if n /∈ M is weak-convergent to y. Then the sequence of scalars {f(yn)} converges to f(y) for all f ∈ X∗ i.e. {n : |f(yn)−f(y)| ≥ ϵ} is a finite set. So {n : |f(yn)−f(y)| ≥ ϵ} ∈ K as K is an admissible ideal. Therefore {yn} is weak K-convergent se- quence. Again M /∈ I and K ⊂ I, so M /∈ I, K. Thus y is weak IK -limit point of xn. Hence the theorem. In the similar way we can set the definition of weak* IK -limit points for the sequence of functionals. Definition 5.3. Let X be a normed space with its dual X∗ and {fn} be a sequence in X∗. Then h ∈ X∗ is called an weak* I-limit point of {fn} if there exists a set M /∈ I such that the sequence {gn}n∈N ∈ X∗ defined by gn = { fn if n ∈ M h if n /∈ M is weak*-convergent to h. Definition 5.4. Let X be a normed space with its dual X∗ and I, K be two ideals on N. Then h ∈ X∗ is called an weak* IK -limit point of {fn} ⊂ X∗ if there exists a set M /∈ I, K such that the sequence {gn}n∈N ∈ X∗ defined by gn = { fn if n ∈ M h if n /∈ M is weak* K-convergent to h. We denote I(Lw∗) and IK(Lw∗) the collection of all weak* I and IK -limit points of the sequence fn ∈ X∗. Theorem 5.2. If K is an admissible ideal and K ⊂ I then I(Lw∗) ⊂ IK(Lw∗). The proof is parallel to proof of the Theorem 5.1. Theorem 5.3. Let X be a normed space with its dual X∗ . If h ∈ X∗ be weak IK -limit point of a sequence {fn} ⊂ X∗ then h is also weak* IK -limit point. Proof. Let y be weak IK -limit point of {fn} ∈ X∗ then there exists a set M /∈ I such that the sequence {gn}n∈N ∈ X∗ defined by gn = { fn if n ∈ M h if n /∈ M is weak K-convergent to y. Again by Theorem 4.3 we get {gn} is weak* K- convergent to h. Hence h is weak* IK -limit point. 320 Weak and weak* IK -convergence in normed spaces Remark 5.1. By the theorem 4.4 we get weak* K-convergence implies weak K- convergence when X is reflexive normed space. Therefore converse of above the- orem holds when X is a reflexive normed space. 6 Conclusions The study of weak and weak* convergence plays an important role in func- tional analysis. So investigations on weak and weak* convergence using single ideal or double ideals take place a great position in the study of summability the- ory. In [9] and [1] weak and weak* convergence have been studied using single ideal respectively. So it quite natural to investigate whether such these results hold for double ideals. So it would be worthwhile to consider extending the notion of weak and weak* convergence by incorporating double ideals in a normed space, and to examine how various fundamental properties are impacted as a result. In case of double ideals, weak I-convergence implies weak IK -convergence if the modified AP condition i.e. AP(I, K) condition holds where as in case of single ideal AP condition is sufficient to hold this result. It is observed citing suitable ex- ample that weak IK -convergence may not imply weak I-convergence in general although weak I∗-convergence always imply weak I-convergence. Besides, it is verified that where X is a reflexive Banach space the idea of weak IK -convergence and weak* IK -convergence coincide. Due to the inherent importance of concept of weak and weak* convergence in function analysis, we recognize that the con- cept of weak and weak* convergence via double ideals in normed spaces would provide a more comprehensive framework for function analysis. 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