Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 754 https://internationalpubls.com Bicomplex Sequence Spaces: Duality via Idempotent Decomposition Mamta Amol Wagh Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi Email – mamtanigam@ddu.du.ac.in Article History: Received: 14-02-2024 Revised: 15-03-2024 Accepted: 21-04-2024 Abstract: This paper investigates the duals of some bicomplex sequence spaces corresponding to bicomplex functions that are holomorphic in the bicomplex space ℂ2, or entire bicomplex sequence spaces. We investigate these spaces through their idempotent decompositions and examine the β-dual, γ-dual, and δ-dual. Precise definitions and analyses of these duals are presented. Our results demonstrate that the duals of the original sequence spaces are strictly contained within the duals of their corresponding idempotent subclasses. These findings are also discussed in the context of algebra homomorphisms between the original sequence space ℵ and its idempotent subclasses 1 ℵ and 2 ℵ. Keywords: bicomplex numbers, entire bicomplex sequence spaces, idempotent sequence spaces, Köthe – Toeplitz duals 2020 Mathematics Subject Classification: 46E10, 46E15, 46E25 1. Introduction 1.1 DUALS OF SEQUENCE SPACES There are two primary types of duals associated with a sequence space: the algebraic dual and the topological dual. The algebraic dual of a linear space V is the set of all linear functionals from V to a scalar field K, and is denoted by L(V, K) = V#. On the other hand, the topological dual consists of all continuous linear functionals on V and is denoted by V*. The only sequence space with a well-behaved algebraic dual consisting of sequences is ϕ, whose dual is ω. Therefore, in duality theory, it is more effective to study sequence spaces with linear topologies, though finding their topological duals is often challenging. To address this, Köthe and Toeplitz [3] introduced the α-dual and β-dual, which facilitate a more practical dual system. Later, Garling [1] proposed the more general γ-dual, and for symmetric sequence spaces, the δ-dual was introduced by Garling [2] and Ruckle [5]. 1.2 BICOMPLEX SPACE ℂ2 Bicomplex Numbers were defined by Corrado Segre (1860 – 1924) in 1892. Infinite set of algebras and the concept of multicomplex numbers was given in [6]. The set of bicomplex numbers is given by ℂ2 = {𝜇1 + 𝑖1𝜇2 + 𝑖2𝜇3 + 𝑖1𝑖2𝜇4: 𝜇1, 𝜇2, 𝜇3, 𝜇4 ∈ ℂ0}, where 𝑖1 2 = 𝑖2 2 = −1 , 𝑖1𝑖2 = 𝑖2𝑖1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 755 https://internationalpubls.com The binary operations of addition and scalar multiplication on ℂ2 are defined coordinate-wise, and multiplication is defined component-wise (i.e., term by term). With these operations, ℂ2 forms a commutative algebra with identity. There are several notable differences between the algebraic structures of ℂ2 and ℂ1, as outlined by Price in [4]. Although bicomplex numbers, like quaternions, form a four-dimensional algebra, they differ in that bicomplex numbers are commutative, whereas quaternions are not. Idempotent Elements – Apart from 0 and 1, the structure contains two distinct nontrivial idempotent elements given by 𝑒1 = 1+𝑖1𝑖2 2 and 𝑒2 = 1−𝑖1𝑖2 2 . The addition of these two idempotent elements is 1 and their product is zero. There are two principal ideals generated by these idempotent elements. Intersection of these ideals is zero and their union is the set of all singular elements of ℂ2. Two bicomplex numbers are zero divisors precisely when one is a complex multiple of one idempotent element and the other is a complex multiple of the other idempotent element. The detailed study of ℂ2 is provided in [13]. 2. Objectives The objective of this paper is to investigate the β-, γ-, and δ-duals of certain classes of bicomplex sequences, and to explore their relationships with the duals of corresponding idempotent subclasses, supported by illustrative examples and counterexamples 3. Methods Idempotent technique has been used to investigate the duals of bicomplex sequence spaces and their subclasses. 4. Results BICOMPLEX KÖTHE – TOEPLITZ DUALS If ω’ is the family of all bicomplex sequences 𝜉 = (𝜉𝑘) with 𝜉𝑘 ∈ ℂ2 , k ≥ 1, where ℂ2 is the space of all bicomplex numbers. If 𝜓 be a bicomplex sequence space, then we denote α –, β –, γ –, and δ – duals of 𝜓 by 𝜓𝛼, 𝜓𝛽, 𝜓𝛾 and 𝜓𝛿 respectively. These duals have been defined in [10]. Let us see these definitions for our ready reference. 𝜓𝛼 = {𝜉: 𝜉 ∈ 𝜔′, ∑‖𝜉𝑖𝜂𝑖‖ < ∞, ∀ 𝜂𝑖 ∈ 𝜆 𝑖≥1 } 𝜓𝛽 = {𝜉: 𝜉 ∈ 𝜔′, ‖∑ 𝜉𝑖𝜂𝑖𝑖≥1 ‖ < ∞, ∀ 𝜂𝑖 ∈ 𝜆} 𝜓𝛾 = {𝜉: 𝜉 ∈ 𝜔′, 𝑠𝑢𝑝 𝑛 ‖∑ 𝜉𝑖𝜂𝑖 𝑛 𝑖=1 ‖ < ∞, ∀ 𝜂𝑖 ∈ 𝜆} 𝜆𝛿 = {𝜉: 𝜉 ∈ 𝜔′, ∑ ‖𝜉𝑖𝜂𝜌(𝑖)‖ < ∞, ∀𝜂𝑖 ∈ 𝜆𝑖≥1 𝑎𝑛𝑑𝜌 ∈ 𝜋} where 𝜋 is the set of all permutations of ℕ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 756 https://internationalpubls.com 4.1 BICOMPLEX SEQUENC SPACES Now, let us first describe the classes of sequences whose duals we aim to study: (i) ℵ = {𝒻: 𝒻 = {𝒳𝒿} = {1𝒳𝒿. 𝑒1 + 2𝒳𝒿. 𝑒2} : 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |1𝒳𝒿 | ℂ1 < ∞, 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |2𝒳𝒿 | ℂ1 < ∞} (ii) 1ℵ = {𝒻: 𝒻 = {1𝒳𝒿. 𝑒1} : 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |1𝒳𝒿 | ℂ1 < ∞} (iii) 2ℵ = {𝒻: 𝒻 = {2𝒳𝒿. 𝑒1} : 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |2𝒳𝒿 | ℂ1 < ∞} The class ℵ given in (i) has been studied in [8] by Srivastava & Srivastava and the subspaces given in (ii) and (iii) have been studied in [9] by Wagh. The subclass in (ii) have been studied with a functional analytic viewpoint in [11] by Wagh. These subclasses are the subspaces of our space ℵ in the sense that they are formed by idempotent sequences of ℵ. In the above spaces, the notation |. |ℂ1 represents the complex norm. For any bicomplex sequence {𝒳𝒿} = {1𝒳𝒿. 𝑒1 + 2𝒳𝒿. 𝑒2} , the following two conditions (iv) and (v) are equivalent: (iv) 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |1𝒳𝒿 | ℂ1 < ∞ and 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿 |2𝒳𝒿 | ℂ1 < ∞ (v) 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿‖𝒳𝒿‖ ℂ2 < ∞ Where ‖ ‖ℂ2 represents the bicomplex norm, given by ‖𝜍‖ℂ2 = { |1𝜍| ℂ1 2 +|2𝜍| ℂ1 2 2 } 1/2 , where (vi) 𝜍 = 𝑢1 + 𝑖2𝑢2 = (𝑢1 − 𝑖1𝑢2)𝑒1 + (𝑢1 + 𝑖1𝑢2)𝑒2 = 1𝜍𝑒1 + 2𝜍𝑒2 ∈ ℂ2, 1𝜍 , 2𝜍 ∈ ℂ1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 757 https://internationalpubls.com 1𝜍 and 2𝜍 are first idempotent component and second idempotent component of 𝜍 respectively. The idempotent representation given in (vi) is unique and was given by Srivastava in [7]. Thus, the class in (i) has an equivalent representation given by ℵ = {𝒻: 𝒻 = {𝒳𝒿}: 𝑠𝑢𝑝 𝒿≥1 𝒿𝒿‖𝒳𝒿‖ ℂ2 < ∞} (𝜉𝑗) ∈ ℵ can be written as 𝜉𝑗 = 𝑧1𝑗 + 𝑖2𝑧2𝑗 = 𝛽1𝑗𝑒1 + 𝛽2𝑗𝑒2, where, 𝛽1𝑗 = 𝑧1𝑗 − 𝑖1𝑧2𝑗, 𝛽2𝑗 = 𝑧1𝑗 + 𝑖1𝑧2𝑗 (𝛽1𝑗) and (𝛽2𝑗) are complex sequences i.e., 𝛽1𝑗 , 𝛽2𝑗 ∈ ℂ1(𝑖1) . 4.2 ALGEBRA HOMOMORPHISM BETWEEN ℵ AND ITS SUBCLASSES Algebra homomorphism (denoted by T1 and T2) between ℵ and its subclasses 1ℵ and 2ℵ have been investigated in [10] given by: 𝑇1: ℵ → 1ℵ as 𝑇1(𝑓) = 𝑇1({𝜉𝑗}) = {1𝜉𝑗 𝑒1} ∈ 1ℵ and 𝑇2: ℵ → 2ℵ as 𝑇2(𝑓) = 𝑇2({𝜉𝑗}) = {2𝜉𝑗 𝑒2} ∈ 2ℵ , {𝜉𝑘} ∈ 𝐵 Its pictorial representation is given below Thus we can say that (𝜂𝑗) ∈ ℵ𝛼 if and only if 𝑇1(𝜂𝑗) ∈ (1ℵ)𝛼, 𝑇2(𝜂𝑗) ∈ (2ℵ)𝛼 T1 ℵ 2ℵ 1ℵ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 758 https://internationalpubls.com Or in words we can also say that a sequence belongs to 𝛼 - dual of the class ℵ if and only if its T1 – image belongs to 𝛼 - dual of the first idempotent component of ℵ or its first subclass and the T2 – image belongs to the 𝛼 - dual of its second idempotent component or its second subclass. This is to be noted that the 𝛼 – dual of the class ℵ has been studied in [12]. In this paper, we are going to analyze other duals of these spaces. β – dual of the class ℵ and its subclasses 1ℵ and 2ℵ are given by ℵ𝛽 = {(𝜂𝑗) ∈ 𝜔′: ‖∑ 𝜂𝑗𝜉𝑗 𝑗≥1 ‖ < ∞, ∀(𝜉𝑗) ∈ ℵ} Or {(𝜂𝑗) ∈ 𝜔′: |∑ 1𝜂𝑗 1𝜉𝑗 𝑗≥1 | < ∞, |∑ 2𝜂𝑗 2𝜉𝑗 𝑗≥1 | < ∞, ∀(𝜉𝑗) ∈ ℵ} (1ℵ)𝛽 = {(𝜂𝑗) ∈ 𝜔′: |∑ 1𝜂𝑗 1𝜉𝑗 𝑗≥1 | < ∞, ∀(𝜉𝑗) ∈ ℵ} (2ℵ)𝛽 = {(𝜂𝑗) ∈ 𝜔′: |∑ 2𝜂𝑗 2𝜉𝑗 𝑗≥1 | < ∞, ∀(𝜉𝑗) ∈ ℵ} Theorem 4.1: (i) ℵ𝛽 ⊂ (1ℵ)𝛽. (ii) ℵ𝛽 ⊂ (2ℵ)𝛽 Proof: (i) We know 1ℵ ⊆ ℵ and ℵ𝛽 ⊆ (1ℵ)𝛽 (since dual of a set is contained in the dual of its subset) To establish that the inclusion is proper, we construct a sequence contained in the β – dual of 1ℵ which does not belong to β – dual of ℵ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 759 https://internationalpubls.com Consider the sequence, (𝜂𝑗) = ( 1 𝑗3 𝑒1 + 𝑗3+𝑗𝑒2). First we show that(𝜂𝑗) ∈ (1ℵ)𝛽. Let (1𝜉𝑗 𝑒1) ∈ 1ℵ ⇒ 𝑠𝑢𝑝 𝑗 𝑗𝑗 |1𝜉𝑗 | < ∞ ⇒ 𝑗𝑗 |1𝜉𝑗 | < 𝑀, ∀𝑗 ≥ 1 for some M. ⇒ |1𝜉𝑗 | < 𝑀 𝑗𝑗 , ∀𝑗 ≥ 1 (I) 𝑗𝑗 > 𝑗3, ∀𝑗 ≥ 3 ⇒ 1 𝑗𝑗 < 1 𝑗3 , ∀𝑗 ≥ 3 and ∑ 1 𝑗3 is convergent ⇒ ∑ 1 𝑗𝑗 is convergent, by comparison test. ∑ 𝑀 𝑗𝑗 is a convergent series. (II) ∴ from (I) and (II) ∑ |1𝜉𝑗 |is convergent. And |∑ 1𝜉𝑗 | ≤ ∑ |1𝜉𝑗 | < ∞ In (𝜂𝑗) = ( 1 𝑗3 𝑒1 + 𝑗3+𝑗𝑒2), |∑ 1𝜂𝑗 | ≤ ∑ |1𝜂𝑗 | = ∑ 1 𝑗3 < ∞ and|∑ 1𝜉𝑗 | < ∞, The term-by-term product of two convergent series is itself convergent, Therefore |∑ 1 𝑗3 1 𝜉𝑗| < ∞, ∀ (1𝜉𝑗 𝑒1) ∈ 1ℵ. Thus(𝜂𝑗) ∈ (1ℵ)𝛽. (a) To show that(𝜂𝑗) ∉ ℵ𝛽, we must show that for some element(𝜉𝑗) ∈ ℵ, ∑ 𝜉𝑗𝜂𝑗 is not convergent. Consider the sequence(𝜉𝑗) = 𝑗−𝑗 ( 1 𝑗2 𝑒1 + 1 𝑗3 𝑒2) = (𝑗−𝑗−2𝑒1 + 𝑗−𝑗−3𝑒2). Note first that, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 760 https://internationalpubls.com 𝑠𝑢𝑝 𝑗 𝑗𝑗‖𝜉𝑗‖ = 𝑠𝑢𝑝 𝑗 𝑗𝑗 ( |1𝜉𝑗 | 2 + |2𝜉𝑗 | 2 2 ) 1/2 = 𝑠𝑢𝑝 𝑗 𝑗𝑗 ( |𝑗−2𝑗−4|+|𝑗−2𝑗−6| 2 ) 1/2 = 𝑠𝑢𝑝 𝑗 { 1 2 ( 1 𝑗4 + 1 𝑗6)} 1/2 < ∞ So that(𝜉𝑗) ∈ ℵ. Now, ‖∑ 𝜉𝑗𝜂𝑗𝑗≥1 ‖ ≤ ∑ ‖𝜉𝑗𝜂𝑗‖ = ∑ ‖( 1 𝑗2+𝑗 𝑒1 + 1 𝑗3+𝑗 𝑒2) . ( 1 𝑗2 𝑒1 + 𝑗2+𝑗𝑒2)‖𝑗≥1𝑗≥1 = ∑ ‖( 1 𝑗4+𝑗 𝑒1 + 1 𝑗 𝑒2)‖𝑗≥1 𝑗4+𝑗 > 𝑗 ⇒ 1 𝑗4+𝑗 < 1 𝑗4 and ∑ | 1 𝑗4 |𝑗≥1 is convergent therefore by comparison test ∑ | 1 𝑗4+𝑗 |𝑗≥1 is also convergent, but ∑ | 1 𝑗4 |𝑗≥1 is not convergent. Therefore, (𝜂𝑗) ∉ ℵ𝛽 (b) Hence from (a) and (b) we get (𝜂𝑗) ∈ (1ℵ)𝛽 but(𝜂𝑗) ∉ ℵ𝛽. Note 1: Beta dual of the class ℵ is properly contained in the beta dual of its T1 image. Next, gamma dual of the class ℵ and its subclasses are given by ℵ𝛾 = {(𝜂𝑗) ∈ 𝜔′: 𝑠𝑢𝑝 𝑛 ‖∑ 𝜂𝑗𝜉𝑗 𝑛 𝑗=1 ‖ < ∞, ∀(𝜉𝑗) ∈ ℵ} (III) Or {(𝜂𝑗) ∈ 𝜔′: 𝑠𝑢𝑝 𝑛 |∑ 1𝜂𝑗 1𝜉𝑗 𝑛 𝑗=1 | < ∞, 𝑠𝑢𝑝 𝑛 |∑ 1𝜂𝑗 2𝜉𝑗 𝑛 𝑗=1 | < ∞, ∀(𝜉𝑗) ∈ ℵ}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 761 https://internationalpubls.com (1ℵ)𝛾 = {(𝜂𝑗) ∈ 𝜔′: 𝑠𝑢𝑝 𝑛 |∑ 1𝜂𝑗 1𝜉𝑗 𝑛 𝑗=1 | < ∞, ∀(𝜉𝑗) ∈ ℵ} (IV) (2ℵ)𝛾 = {(𝜂𝑗) ∈ 𝜔′: 𝑠𝑢𝑝 𝑛 |∑ 2𝜂𝑗 2𝜉𝑗 𝑛 𝑗=1 | < ∞, ∀(𝜉𝑗) ∈ ℵ} (V) Theorem 4.2: A sequence (𝜂𝑘) belongs to 𝛾 - dual of the class ℵ if and only if its first idempotent sequence belongs to 𝛾 - dual of the class 1ℵ and the second idempotent sequence belongs to 𝛾 - dual of the class 2ℵ that is (𝜂𝑗) ∈ 𝐵𝛾 ⇔ (1𝜂𝑗 𝑒1) ∈ (1ℵ)𝛾 𝑎𝑛𝑑 (2𝜂𝑗 𝑒2) ∈ (2ℵ)𝛾. Proof: Let (𝜂𝑗) ∈ ℵ𝛾 be any sequence. By (III), (𝜂𝑗) ∈ 𝐵𝛾 ⇔ 𝑠𝑢𝑝 𝑛 ‖∑ 𝜂𝑗𝜉𝑗 𝑛 𝑗=1 ‖ < ∞, ∀(𝜉𝑗) ∈ ℵ ⇔ {𝑠𝑢𝑝 𝑛 |∑ 1𝜂𝑗 1𝜉𝑗 𝑛 𝑗=1 |} 𝑒1 + {𝑠𝑢𝑝 𝑛 |∑ 2𝜂𝑗 2𝜉𝑗 𝑛 𝑗=1 |} 𝑒2 < ∞, ∀(𝜉𝑗) ∈ ℵ ⇔ 𝑠𝑢𝑝 𝑛 |∑ 1𝑗1𝜉𝑗 𝑛 𝑗=1 | < ∞ and 𝑠𝑢𝑝 𝑛 |∑ 2𝜂𝑗 2𝜉𝑗 𝑛 𝑗=1 | < ∞, ∀(𝜉𝑗) ∈ ℵ Hence, (𝜂𝑗) ∈ ℵ𝛾 ⇔ (1𝜂𝑗𝑒1) ∈ (1ℵ)𝛾 and (2𝜂𝑗𝑒2) ∈ (2ℵ)𝛾, by (III) and (IV). (c) Corollary 1: Thus from (b) and (c) we can say that (𝜂𝑗) ∈ ℵ𝛾if and only if 𝑇1(𝜂𝑗) ∈ (1ℵ), 𝑇2(𝜂𝑗) ∈ (2ℵ)𝛾 Or in words we can also say that a sequence belongs to 𝛾 - dual of the class ℵ if and only if its T1 – image belongs to 𝛾 - dual of the first idempotent component of ℵ or its first subclass and the T2 – image belongs to the 𝛾 - dual of its second idempotent component or its second subclass. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 762 https://internationalpubls.com Theorem 4.3: (i)ℵ𝛾 ⊂ (1ℵ)𝛾. (ii) ℵ𝛾 ⊂ (2ℵ)𝛾 Proof: Similar example may be considered. Note 2: Gamma dual of the class ℵ is properly contained in the gamma dual of its idempotent parts/ T1\] ,and T2 – images. δ – dual of the class ℵ and its subclasses ℵ𝛿 = {(𝜂𝑗) ∈ 𝜔′: ∑ ‖𝜂𝑗𝜉𝜌(𝑗)‖ < ∞, ∀(𝜉𝑗) ∈ ℵ 𝑗≥1 , 𝜌 ∈ 𝜋} (VI) = {(𝜂𝑗) ∈ 𝜔′: ∑ |1𝜂𝑗 1𝜉𝜌(𝑗) | < ∞, 𝑗≥1 ∑ |2𝜂𝑗 2𝜉𝜌(𝑗) | < ∞, 𝑗≥1 ∀(𝜉𝑗) ∈ ℵ , 𝜌 ∈ 𝜋} (1ℵ)𝛿 = {(𝜂𝑗) ∈ 𝜔′: ∑ |1𝜂𝑗 1𝜉𝜌(𝑗)| < ∞, ∀(𝜉𝑗) ∈ ℵ𝑗≥1 𝑎𝑛𝑑 𝜌 ∈ 𝜋} (VII) (2ℵ)𝛿 = {(𝜂𝑗) ∈ 𝜔′: ∑ |2𝜂𝑗 2𝜉𝜌(𝑗)| < ∞, ∀(𝜉𝑗) ∈ ℵ𝑗≥1 𝑎𝑛𝑑 𝜌 ∈ 𝜋} (VIII) Theorem 4.4: A sequence (𝜂𝑗) belongs to 𝛿 - dual of the class ℵ if and only if its first idempotent sequence belongs to 𝛿 - dual of the class 1ℵ and the second idempotent sequence belongs to 𝛿 - dual of the class 2ℵ, that is (𝜂𝑗) ∈ 𝐵𝛿 ⇔ (1𝜂𝑗𝑒1) ∈ (1ℵ)𝛿 and (2𝜂𝑗𝑒2) ∈ (2ℵ)𝛿 Proof: Let (𝜂𝑘) ∈ 𝐵𝛿be any sequence. From (VI), (𝜂𝑗) ∈ ℵ𝛿 ⇔ ∑‖𝜂𝑗𝜉𝜌(𝑗)‖ < ∞, ∀(𝜉𝑗) ∈ ℵ 𝑗≥1 and 𝜌 ∈ 𝜋 ⇔ {{∑ |1𝜂𝑗 1𝜉𝜌(𝑗)|𝑗≥1 }𝑒1 + {∑ |2𝜂𝑗 2𝜉𝜌(𝑗)|𝑗≥1 }𝑒2} < ∞, ∀(𝜉𝑗) ∈ ℵ ⇔ ∑ |1𝜂𝑗 1𝜉𝜌(𝑗)| < ∞𝑗≥1 and ∑ |2𝜂𝑗 2𝜉𝜌(𝑗)| < ∞𝑗≥1 , ∀(𝜉𝑗) ∈ ℵ Hence, (𝜂𝑗) ∈ ℵ𝛿 ⇔ (1𝜂𝑗𝑒1) ∈ (1ℵ)𝛿 and (2𝜂𝑗𝑒2) ∈ (2ℵ)𝛿, by (VII) and (VIII). (d) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 763 https://internationalpubls.com Corollary 2: From (d) and the definitions of T1 and T2 we can say that (𝜂𝑗) ∈ ℵ𝛿 if and only if 𝑇1(𝜂𝑗) ∈ (1ℵ), 𝑇2(𝜂𝑗) ∈ (2ℵ) Or in words we can also say that a sequence belongs to 𝛿 - dual of the Class ℵ if and only if its T1 – image belongs to 𝛿 - dual of the first idempotent component of ℵ or its first subclass and the T2 – image belongs to the 𝛿 - dual of its second idempotent component or its second subclass. Theorem 4.5: (i)ℵ𝛿 ⊂ (1ℵ)𝛿. (ii) ℵ𝛿 ⊂ (2ℵ)𝛿 Proof: Can be shown easily with the help of similar example. Note 3: Delta dual of the class ℵ is properly contained in the delta dual of its idempotent parts/T1 and T2 images. Constructing other counter – examples We can also construct other counter – examples. For instance, for part (i) of the theorems 4.1, 4.3 and 4.5, we can take (𝜉𝑗) = 𝑗−𝑗( 1 𝑗3 𝑒1 + 1 𝑗4 𝑒2) = 1 𝑗3+𝑗 𝑒1 + 1 𝑗4+𝑗 𝑒2 , (𝜂𝑗) = ( 1 𝑗2 𝑒1 + 𝑗𝑗+3𝑒2) For part (ii), take (𝜂𝑗) = (𝑗𝑗+2𝑒1 + 1 𝑗2 𝑒2) or (𝑗𝑗+2𝑒1 + 1 𝑗3 𝑒2) Generalized counter – example, for part (i) of the theorems 4.1, 4.3 and 4.5, take (𝜉𝑗) = 𝑗−𝑗( 1 𝑗𝑛 𝑒1 + 1 𝑗𝑛+1 𝑒2) = 1 𝑗𝑛+𝑗 𝑒1 + 1 𝑗𝑛+1+𝑗 𝑒2, 𝑛 ≥ 2 (𝜂𝑗) = ( 1 𝑗𝑛−1 𝑒1 + 𝑗𝑗+𝑛𝑒2) For part (ii), take (𝜉𝑗) = 𝑗−𝑗( 1 𝑗𝑛 𝑒1 + 1 𝑗𝑛+1 𝑒2) = 1 𝑗𝑛+𝑗 𝑒1 + 1 𝑗𝑛+1+𝑗 𝑒2, 𝑛 ≥ 2 (𝜂𝑗) = (𝑗𝑗+𝑛−1𝑒1 + 1 𝑗𝑛−2 𝑒2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 764 https://internationalpubls.com Conclusion: we have studied beta, gamma and delta duals of the idempotent subspaces of an entire bicomplex sequence space and shown that these duals of idempotent sequences are properly contained in the dual of our original space. We have also given the generalized example for this proper containment. Conflict of interest: there is no conflict of interest. References [1] Garling, D.G.H., “The β - and γ - duality”, Proc. Cambridge Philos. Soc. ,1967, 963 – 981. [2] Garling, D.G.H, “On symmetric sequence spaces”, Proc. London Math. Soc. 16 (3), 1966, 85 – 106. [3] Köthe, G & Toeplitz, O., “Lineare Raume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen”, Jour. Reine angew. Math. 171, 1934, 193 – 226. [4] Price, G. Baley, “An Introduction to Multicomplex Spaces And functions”, Marcel Dekker, Inc., 1991. 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