IHJPAS. 36(1)2023 311 This work is licensed under a Creative Commons Attribution 4.0 International License The Completion of Generalized 2-Inner Product Spaces Abstract A complete metric space is a well-known concept. Kreyszig shows that every non-complete metric space 𝑊 can be developed into a complete metric space Ŵ, referred to as completion of 𝑊. We use the b-Cauchy sequence to form �̂� which “is the set of all b-Cauchy sequences equivalence classes”. After that, we prove �̂� to be a 2-normed space. Then, we construct an isometric by defining the function from W to Ŵ0; thus Ŵ0 and W are isometric, where Ŵ0 is the subset of Ŵ composed of the equivalence classes that contains constant b-Cauchy sequences. Finally, we prove that �̂�0 is dense in �̂�, �̂� is complete and the uniqueness of �̂� is up to isometrics. Keywords: b-Cauchy Sequence, Equivalent Class, Metric space, Completion Generalized 2-Inner Product Space. 1. Introduction Cho and Freese [3-4] introduced 2-normed space by: Let W be a real linear space with a dimension greater than 1. Suppose that ‖, ‖ is a real-valued function on W × W for all w, y, z in W and α ∈ ℝ satisfying the following requirements: 1. ‖w, y‖ = 0 if and only if w and y are linearly dependent. 2. ‖w, y‖ = ‖y, w‖ 3. ‖αw, y‖ = |α|‖w, y‖ 4. ‖w + y, z‖ ≤ ‖w, z‖ + ‖y, z‖ Then ‖, ‖ is called a 2-norm on 𝑊 and the pair (𝑊, ‖, ‖) is called a linear 2-normed space or 2-normed space. For more details, see [11-12] doi.org/10.30526/36.1.2952 Article history: Received 14 Augest 2022, Accepted 11 September 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Safa L. Hamad Department of Mathematics , College of Sciences, University of Baghdad- Iraq. safalafta2019@gmail.com Zeana Z. Jamil Department of Mathematics , College of Sciences, University of Baghdad- Iraq. zina.z@sc.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.30526/36.1.3212 mailto:safalafta2019@gmail.com mailto:zina.z@sc.uobaghdad.edu.iq IHJPAS. 36(1)2023 312 Riys and Ravinderan [10] defines the generalized 2-inner product space as a complex vector space W, called a generalized 2-inner product space if there exists a complex valued function 〈(,),(,)〉 on W2 × W2 such that a, b, c, d ∈ W, and α ∈ C, as the following: 1. 〈(a, b), (c, d)〉 = 〈(c, d), (a, b)〉̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ 2. If a and b are linear independent in W, then 〈(a, b), (c, d)〉 > 0. 3. 〈(a, b), (c, d)〉 = −〈(b, a), (c, d)〉. 4. 〈(αa + e, b), (c, d)〉 = α〈(a, b), (c, d)〉 + 〈(e, b), (c, d)〉. For more details, see [8][2][5]. Ghafoor [6], shows that the generalized 2-inner product space is a 2-normed space with ‖w, y‖ = 〈(w, y), (w, y)〉 1 2. After many failures to define orthogonal vectors in 2-normed space, Riyas and Ravindran, in 2014, [10] defined orthogonal vectors in a 2-normed space by restriction space W × W to W × 〈b〉, where < b > is a non-zero subspace in W. Thus, the domain of the generalized 2-inner product restriction with space W2 × 〈b〉2. It is well-known that there are incomplete metric spaces. Kreyszig, in 1978 [7] discussed the strategy for completing every metric space by defining an equivalent relation on Cauchy sequences and the metric on it. In 2001, Cho and Freese [3] used the same strategy for the completion of a 2- normed space, but some difficulties appeared when defining a metric on it, and then, he had to give another condition on the space. Despite that all generalized 2-inner product space is 2-normed space, but in this paper, we give a completion of the generalized 2-inner product space without need any condition using the b-Cauchy sequences. This paper includes two sections. The first section discusses some of the properties of the b- Cauchy sequence in a generalized 2-inner product space. The second section proves the completion of the generalized 2-inner product. We will abbreviate ‖w, b‖by ‖w‖b in the sequel. 2. b-Cauchy sequences. This section discusses some of the properties of the b-Cauchy sequence in a generalized 2- inner product space. Mazaheri and Kazemi in [9] introduce a b-Cauchy sequence concept as follows Definition (2.1)[9]: Let 𝑊 be a generalized 2-inner product space, 0 ≠ 𝑏 ∈ 𝑊, {𝑤𝑛} be a sequence in 𝑊, then, it is called a b-Cauchy sequence if lim 𝑛,𝑚→∞ ‖𝑤𝑛 − 𝑤𝑚, 𝑏‖ = 0. The following definition has been devised from [9] Definition (2.2): An open ball of radius 𝑟 and centered at y in a generalized 2-inner product space W is defined as: Br(y) ≔ {w ∈ W: ‖w − y‖b < r}. The following proposition is a characterization of b-Cauchy sequences in a generalized 2- inner product space. But first, we define a neighborhood of 0. Definition (2.3): If w is a point in a generalized 2-inner product space W, then a neighborhood U of w is a set containing Br(w) for some r > 0, i.e, there exists r > 0, such that w ∈ Br(w) ⊂ U. IHJPAS. 36(1)2023 313 Proposition (2.4): Let 𝑊 be a generalized 2-inner product space, {wn} is a b-Cauchy sequence in W if and only if for any neighborhood U of 0; there is an integer M(U) such that for all n, m ≥ M(U) implies that wn − wm ∈ U. Proof: Let U be a neighborhood of 0, then there exists ε > 0 such that B0(ε) ⊆ U. Since {wn} is a b-Cauchy sequence in W, thus 𝑙𝑖𝑚 𝑛,𝑚→∞ ‖𝑤𝑛 − 𝑤𝑚‖𝑏 = 0 . It implies that there exists M(ε) > 0 such that ‖wn − wm‖b < ε; n, m ≥ M(ε). Then, wn − wm ∈ B0(ε) ⊆ U. Conversely, let {wn} be a sequence in W such that for every neighborhood U of 0 there is an integer M(U) > 0; wn − wm ∈ U where n, m ≥ M(U). Then, there exists δ(U) > 0 such that ‖wn − wm‖b < δ, where n, m ≥ M(U). It implies that for all ε > 0, there exists M(ε) such that ‖wn − wm‖b < ε for n, m ≥ M(ε), then , 𝑙𝑖𝑚 𝑛,𝑚→∞ ‖𝑤𝑛 − 𝑤𝑚‖𝑏 = 0 . Thus, {wn} is a b-Cauchy sequence in W.∎ 3. Completion of the generalized 2-inner product spaces. Kreyszig [1] states few steps to prove that an arbitrary incomplete metric space can be completed. In this section, we follow Kreyszig strategy to prove the completion of the generalized 2-inner product space: Step1: Forming �̂� is the set of all b-Cauchy sequence equivalence classes. Definition (3.1): Two b-Cauchy sequences {wn} and {yn} in a generalized 2-inner product space W have a relation denoted by {wn}~{yn}, if for every neighborhood U of 0 there is an integer M(U) such that n ≥ M(U) implies that wn − yn ∈ U. It is clear that ~ is a reflexive and symmetric relation. The following proposition shows that this relation is equivalent. Proposition (3.2): The relation ~ on the set of b-Cauchy sequences in W is an equivalent relation on W. Proof: Let {wn}~{yn} and {yn}~{zn}. Let, U is an arbitrary neighborhood of 0. There exists a neighborhood V of 0 such that V + V ⊂ U. By Definition (2.1) and for this V, there exists an integer M such that wn − yn, yn − zn ∈ V for n ≥ M. Hence , wn − zn = (wn − yn) + (yn − zn) is an element of U for n ≥ M. Therefore, {wn}~{zn}.∎ Define Ŵ: ={ŵ: ŵ is equivalent class of b-Cauchy sequences}. Step 2: Proof �̂� vector space. Let ŵ,ŷ in Ŵ. Define the terms addition and scalar multiplication. On Ŵ where {wn} ∈ ŵ and {yn} ∈ ŷ , as shown below: • �̂� + �̂� = {𝑤𝑛 + 𝑦𝑛} • 𝛼ŵ = {𝛼wn} IHJPAS. 36(1)2023 314 The following proposition explains that the two operations defined on Ŵ are well-defined because they are unaffected by the elements chosen from {ŵn} and {ŷn}. But first, we need the following proposition: Proposition (3.3): A b-Cauchy sequence {wn} is equivalent to {an} in a generalized 2-inner product space W if and only if lim n→∞ ‖wn − an‖b = 0. Proof: Let U be a neighborhood of zero, then, there exists M(U) > 0, such that wn − an ∈ U for n ≥ M(U). Hence, for all neighborhood U of 0, there exists ε = ε(U) > 0 such that ‖wn − an‖b < ε; n ≥ M(U). Then, for every δ > 0, there exists M(δ) > 0 such that ‖wn − an‖b < δ for all n ≥ M(δ), therefore, limn→∞ ‖wn − an‖b = 0 for n ≥ M(δ). Conversely, let {wn}, {an} be b-Cauchy sequences in W such that for every neighborhood U of 0, there exists ε > 0 such that Bε(0) ⊂ U. By our hypothesis limn→∞ ‖wn − an‖b = 0, then there exists M(ε) > 0 such that ‖wn − an‖b < ε for n ≥ M(ε). Hence, wn − an ∈ Bε(0) ⊂ U for n ≥ M(ε), then {wn}~{an}.∎ Proposition (3.4): If {an} and {bn} are equivalent to {wn} and {yn} in a generalized 2-inner product space W. Then, {an + bn} is equivalent to {wn + yn} and {αan} is equivalent to {αwn}. Moreover, Ŵ is a linear space. Proof: Since {an}~{wn} and {bn}~{yn} thus we get ‖(wn + yn) − (an + bn)‖b ≤ ‖wn − an‖b + ‖yn − bn‖b Then lim 𝑛,𝑚→∞ ‖(wn + yn) − (an + bn)‖b = 0 … (1) On the other hand, lim 𝑛→∞ ‖αwn − αan‖b = 0 ... (2) It implies that {an + bn}~{wn + yn} and {αan}~{αwn}. Therefore, from (1), (2) and Proposition (2.3), Ŵ is a linear space.∎ Step3: Proof �̂� is a 2-normed space. We will define a 2-norm function on the space Ŵ. as: ‖. ‖b̂: Ŵ ×< b̂ >→ R + is defined as: ‖ŵ − ŷ‖b̂ = lim 𝑛→∞ ‖wn − yn‖b … (3) where {wn} ∈ ŵ, {yn} ∈ ŷ. The function is well-defined as follows: Proposition (3.5): If W is a generalized 2-inner product space, then for any two b-Cauchy sequences {wn} and {yn} in W: IHJPAS. 36(1)2023 315 1. lim 𝑛→∞ ‖wn − yn‖b exists. 2. For pairs of equivalent b-Cauchy sequences {an}~{wn} and {bn}~{yn}, lim 𝑛→∞ ‖wn − yn‖b = lim 𝑛→∞ ‖an − bn‖b . Proof: 1. Let {wn} ∈ ŵ , {yn} ∈ ŷ be any two b-Cauchy sequences, then ‖wn − yn‖b ≤ ‖wn − wm‖b + ‖wm − ym‖b + ‖ym − yn‖b Hence, ‖wn − yn‖b − ‖wm − ym‖b ≤ ‖wn − wm‖b + ‖ym − yn‖b On the other hand, if we change m by n, ‖wm − ym‖b − ‖wn − yn‖b ≤ ‖wm − wn‖b + ‖yn − ym‖b It implies that |‖wn − yn‖b − ‖wm − ym‖b| ≤ ‖wn − wm‖b + ‖yn − ym‖b … (4) Thus, by taking n, m → ∞ and Definition (1.1), it follows that lim 𝑛→∞ |‖wn − yn‖b − ‖wm − ym‖b| = 0 Then, {‖wn − yn‖b} is a Cauchy sequence in R. But, R is complete, thus lim 𝑛→∞ ‖wn − yn‖b exists. 2. Let {an}~{wn} and {bn}~{yn}. By the same argument of (4), it implies that |‖wn − yn‖b − ‖an − bn‖b| ≤ ‖wn − an‖b + ‖yn − bn‖b By taking n, m → ∞ and proposition (2.3), we get lim 𝑛→∞ ‖wn − yn‖b = lim 𝑛→∞ ‖an − bn‖b .∎ From equation (3) and Proposition (2.3), the conditions (1-3) of a 2-normed space are done. Proposition (3.6): (Ŵ, ‖. ‖b̂) is a 2-normed space. Proof: since ‖ŵ − ẑ‖b̂ = lim 𝑛→∞ ‖wn − zn‖ b ≤ lim 𝑛→∞ ‖wn − yn‖b + lim 𝑛→∞ ‖yn − zn‖b = ‖ŵ − ŷ‖b̂ + ‖ŷ − ẑ‖b̂, then (Ŵ, ‖, ‖b̂) is a 2-norm.∎ Step4: Construction of an isometric 𝑇: 𝑊 → �̂�0 ⊂ �̂�. Let Ŵ0 be the subset of Ŵ composed of the equivalence classes containing constant b-Cauchy sequences. Define a function T: W → Ŵ0 ⊂ Ŵ by T(w) = ŵ = (w, w, … ). It is clearly that T is a well- defined onto and one to one. In fact, ‖Tw − Ty‖b = ‖ŵ − ŷ‖b̂ = lim 𝑛→∞ ‖w − y‖b = ‖w − y‖b, Thus, Ŵ0 and W are isometric. IHJPAS. 36(1)2023 316 Step 5: Proof �̂�0 is dense in �̂�. Proposition (3.7): If W is a generalized 2-inner product space, then, Ŵ0 is dense in Ŵ. Proof: Let ŵ ∈ Ŵ − Ŵ0 , then, there exists a b-Cauchy sequence {wn} ∈ ŵ where {wn} = {w1, w2, … }. Define ŵ m = {wm, wm, … } for all m ∈ N, thus �̂� m ∈ X̂0. Hence, by Definition (1.1) ‖ŵm − ŵ‖b̂ = lim 𝑛,𝑚→∞ ‖wn − wm‖b = 0 . Then, Ŵ0 is dense in Ŵ.∎ Step 6: Proof completeness of �̂�. Theorem (3.8): If W is a generalized 2-inner product space, then Ŵ is complete. Proof: Let {ŵn} be a b-Cauchy sequence in Ŵ. Since Ŵ0 is dense in Ŵ, thus there exists {ẑn} ∈ Ŵ0 such that ‖ŵn − ẑn‖b̂ = 0 . But ‖ẑn − ẑm‖b̂ ≤ ‖ẑn − �̂�n‖b̂ + ‖ŵn − ŵm‖b̂ + ‖ŵm − ẑm‖b̂. Then, by equation (3) and Definition (1.1) and if we take n, m → ∞, we get lim 𝑛,𝑚→∞ ‖ẑn − ẑm‖b̂ = 0 , it implies that {ẑn} is a b-Cauchy sequence in Ŵ0. But W and Ŵ are isometric. Thus, there exists a b-Cauchy sequence {zn} in W which is contained in an equivalent class in Ŵ, say ŵ. Note that, ‖ŵn − ŵ‖b̂ ≤ ‖ŵn − ẑn‖b̂ + ‖ẑn − ŵ‖b̂ = ‖ŵn − ẑn‖b̂ + ‖ẑn − ẑn‖b̂. Thus, lim 𝑛→∞ ‖ŵn − ŵ‖b̂ = 0 . Therefore, Ŵ is complete.∎ Step7: Proof uniqueness of �̂� up to isometrics. Theorem (3.9): The space Ŵ is unique up to isometrics. Proof: Let Ŷ be another completion to W with a dense subset Ŷ0 in Ŷ. Then, there exists S: W → Ŷ0 is isometric by step 4 defined by S(w) = ŷ = (y, y, … ). We will define h: Ŵ0 → Ŷ0 by h(ŵ) = ST −1(ŵ). It implis that Ŵ0 isometric to Ŷ0. For ŷ1,ŷ2 in Ŷ there exists b-Cauchy sequences {ŷ1n}, {ŷ2n} in Ŷ0 such that ŷ1n → ŷ1 and ŷ2n → ŷ2. Thus, by equation (4) |‖�̂�1 − �̂�2‖�̂� − ‖�̂�1𝑛 − �̂�2𝑛 ‖𝑏 | ≤ ‖�̂�1 − �̂�1𝑛‖𝑏 − ‖�̂�2 − �̂�2𝑛‖𝑏 → 0 By taking n → ∞ ‖�̂�1 − �̂�2‖�̂� = lim 𝑛→∞ ‖�̂�1𝑛 − �̂�2𝑛‖𝑏 (5) by the same argument ‖�̂�1 − �̂�2‖�̂� = lim 𝑛→∞ ‖�̂�1𝑛 − �̂�2𝑛‖𝑏 (6) IHJPAS. 36(1)2023 317 Thus, by (5) and (6) we get ‖�̂�1 − �̂�2‖�̂� = lim 𝑛→∞ ‖�̂�1𝑛 − �̂�2𝑛‖𝑏 = lim 𝑛→∞ ‖�̂�1𝑛 − �̂�2𝑛‖𝑏 = ‖�̂�1 − �̂�2‖�̂� It implies that Ŵ is isometric to Ŷ.∎ 4. Discussion and Conclusion A complete metric space is a well-known concept. Every non-complete metric space W can be built into a complete metric space Ŵ, which is known as a completion of W. In this paper, we construct equivalent classes of b-Cauchy sequences to complete a generalized 2-inner product space. References 1.Anshul, R.; Ravinder, K., S., ; Sumit, C., Stability of Complex Functional Equations in 2-Banach Spaces. 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