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FUZZY INFORMATION AND ENGINEERING 2021, VOL. 13, NO. 2, 236–247 https://doi.org/10.1080/16168658.2021.1943876 On the Completion of Fuzzy Normed Linear Spaces in the Sense of Bag and Samanta Conghua Yan Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, People’s Republic of China ABSTRACT KEYWORDS Fuzzy metric spaces; fuzzy In this paper, the completion of fuzzy normed linear space (in the normed linear space; Cauchy sense of Bag and Samanta) is studied. First, some properties of con- fuzzy point sequence; vergence fuzzy point sequences are discussed. Specially, we give uniformly dense in every another characterisation of Q-neighbourhood base of θ (λ ∈ (0, 1]) stratum; completion for I-topology introduced by Saheli. Then we show that each fuzzy normed linear space has an (up to isomorphism) unique complete fuzzy normed linear space which contains an uniformly dense in every stratum subspace isomorphic to it. 1. Introduction The notion of fuzzy norm on a linear space was first introduced by Katsaras [1]. Felbin [2] gave an idea of fuzzy norm on a linear space whose associated metric is Kaleva type [3]. Influenced by the work given by Karmosil and Michalek [4], Cheng and Menderson [5]intro- duced another definition of fuzzy norm on a linear space. Bag and Samanta [6] modified slightly the notion of fuzzy norm determined by Cheng and Menderson. The relationships between above three types of fuzzy norms are discussed by Bag and Samanta [7]. Based on the notion of fuzzy norm in the sense of Bag and Samanta, the theory of fuzzy normed lin- ear spaces is studied systematically [8–12]. Moreover, some notions and properties of finite dimensional fuzzy cone normed linear spaces are discussed in [13–15]. Among them, Saheli [12] introduced a new I-topology on fuzzy normed linear space, and show that this I-topology is compatible with the vector structure. The Q-neighbourhood base of θ (λ ∈ (0, 1]) for this I-vector topology is obtained. Moreover, a comparative study of I-topologies which obtained by Saheli in [11, 12] on fuzzy normed linear spaces is presented. The study of completion of fuzzy metric space and fuzzy normed linear space consti- tutes a natural and interesting open question in the analysis of such spaces. The first effort is due to Kaleva [16] in the frame of fuzzy metric space introduced by Kaleva. From then on, many authors devoted to study the completion of fuzzy metric spaces or fuzzy normed linear spaces in the sense of Kaleva type or Felbin type, and several important results are dis- cussed ([17–20]). The study of completion on the fuzzy metric space introduced by Karmosil CONTACT Conghua Yan chyan@njnu.edu.cn © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 237 and Michalek is originally Gregori and Romaguera [21], an ordinary topology is considered in their study. They show that for each fuzzy metric space there is an (up to uniform iso- morphism) unique complete fuzzy metric space that contains a dense subspace uniformly isomorphic to it. The original research of completion on the fuzzy normed linear spaces wasFelbin’ sworkin[19] with help of usual classical topology. As above claimed, Saheli [12] introduced a new I-topology on fuzzy normed linear space in the sense of Bag and Samanta recently. From the properties discussed by the author, we may consider this new I-topology is more suitable for the further study in fuzzy normed linear space. The main purpose of this paper is to study the completion of fuzzy normed linear space with respect to the I-topology determined by Saheli. At first, we study some prop- erties of fuzzy point sequences and give another structure of Q-neighbourhood base of θ (λ ∈ (0, 1]) for the I-topology. Then we prove that each fuzzy normed linear space has a completion with respect to I-vector topology. First we fix some notations, throughout this paper, I = [0, 1] and I denotes the family of all fuzzy sets on the nonempty set X. The notation Pt(I ) denotes the set of all fuzzy points X X on X. For every x ∈ Pt(I ), A ∈ I , the notation x ∈A denotes the relationship A(x) + λ> 1. λ λ According to the terminology introduced by Rodabaugh [22], for r ∈ [0, 1], r denotes the fuzzy set on X which takes the constant value r. Definition 1.1 ([6]): Let X be a vector space over R (real number), N a fuzzy set of R such that for all x, u ∈ X and c ∈ R: (N1) N(x, t) = 0 for all t ≤ 0; (N2) x = θ if and only if N(x, t) = 1 for all t > 0; (N3) If c = 0, then N(cx, t) = N(x, ) for all t ∈ R; |t| (N4) N(x + u, s + t) ≥ N(x, s) ∧ N(u, t) for all s, t ∈ R; (N5) N(x, ·) is nondecreasing function of R and lim N(x, t) = 1. t→∞ Then N is called a fuzzy norm on X and the pair (X, N) is called a fuzzy normed linear space. Definition 1.2 ([23]): An I-topology on a set X is a family τ of fuzzy subsets of X satisfying the following: (1) For each λ ∈ [0, 1], λ ∈ τ (2) τ is closed under finite intersection of fuzzy subsets (3) τ is closed under arbitrary union of fuzzy subsets. The pair (X, τ) is called an I-topological space. Definition 1.3 ([24]): An I-topology τ on a vector space X is said to be an I-vector topology, if the following two mappings f : X × X → X, (x, y) → x + y and g : K × X → X, (k, x) → kx are continuous, where K is equipped with the I-topology induced by the usual topology and X × X, K × X are equipped with the corresponding product I-topologies. At this time, the pair (X, τ) is called an I-topological vector space. 238 C. YAN Definition 1.4 ([24]): Let (X, τ) be an I-topological space and x ∈ Pt(I ). (1) A fuzzy set U on X is called Q-neighbourhood of x iff there exists G ∈ τ such that x ∈G ⊆ U. (2) A family U of Q-neighbourhoods of x is called Q-neighbourhood base of x iff for x α α every Q-neighbourhood A of x , there exists U ∈ U such that U ⊆ A. α x (3) (X, τ) is called first countable, if for each x ∈ Pt(I ), there exists Q-neighbourhood base U of x such that U has countable fuzzy sets. x α x α α Definition 1.5 ([24]): An I-topological vector space is said to be a QL-type, if there exists a family U of fuzzy sets on X such that for each λ ∈ (0, 1], U ={U r | U ∈ U, r ∈ (1 − λ,1]}. is a Q-neighbourhood base of θ in (X, τ). The family U is called a Q-prebase for τ . Theorem 1.6 ([12]): Let (X, N) be a fuzzy normed linear space. Then the family τ ={μ ∈ I |∀ x ∈ σ (μ), r ∈ (0, μ(x)), there is ε> 0, s.t. x + B r ⊆ μ} N 0 ε is an I-topology on X. Here C (x) = {α ∈ (0, 1] | N(x, ε) ≥ α}, ∀x ∈ X. Theorem 1.7 ([12]): Let (X, N) be a fuzzy normed linear space. Then (X, τ ) is an I-topological vector space and for each λ ∈ (0, 1], U ={C r | ε> 0, r ∈ (1 − λ,1]} λ ε is a Q-neighbourhood base of θ . 2. Some Basic Properties in Fuzzy Normed Linear Spaces Definition 2.1: Let (X, N) be a fuzzy normed linear space, ε> 0. The fuzzy set B on X is defined as follows: B (x) = {1 − α | { t > 0: N(x, t)> 1 − α} <ε}, ∀ x ∈ X. Lemma 2.2: Let (X, N) be a fuzzy normed linear space, x ∈ Pt(I ). Then x ∈B iff { t > 0: α α ε N(x, t)> 1 − α} <ε. Proof: Necessity. Since x ∈B , then there exists β ∈ (0, α) such that { t > 0: N(x, t)> α ε 1 − β} <ε.Sowehave t <ε which implies N(x, t )> 1 − β> 1 − α. This deduces that 0 0 { t > 0: N(x, t)> 1 − α}≤ t <ε. Sufficiency. First we want to prove lim { t > 0: N(x, t)> 1 − β}= { t > 0: N(x, t)> 1 − α}. β→α− In fact, for each sequence {β } which increases and convergence to α, since { t > 0: N(x, t)> 1 − β }⊆{ t > 0: N(x, t)> 1 − α}, ∀n ∈ N, we have { t >0: N(x, t)> 1 − β }≥ { t >0: N(x, t)> 1 − α}, ∀n ∈ N.Thus lim { t >0: N(x, t)> 1 − β }≥ { t >0: N(x, t)> 1 − α}.Iflim { t >0: n n n FUZZY INFORMATION AND ENGINEERING 239 N(x, t)> 1 − β } > { t >0: N(x, t)> 1 − α}, then there exists k >0suchthat lim { t >0: N(x, t)> 1 − β } > k > { t >0: N(x, t)> 1 − α}. This implies that n n there exists q ∈ N such that { t > 0: N(x, t)> 1 − β } > k for all n > p. Hence N(x, k) ≤ 1 − β for all n > p.Put n →∞,wehave N(x, k) ≤ 1 − α.So { t > 0: N(x, t)> 1 − α}≥ k, this contradicts with the fact k > { t >0: N(x, t)> 1 − α}. Then lim { t >0: N(x, t)> 1 − β }= { t > 0: N(x, t)> 1 − α}.Because { t > 0: N(x, t)> 1 − α} is decrease for the variable α, it deduces lim { t > 0: N(x, t)> 1 − β}= { t > 0: β→α− N(x, t)> 1 − α}. From the assumption { t > 0: N(x, t)> 1 − α} <ε and above proof, there exists β ∈ (0, α) such that { t > 0: N(x, t)> 1 − β} <ε. Then B (x) ≥ 1 − β> 1 − α. Hence x ∈B . α ε Theorem 2.3: Let (X, N) be a fuzzy normed linear space, τ an I-topology determined by fuzzy norm. Then τ can be determined by the Q-neighbourhood baseB ={B r |ε> 0, r ∈ (1 − λ ε λ,1]} of θ , λ ∈ (0, 1]. Proof: For each ε> 0, we may prove the following B ⊆ C ⊆ B . ε ε 2ε In fact, for each x ∈B , there is β ∈ (0, α) such that x ∈B . Then { t > 0: N(x, t)> 1 − α ε β ε β} <ε. This implies N(x, ε) > 1 − β.Thus C (x) ≥ 1 − β> 1 − α.So x ∈C . Hence B ⊆ C . ε α ε ε ε On the other hand, for each x ∈C , there exists μ ∈ (1 − λ,1) such that N(x, ε) ≥ μ> λ ε 1 − λ. Then { t > 0: N(x, t)> 1 − λ}≤ ε< 2ε.So x ∈B . This implies C ⊆ B . λ 2ε ε 2ε By Theorem 1.7, the family B (λ ∈ (0, 1]) of fuzzy sets is Q-neighbourhood base of θ λ λ which determined the I-topology is equivalent to τ . Theorem 2.4: Let (X, N) be a fuzzy normed linear space, τ an I-topology determined by fuzzy norm. Then (X, τ ) is first countable I-topological vector space. Proof: By Theorem 1.7 and Theorem 2.3, (X, τ ) is an I-topological vector space, and for any λ ∈ (0, 1], U ={B r | r ∈ (1 − λ,1], ε> 0} is a Q-neighbourhood base of θ .Since λ ε λ for each ε> 0, there exists n ∈ N such that B ⊆ B , hence the family of fuzzy sets B = ε λ {B r | r ∈ (1 − λ,1] Q, n ∈ N} is a Q-neighbourhood base of θ andB has countable 1 λ λ elements. So (X, τ ) is first countable an I-topological vector space. Theorem 2.5: Let (X, N) be a fuzzy normed linear space, τ an I-topology determined by fuzzy (n) norm. Then the fuzzy sequence {x } is convergent to x with respect to τ if and only if for any λ N (n) ε ∈ (0, λ), there exist t ∈ (0, ε), p ∈ N such that N(x − x, t)> 1 − λ , λ >λ − ε. n n Proof: Necessity. For any ε ∈ (0, λ), B 1 − λ + ε is a Q-neighbourhood of θ .Since ε λ (n) (n) (n) x → x , there exists p ∈ N such that x ∈ x + B 1 − λ + ε for all n > p. Then (x − λ ε λ λ n n (n) x) ∈ B and λ >λ − ε.Sowehave α which satisfies N(x − x, ε) ≥ α and α> 1 − λ . λ ε n n From the fact N(x, t) holds the condition (N7∗), there is t <ε such that N(x, t)> 1 − λ . Sufficiency. Let W be a Q-neighbourhood of x , then we have ε ∈ (0, λ) such that x + B 1 − λ + ε ⊆ W. From the assumption of sufficiency, there exist t ∈ (0, ε), p ∈ N such ε 240 C. YAN (n) that N(x − x, t)> 1 − λ , λ >λ − ε for all n ≥ p. Then there is δ ∈ (0, λ ) such that n n n n (n) (n) (n) N(x − x, t)> 1 − λ + δ . This implies N(x − x, ε) > 1 − λ + δ ,so B (x − x) ≥ n n n n ε (n) (n) 1 − λ + δ > 1 − λ .Thus (x − x) ∈B . This deduces that x ∈x + B 1 − λ + ε ⊆ n n n λ ε ε n λ (n) W. Hence {x } converges to x with respect to τ . λ N (n) Definition 2.6: Let (X, N) be a fuzzy normed linear space, {x } a sequence of fuzzy points in X. Then (n) (1) {x } is called λ-Cauchy sequence, if for each W ∈ U , there exists p ∈ N such that (n) (m) x − x ∈W for all n, m ≥ p. λ ∧λ λ ∧λ n m (n) (n) (2) {x } is called Cauchy sequence, if for each λ ∈ (0, lim λ ), {x } is λ-Cauchy sequence. n n λ λ n n (n) (3) (X, N) is called fuzzy complete if for each Cauchy sequence {x }, there exists x ∈ X (n) such that x converges to x with respect to I-topology τ , where μ = lim λ . μ N n n (n) Theorem 2.7: Let (X, N) be a fuzzy normed linear space, {x } a sequence of fuzzy points in X. (n) Then {x } is a Cauchy sequence if and only if lim λ = μ and for any λ ∈ (0, μ), ε ∈ (0, λ), n n (n) (m) there exist t ∈ (0, ε), p ∈ N such that N(x − x , t )> 1 − λ for all n, m ≥ p. 0 0 (n) Proof: Necessity. Suppose that {x } is a Cauchy sequence and lim λ = μ. For any n n (n) λ ∈ (0, μ) and ε ∈ (0, λ), B 1 − λ + ε ∈ U , then there exists p ∈ N such that x − ε λ λ ∧λ (m) (n) x ∈B 1 − λ + ε for all n, m ≥ p. This deduces that λ >λ ∧ λ>λ − ε and (x − ε n n λ ∧λ (m) x ) ∈B .So λ >λ − ε for all n ≥ p and λ ∧λ ∧λ ε n n m (n) (m) { t > 0: N(x − x , t)> 1 − λ} (n) (m) ≤ { t > 0: N(x − x , t)> 1 − (λ ∧ λ ∧ λ)} <ε. n m (n) (m) Thus there is t ∈ (0, ε) such that N(x − x , t )> 1 − λ. In addition, lim λ ≥ λ, since 0 0 n the arbitrariness of λ,wehavelim λ ≥ μ. Hence lim λ = μ. n n n Sufficiency. If lim λ = μ and for any λ ∈ (0, μ), ε ∈ (0, λ), there exist t ∈ (0, ε), p ∈ N n n 0 (n) (m) such that N(x − x , t )> 1 − λ for all n, m ≥ p.Fromthe factlim λ = μ>λ,wehave 0 n n (n) (m) q ∈ N with q > p which implies λ >λ for all n ≥ q. Then { t > 0: N(x − x , t)> (n) (m) (n) (m) 1 − λ}≤ t <ε.Sowehave (x − x ) = x − x ∈B 1 − λ + ε for all n, m ≥ 0 λ ε λ ∧λ λ ∧λ n m (n) q. This means that {x } is a Cauchy sequence. Remark 2.8: The notion of Cauchy sequences and fuzzy complete is based on I-topology in this paper. This notions is not different from the corresponding notions introduced by Felbin [19]. In fact, the notion of Cauchy sequences and complete given by Felbin [19]is based on crisp topology, equivalently, every Cauchy sequence {x } is convergent in every stratum (X, · ) for all α ∈ (0, 1]. In addition, the notions of fuzzy normed linear spaces are not completely same. By Theorem 2.7, the notion of Cauchy sequences in this paper is for fuzzy points (not crisp points). FUZZY INFORMATION AND ENGINEERING 241 3. The Completion of Fuzzy Normed Linear Spaces Definition 3.1: Let (X, N) be a fuzzy normed linear space, A ⊆ X is called uniformly dense in every stratum if for any x ∈ X, there exists a sequence {x }⊆ A such that for each λ ∈ (0, 1], ε> 0, there is t <ε, which deduces that N(x − x, t)> 1 − λ. Definition 3.2: Let (X, N), (Y, N ) be two fuzzy normed linear spaces, then they are called isomorphic if there exists a linear operator T : X → Y such that for any x ∈ X, λ ∈ (0, 1], the next equality holds. ∧{t > 0: N(x, t)> 1 − λ}=∧{t > 0: N (Tx, t)> 1 − λ}. Definition 3.3: A complete fuzzy normed linear space (X, N ) is said to a completion of the fuzzy linear space (X, N) if (X, N ) has an uniformly dense subspace in every stratum (W, N ) 1 1 being isometric to (X, N). Theorem 3.4: Any fuzzy normed linear space has a completion. Proof: By the Definition 3.3, the whole proof is divided into following four steps. Step 1. Construct a fuzzy normed linear space (X, N ). At first we define the sets X and 1 c θ as follows: (n) (n) (m) X ={{x } : ∀ λ ∈ (0, 1], ε> 0, ∃ t <ε, p ∈ N, N(x − x , t)> 1 − λ(for all n, m > p)}. (n) (n) (n) θ ={{x } : {x }∈ X , ∀λ ∈ (0, 1], ε> 0, ∃ t <ε, p ∈ N, N(x , t)> 1 − λ(for any n > p)}. ˜ ˜ It is easy to find X =∅ and θ =∅. The relation ∼ on X \ θ is defined as follows: c c (n) (n) (n) (n) {x }∼{y }⇐⇒∀ λ ∈ (0, 1], ε> 0, ∃ t <ε, p ∈ N, N(x − y , t)> 1 − λ(for each n > p). (n) By the definition of fuzzy norm, the above relation is equivalent. For each ξ ={x }∈ (n) (n) ˜ ˜ ˜ X \ θ, its equivalent class is denoted by ξ = {x }. Specially, if {x }∈ θ, then we claim that (n) ˜ ˜ ˜ {x }= θ. Denote X = (X \ θ)/ ∼,and X = X {θ }. The addition and scalar multiplica- 0 c 0 tion in X are well-defined as follows: (n) (n) For all {x }, {y }∈ X , k ∈ K, (n) (n) (n ) (n) {x }+ {y }= {x + y }; (n) (n) (n) ˜ ˜ {x }+ θ = θ + {x }= {x }; ˜ ˜ kθ = θ; θ, k = 0 (n) k{x }= . (n) {kx }, k = 0 It is easy to verify that X is a linear space. The mapping N : X × R → [0, 1] is defined as follows: ˜ ˜ 0, (ξ, t) = (θ,0) N (ξ, t) = . ˜ ˜ ˜ 1 − {λ : ξ ≤ t}, (ξ, t) = (θ,0) λ∈(0,1] λ 242 C. YAN ˜ ˜ 0, ξ = θ X (n) ˜ ˜ Here ξ = ,and {x }∈ ξ. λ (n) ˜ ˜ lim { t > 0: N(x , t)> 1 − μ}, ξ = θ n→∞ μ<λ (n) (n) Since {x }∈ ξ, the sequence { t > 0: N(x , t)> 1 − μ} is a real Cauchy sequence. Thus the limit of this sequence exists. In what follows, we must prove that ξ is deter- (n) (n) mined by ξ uniquely. In fact, if {x }∼{y }, then for each λ ∈ (0, 1], ε> 0, there exists (n) (n) t <ε such that N(x − y , t)> 1 − λ when n →∞. This implies lim { t >0: n→∞ (n) (n) N(x − y , t)> 1 − μ}= 0. On the other hand, since (n) (n) { t > 0: N(x , t)> 1 − μ}− { s > 0: N(y , s)> 1 − μ} (n) (n) ≤ { s + t > 0: N(x − y , s + t)> 1 − μ}. (n) X (n) X So {x } = {y } . This means that the mappings N is well-defined. λ λ In the following, it needs to verify that (X, N ) is a fuzzy normed linear space. (N1) Clearly, N (ξ, t) = 0for each t ∈ (−∞,0]. ˜ ˜ (N2) If N (ξ, t) = 1foreach t > 0, then ξ = 0 for any λ ∈ (0, 1]. For any ε> 0, λ ∈ (0, 1], (n) there exists μ<λ such that lim { t > 0: N(x , t)> 1 − μ} <ε. Then there is n→∞ (n) (n) ˜ ˜ t <ε, p ∈ N such that N(x , t)> 1 − μ for each n > p.So ξ = {x }= θ. Conversely, ˜ ˜ ˜ if ξ = θ, the equality N (θ, t) = 1for each t > 0 holds obviously. (N3) For any t > 0, c ∈ R, c = 0, ˜ ˜ N (cξ, t) = 1 − {λ : cξ ≤ t} λ∈(0,1] t t ˜ ˜ = 1 − λ : ξ ≤ = N ξ, . |c| |c| λ∈(0,1] ˜ ˜ (N4) Suppose that N (ξ, t) = 1 − {λ : ξ ≤ t}= a and N (η ˜, s) = 1 − 1 1 λ∈(0,1] {λ : ˜ η ≤ s}= b. Without loss of generality, let a ∧ b = 0, then for any λ∈(0,1] λ r ∈ (0, a ∧ b), there exist α, β ∈ (0, 1] with α< 1 − r, β< 1 − r such that ξ ≤ t X X X X X X ˜ ˜ and ˜ η ≤ s. From the definition of ˜ η , ˜ η ≤ η˜ and ξ ≤ ξ .Thus β β 1−r β 1−r α we have X X X X X ˜ ˜ ˜ ξ +˜ η ≤ ξ + η˜ ≤ ξ + η˜ ≤ t + s. 1−r 1−r 1−r α β ˜ ˜ Hence N (ξ +˜ η, s + t) = 1 − {λ : ξ +˜ η ≤ s + t}≥ 1 − (1 − r) = r. λ∈(0,1] λ ˜ ˜ By the arbitrariness of r, it deduces that N (ξ +˜ η, s + t) ≥ a ∧ b = N (ξ, t) ∧ 1 1 N (η ˜, s). (N5) For all t , t ∈ (0, +∞), t < t , 1 2 1 2 X X ˜ ˜ ˜ ˜ N (ξ, t ) = 1 − {λ : ξ ≤ t }≤ 1 − {λ : ξ ≤ t }= N (ξ, t ). 1 1 1 2 1 2 λ λ λ∈(0,1] λ∈(0,1] (n) X X ˜ ˜ ˜ In addition, for any ξ = {x }∈ X, λ ∈ (0, 1], since ξ ∈ [0, +∞) and ξ ≤ s λ λ X X ˜ ˜ implies {λ : ξ ≤ s}≤ λ. Then we have 1 − {λ : ξ ≤ s}≥ 1 − λ∈(0,1] λ λ∈(0,1] λ λ.Solim N (ξ, t) = 1. t→+∞ 1 FUZZY INFORMATION AND ENGINEERING 243 Step 2. Define a mapping: T : (X, N) → (X, N ) and prove that T(X) is uniformly dense in every stratum. Let T : X → X be defined as follows. θ, x = θ Tx = , ∀ x ∈ X. x ˜, x = θ Here the notation is the equivalent class of the element {x, x, x, x, ...}. Denote T(X) = W,it is easy to find T isalinearoperatorfrom X onto W. In the next we prove that T is an isometric mapping. In fact, if x = θ, both N(θ, t) = 1and N (θ, t) = 1 for any t > 0. Then { t > 0: N(θ, t)> 1 − λ}= { t > 0: N(Tθ, t)> 1 − λ} for any λ ∈ (0, 1]. If x = θ,for each λ ∈ (0, 1] and s > { t > 0: N (Tx, t)> 1 − λ}, there is t < s such that N (Tx, t)> 1 − λ. Then we have μ<λ which satisfies x ˜ ≤ t < s. By the definition of x , there exists ν< μ and r < s such that N(x, r)> 1 − ν> 1 − μ> 1 − λ.So { ω> 0: N(x, ω) > 1 − λ}≤ r < s. This means that { t >0: N (Tx, t)> 1 − λ}≥ { ω> 0: N(x, ω) > 1 − λ}. On the other hand, for each s > { ω> 0: N(x, ω) > 1 − λ}, there is ω< s such that N(x, ω) > 1 − λ. Further we have δ> 0 with N(x, ω) > 1 − (λ − δ). Then x ˜ = lim { t >0: N(x, t)> 1 − μ}≤ ω.Thus N (x ˜, ω) ≥ 1 − n→∞ 1 μ<λ−δ λ−δ (λ − δ) > 1 − λ. This implies that { t > 0: N (Tx, t)> 1 − λ}≤ ω< s. So we can obtain { t > 0: N (Tx, t)> 1 − λ}≤ { t > 0: N(x, t)> 1 − λ}. Therefore T is an isometric mapping. In what follows, it needs to prove T(X) = W is uniformly dense in every stratum with (n) (n) respect to fuzzy normed linear space (X, N ). For any ξ = {x }∈ X,itisclear x ∈ W for (n) (n) (n) (n) any n ∈ N, here the notation x ∈ W is the equivalent class of {x , x , x , ...}. For all (n) λ ∈ (0, 1], ε> 0, from the fact {x ∈ X and let μ ∈ (0, λ), there exist t <ε, p ∈ N such c 0 (n) (m) that N(x − x , t)> 1 − μ for all n, m ≥ p. Thus we have the following X X X (n) (n) (p) (p) ˜ ˜ x − ξ ≤ x − x + x − ξ λ λ λ (n) (p) = { s > 0: N(x − x , s)> 1 − μ} μ<λ (p) (n) + lim { s > 0: N(x − x , s)> 1 − μ} n→∞ μ<λ (n) (p) ≤ { s > 0: N(x − x , s)> 1 − μ } (p) (n) + lim { s > 0: N(x − x , s)> 1 − μ }. n→∞ (n) (n) ˜ ˜ Then x − ξ ≤ t for all n ≥ p.So N (x − ξ, t)> 1 − λ for all n ≥ p. This means that W is uniformly dense in every stratum. (n) Step 3. We prove that (X, N ) is complete. Suppose that {ξ } is Cauchy fuzzy point sequence in (X, N ), from Theorem 2.7, lim λ = μ> 0. In addition, for each λ ∈ 1 n→∞ n (m) (n) ˜ ˜ (0, 1], ε> 0, there exist t <ε, p ∈ N such that N (ξ − ξ , t)> 1 − λ for all n, m ≥ p. (k) Since W is uniformly dense in every stratum, and for any k ∈ N, ξ ∈ X,wehavea 1 1 (n) sequence {x }⊆ W such that for above λ and > 0, there is s < and q ∈ N, q > p, k k (n) (k) which implies that N (x − ξ , s )> 1 − λ for all n ≥ q. 1 k 244 C. YAN (k) (n ) k k ˜ For each k ∈ N, there exists n ≥ q such that N (x − ξ , s )> 1 − λ. Denote k k (n ) (k) k k x = x . Then for m, n ≥ q, the following inequality holds: 1 1 (m) (n) N x − x , t + + m n 1 1 (m) (m) (n) (n) (m) (n) ˜ ˜ ˜ ˜ ≥ N x − ξ , N (ξ − ξ , t) N ξ − x , 1 1 1 m n (m) (m) (n) (n) (m) (n) ˜ ˜ ˜ ˜ ≥ N (x − ξ , s ) N (ξ − ξ , t) N (ξ − x , s )> 1 − λ. 1 m 1 1 n (n) (n) So {x } is Cauchy sequence in W. Notice that T is isometric, we have {x }∈ X .Let (n) (m) ˜ ˜ ˜ ˜ ξ = {x }, clearly ξ ∈ X.Wewillprove ξ → ξ as n →∞. In fact, for each ε ∈ (0, μ), (m) (n) λ ∈ ( , μ), since lim λ = μ and N(x − x , t) → 1(m, n →∞), there is p ∈ N, t < n→∞ n (m) (n) ε such that λ >λ and N(x − x , t)> 1 − for all n, m ≥ p. Thus for any n ≥ p, X (n) (m) (n) x − ξ = lim { s > 0: N(x − x , s)> 1 − ν} m→∞ ν<λ (n) (m) ≤ lim s > 0: N(x − x , s)> 1 − ≤ t. m→∞ (n) This implies N (x − ξ, t) ≥ 1 − > 1 − λ for all n ≥ p. Furthermore, 1 1 (n) (n) (n) (n) ˜ ˜ ˜ ˜ N ξ − ξ, t + ≥ N ξ − x , N (x − ξ, t)> 1 − λ. 1 1 1 n n (n) For above t <ε, there is q ∈ N, q > p such that t + <ε for all n > q. So the proof of ξ → n n ξ is completed. This means that (X, N ) is complete fuzzy normed linear space. μ 1 Step 4. The completion of fuzzy normed linear space (X, N ) is unique except for isomet- rics. Suppose that (Y, N ) is also a completion of (X, N) and S is an isometric linear operator from X onto Y.Since W = TX is uniformly dense in every stratum, then for each ξ ∈ X, there (n) exists a sequence {x }⊆ W such that for each λ ∈ (0, 1], ε> 0, there exist t < , p ∈ N, (n) which deduces that N (x − ξ, t)> 1 − λ for all n > p. Thus for all m, n > p, (n) (m) (n) (m) N (Tx − Tx ,2t) = N (x − x ,2t) 1 1 (n) (m) ˜ ˜ ≥ N (x − ξ, t) N (ξ − x , t)> 1 − λ. 1 1 (n) That is to say {Tx }, (α ∈ (0, 1]) is a Cauchy fuzzy points sequence in W.Since T and S are isometrics, we have (n) (m) { s > 0: N (Tx − Tx , s)> 1 − λ} (n) (m) = { s > 0: N(x − x , s)> 1 − λ} (n) (m) = { s > 0: N (Sx − Sx , s)> 1 − λ}. (n) Thus deduces that {Sx }, (α ∈ (0, 1]) is a Cauchy fuzzy point sequence in SX.Fromthe (n) fact Y is complete, there is a unique y ∈ Y such that Sx → y . It may be proved that the α α FUZZY INFORMATION AND ENGINEERING 245 (n) element y has nothing to do with the choice of {x }. In reality, if there exists a sequence (n) {z }⊆ W such that for each λ ∈ (0, 1], ε> 0, there exist t < , p ∈ N, which deduces that (n) N (z − ξ, t)> 1 − λ for all n > p. Taking the same method, we may prove that there exists unique z ∈ Y such that for each λ ∈ (0, 1], ε> 0, ∃t <ε, q > p, q ∈ N, which implies (n) N (Sz − z, t)> 1 − λ for all n > q.Since (n) (n) { s > 0: N (Sx − Sz , s)> 1 − λ} (n) (n) = { s > 0: N(x − z , s)> 1 − λ} (n) (n) = { s > 0: N (Tx − Tz , s)> 1 − λ}. (n) (n) (n) (n) ˜ ˜ Moreover, N (Tx − Tz ,2t) ≥ N (Tx − ξ, t) N (ξ − Tz , t)> 1 − λ. 1 1 1 (n) (n) Then N (Sx − Sz ,2t)> 1 − λ.Sowehave (n) (n) (n) (n) N (y − z,4t) ≥ N (y − Sx , t) N (Sx − Sz ,2t) N (Sz − z, t)> 1 − λ. 2 2 2 2 By the arbitrariness of λ, y = z. Define a mapping ϕ : X → Y as follows: ϕ(ξ) = y. In the next we prove the mapping ϕ is isometric from X onto Y. Since S and T is isometric, then for any λ ∈ (0, 1], (n) (n) { t > 0: N (Sx , t)> 1 − λ}= { t > 0: N(x , t)> 1 − λ} (n) = { t > 0: N (Tx , t)> 1 − λ}. Thus { t + t + t > 0: N (ϕ(ξ), t + t + t )> 1 − λ} 1 2 3 2 1 2 3 (n) ≤ { t > 0: N (y − Sx , t )> 1 − λ} 1 2 1 (n) + { t + t > 0: N (Sx , t + t )> 1 − λ} 2 3 2 2 3 (n) = { t > 0: N (y − Sx , t )> 1 − λ} 1 2 1 (n) + { t + t > 0: N (Tx , t + t )> 1 − λ} 2 3 1 2 3 (n) ≤ { t > 0: N (y − Sx , t )> 1 − λ} 1 2 1 (n) + { t > 0: N (Tx − ξ, t )> 1 − λ} 2 1 2 + { t > 0: N (ξ, t )> 1 − λ}. 3 1 3 ˜ ˜ Put n →∞,wehave { t > 0: N (ϕ(ξ), t)> 1 − λ}≤ { t > 0: N (ξ, t)> 1 − λ}. Simi- 2 1 ˜ ˜ larly, we may prove { t > 0: N (ϕ(ξ), t)> 1 − λ}≥ { t > 0: N (ξ, t)> 1 − λ}.Sothe 2 1 mapping ϕ is isometric. (n) At last, we prove the mapping ϕ is surjective. For any z ∈ Y, there exists {z }∈ SX such (n) that for each λ ∈ (0, 1], ε> 0, p ∈ N, N (z − z, t)> 1 − λ holds for all n > p. Then we have (n) (n) (n) sequence {x } satisfies S(x ) = z , n = 1, 2, .... From the above proof, there is unique 246 C. YAN (n) η ˜ ∈ X such that for each λ ∈ (0, 1], ε> 0, q ∈ N,N (x −˜ η, t)> 1 − λ holds for all n > q. Since ϕ is isometric, then (n) (n) { t > 0: N (ϕ(Tx ), t)> 1 − λ}= { t > 0: N (Tx , t)> 1 − λ} 2 1 (n) (n) = { t > 0: N(x , t)> 1 − λ}= { t > 0: N (Sx , t)> 1 − λ}. Furthermore, (n) { t > 0: N (Sx − ϕ(η) ˜ , t)> 1 − λ} (n) = { t > 0: N (ϕ(Tx ) − ϕ(η) ˜ , t)> 1 − λ} (n) = { t > 0: N (ϕ(Tx −˜ η), t)> 1 − λ} (n) = { t > 0: N (Tx −˜ η, t)> 1 − λ} (n) (n) Thus N (z − ϕ(η) ˜ ,2t) ≥ N (z − z , t) N (Sx − ϕ(η) ˜ , t)> 1 − λ. 2 2 2 So z = ϕ(η) ˜ . Therefore we complete the whole proof. Acknowledgments I am grateful to the referees for carefully reading the paper and making some critical remarks which allowed me to improve the exposition of the paper. Disclosure Statement No potential conflict of interest was reported by the author(s). Funding This project is funded by the Natural Science Foundation of China ( Nos. 11571006, 12071225) and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. Notes on contributor Dr Conghua Yan (1966–) is a professor in school of Math. and Computer Sciences, Nanjing Normal University. He received a B.A. from Nanjing Normal University of Jiangsu Province and PhD also from Nanjing Normal University. He is a member of Fuzzy Mathematics and Systems Association of the Systems Engineering Society of China. His interest includes fuzzy analysis, fuzzy topologies. Dr C. H. Yan published numerous articles and papers concerning fuzzy analysis, fuzzy topologies in the Journal of “Fuzzy Sets and Systems”, “Information Sciences”, “Mathematicae Japonicae” and more. References [1] Katsaras AK. Fuzzy topological vector spaces II. Fuzzy Sets Syst. 1984;12:143–154. [2] Felbin C. Finite dimensional fuzzy normed linear space. Fuzzy Sets Syst. 1992;48:239–248. [3] Kaleva O, Seikkala S. On fuzzy metric spaces. Fuzzy Sets Syst. 1984;12:215–229. [4] Kramosil I, Michálek J. Fuzzy metric and statistical metric spaces. Kybernetica. 1975;11:326–334. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Apr 3, 2021
Keywords: Fuzzy metric spaces; fuzzy normed linear space; Cauchy fuzzy point sequence; uniformly dense in every stratum; completion
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