On the Completion of Fuzzy Normed Linear Spaces in the Sense of Bag and Samanta

In this paper, the completion of fuzzy normed linear space (in the sense of Bag and Samanta) is studied. First, some properties of convergence fuzzy point sequences are discussed. Specially, we give another characterisation of Q-neighbourhood base of 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.


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] introduced 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 linear spaces is studied systematically [8][9][10][11][12]. Moreover, some notions and properties of finite dimensional fuzzy cone normed linear spaces are discussed in [13][14][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 (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 constitutes 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 discussed ( [17][18][19][20]). The study of completion on the fuzzy metric space introduced by Karmosil 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 isomorphism) 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 was Felbin' s work in [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 properties of fuzzy point sequences and give another structure of Q-neighbourhood base of θ λ (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 X denotes the family of all fuzzy sets on the nonempty set X. The notation Pt(I X ) denotes the set of all fuzzy points on X. For every x λ ∈ Pt(I X ), A ∈ I X , the notation x λ ∈A denotes the relationship A(x) + λ > 1. 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: is nondecreasing function of R and lim t→∞ N(x, t) = 1.
Then N is called a fuzzy norm on X and the pair (X, N) is called a fuzzy normed linear space.
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.
(1) A fuzzy set U on X is called Q-neighbourhood of x α iff there exists G ∈ τ such that x α ∈G ⊆ U.
, there exists Q-neighbourhood base U x α of x α such that U x α has countable fuzzy sets.

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], is a Q-neighbourhood base of θ λ in (X, τ ). The family U is called a Q-prebase for τ .
is a Q-neighbourhood base of θ λ .

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 ε on X is defined as follows: Lemma 2.2: Let (X, N) be a fuzzy normed linear space, x α ∈ Pt(I X ) .
In fact, for each sequence {β n } which increases and convergence to α, since

Theorem 2.3: Let (X, N) be a fuzzy normed linear space, τ N an I-topology determined by fuzzy norm. Then τ can be determined by the Q-neighbourhood base
Proof: For each ε > 0, we may prove the following On the other hand, for each By Theorem 1.7, the family B λ (λ ∈ (0, 1]) of fuzzy sets is Q-neighbourhood base of θ λ which determined the I-topology is equivalent to τ N .
Definition 2.6: Let (X, N) be a fuzzy normed linear space, {x (n) λ n } a sequence of fuzzy points in X. Then N) is called fuzzy complete if for each Cauchy sequence {x λ n } is a Cauchy sequence and lim n λ n = μ. For any λ ∈ (0, μ) and ε ∈ (0, λ), In addition, lim n λ n ≥ λ, since the arbitrariness of λ, we have lim n λ n ≥ μ. Hence lim n λ n = μ. Sufficiency. If lim n λ n = μ and for any λ λ n } 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 n } is convergent in every stratum (X, · α ) 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).

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 n } ⊆ A such that for each λ ∈ (0, 1], ε > 0, there is t < ε, which deduces that N(x n − x, t) > 1 − λ. Let (X, N), (Y, N 1 ) 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.

Definition 3.3:
A complete fuzzy normed linear space ( X, N 1 ) is said to a completion of the fuzzy linear space (X, N) if ( X, N 1 ) has an uniformly dense subspace in every stratum (W, N 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 1 ). At first we define the sets X c and θ as follows: It is easy to find X c = ∅ andθ = ∅. The relation ∼ on X c \θ is defined as follows: By the definition of fuzzy norm, the above relation is equivalent. For each ξ = {x (n) } ∈ X c \θ, its equivalent class is denoted byξ = {x (n) }. Specially, if {x (n) } ∈θ, then we claim that {x (n) } =θ. Denote X 0 = (X c \θ)/ ∼, and X = X 0 {θ}. The addition and scalar multiplication in X are well-defined as follows: For all {x (n) }, {y (n) } ∈ X 0 , k ∈ K, It is easy to verify that X is a linear space. The mapping N 1 : X × R → [0, 1] is defined as follows: , and {x (n) } ∈ξ .
Since {x (n) } ∈ξ , the sequence { t > 0 : N(x (n) , t) > 1 − μ} is a real Cauchy sequence. Thus the limit of this sequence exists. In what follows, we must prove that ξ X λ is determined by ξ uniquely. In fact, if {x (n) } ∼ {y (n) }, then for each λ ∈ (0, 1], ε > 0, there exists This means that the mappings N 1 is well-defined. In the following, it needs to verify that ( X, N 1 ) is a fuzzy normed linear space.
Step 2. Define a mapping: T : (X, N) → ( X, N 1 ) and prove that T(X) is uniformly dense in every stratum.
Let T : X → X be defined as follows.
Here the notation is the equivalent class of the element {x, x, x, x, . . .}. Denote T(X) = W, it is easy to find T is a linear operator from X onto W. In the next we prove that T is an isometric mapping.
In what follows, it needs to prove T(X) = W is uniformly dense in every stratum with respect to fuzzy normed linear space ( X, N 1 ). For anyξ = {x (n) } ∈ X, it is clear x (n) ∈ W for any n ∈ N, here the notation x (n) ∈ W is the equivalent class of {x (n) , x (n) , x (n) , . . .}. For all λ ∈ (0, 1], ε > 0, from the fact {x (n) ∈ X c and let μ 0 ∈ (0, λ), there exist t < ε, p ∈ N such that N(x (n) − x (m) , t) > 1 − μ 0 for all n, m ≥ p. Thus we have the following This means that W is uniformly dense in every stratum.
Since W is uniformly dense in every stratum, and for any k ∈ N,ξ (k) ∈ X, we have a sequence { x (n) k } ⊆ W such that for above λ and 1 k > 0, there is s k < 1 k and q ∈ N, q > p, which implies that (k) . Then for m, n ≥ q, the following inequality holds: We will proveξ (n) λ n →ξ μ as n → ∞. In fact, for each ε ∈ (0, μ), λ ∈ ( μ 2 , μ), since lim n→∞ λ n = μ and N( Thus for any n ≥ p, For above t < ε, there is q ∈ N, q > p such that t + 1 n < ε for all n > q. So the proof ofξ (n) λ n → ξ μ is completed. This means that ( X, N 1 ) is complete fuzzy normed linear space.
Step 4. The completion of fuzzy normed linear space ( X, N 1 ) is unique except for isometrics. Suppose that (Y, N 2 ) 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 eachξ ∈ X, there exists a sequence { x (n) } ⊆ W such that for each λ ∈ (0, 1], ε > 0, there exist t < ε 2 , p ∈ N, which deduces that N 1 ( x (n) −ξ , t) > 1 − λ for all n > p. Thus for all m, n > p, That is to say {Tx (n) α }, (α ∈ (0, 1]) is a Cauchy fuzzy points sequence in W. Since T and S are isometrics, we have Thus deduces that {Sx (n) α }, (α ∈ (0, 1]) is a Cauchy fuzzy point sequence in SX. From the fact Y is complete, there is a unique y ∈ Y such that Sx (n) α → y α . It may be proved that the element y has nothing to do with the choice of { x (n) }. In reality, if there exists a sequence { z (n) } ⊆ W such that for each λ ∈ (0, 1], ε > 0, there exist t < ε 2 , p ∈ N, which deduces that N 1 ( z (n) −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 Then N 2 (Sx (n) − Sz (n) , 2t) > 1 − λ. So we have 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.