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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 131868, 14 pages
http://dx.doi.org/10.1155/2010/131868
Research Article

Schauder Basis, Separability, and Approximation Property in Intuitionistic Fuzzy Normed Space

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul 34220, Turkey
3Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia

Received 23 August 2010; Accepted 30 November 2010

Academic Editor: Norimichi Hirano

Copyright © 2010 M. Mursaleen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define and study the concepts of Schauder basis, separability, and approximation property in intuitionistic fuzzy normed spaces and establish some results related to these concepts. We also display here some interesting examples by using classical sequence spaces .

1. Introduction and Background

We will write for the set of all complex sequences . Let , , and denote the sets of all bounded, convergent, and null sequences, respectively. We write for . By and , we denote the sequences such that for ,, and . Note that , , and are Banach spaces with the sup-norm , and are Banach spaces with the norm .

A sequence in a normed linear space is called a Schauder basis [1] if for every , there is a unique sequence of scalars such that , that is, .

Recently, the concept of intuitionistic fuzzy normed space has been introduced and studied by Saadati and Park [2] and further studied by Mursaleen and Mohiuddine [35]. The concept of Schauder basis and its applications have recently been studied by Palomares et al. [6] and by Yılmaz [7]. In this paper, we define and study the concept of Schauder basis, separability, and approximation property in intuitionistic fuzzy normed spaces and establish some results related to these concepts analogous to those of Yılmaz [7]. We also display here some interesting examples by using classical sequence spaces .

In this section, we recall some notations and basic definitions used in this paper.

Definition 1.1. A binary operation is said to be a continuous -norm if it satisfies the following conditions:(a) is associative and commutative, (b) is continuous, (c) for all ,(d) whenever and for each .

Definition 1.2. A binary operation is said to be a continuous -conorm if it satisfies the following conditions: is associative and commutative, is continuous, for all , whenever and for each .

Using the notions of continuous -norm and -conorm, Saadati and Park [2] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition 1.3. The five-tuple is said to be an intuitionistic fuzzy normed spaces (for short, IFNS) if is a vector space, is a continuous -norm, is a continuous -conorm, and are fuzzy sets on satisfying the following conditions. For every and ,(i), (ii), (iii) if and only if , (iv) for each , (v), (vi) is continuous,(vii) and , (viii), (ix) if and only if , (x) for each , (xi),(xii) is continuous, and(xiii) and.
In this case is called an intuitionistic fuzzy norm.

Example 1.4. Let be a normed space, , and for all . For all and every , consider Then is an IFNS.

Remark 1.5 (see [3]). Let be an IFNS with the condition Let , for all . Then is an ascending family of norms on . These norms are called α-norms on corresponding to intuitionistic fuzzy norm .

2. Some Topological Concepts in IFNS

Recently, the strong and weak intuitionistic fuzzy convergence as well as strong and weak intuitionistic fuzzy limit were discussed by Mursaleen and Mohiuddine [3].

Definition 2.1. Let be an IFNS. Then, a sequence is said to be(i)weakly intuitionistic fuzzy convergent to if and only if, for every and , there exists some such that and for all . In this case we write ,(ii)strongly intuitionistic fuzzy convergent to if and only if, for every , there exists some such that and for all . In this case we write .

The following result characterizes the (wif)- and (sif)-limit through α-norms.

Proposition 2.2. Let and be two IFNS satisfying (1.2) and be a mapping. Then(i) if and only if, for each(ii) if and only if where and are α-norms of the intuitionistic fuzzy norms and , respectively.

Proof. Here we prove the case (ii). Suppose that . For a given , there exists some such that for all . For each , if then and . Hence and and so that . Since does not depend on α, we get
Conversely, let uniformly in α. Given , there exists some such that for all and . Choose some and or . Since there exists some such that and . Hence by the hypothesis, that is, So, we get and .

Proposition 2.3. Let be a sequence in the IFNS satisfying (1.2). Then(i) if and only if, for each (ii) if and only if where are α-norms of the intuitionistic fuzzy norms .

The proof of the above theorem directly follows from Propositions 2.2.

We define the following concepts analogous to that of Yılmaz [7].

Definition 2.4. The sif(wif)-closure of a subset in IFNS is the set of all such that there exists a sequence such that . In this case, we write . is said to be sif(wif)-closed whenever .

It is easy to see that . The following example shows that inclusion is strict.

Example 2.5. Let and on . Let and we show that . For every , there exists a sequence such that as , for each . This is accomplished by taking since each and for each . However, . Indeed for , there exists such that as , uniformly in α. This means that, given , there exists an integer such that for every and , On the other hand, for all and . Letting (and hence ), we get . Therefore, .

Definition 2.6. A subset of an IFNS is said to be dense in if and only if .

Definition 2.7. An IFNS is said to separable if it contains a countable dense subset, that is, there is a countable set with the following property: for each and each , there is at least one with for .

Theorem 2.8. Every finite dimension IFNS is separable.

Proof. Let be a finite dimension normed linear space and a basis of . Since is a countable subset of , it follows that is countable subset of . Also, is dense in . To see this, let and . Let be scalars such that . By the denseness of in , there exist in such that for all . Then it follows that Similarly, This implies that is dense in .

Theorem 2.9. Every IFNS having wif-basis is separable.

Proof. Let be IFNS with wif-basis . Since with for all is dense in , it is enough to show that has a countable dense subset. Let for all . Then will be a countable dense subset of (see Theorem 2.8). Thus is a countable dense subset of .

3. Intuitionistic Fuzzy Schauder Bases

In this section, we define strong and weak intuitionistic fuzzy Schauder bases.

Definition 3.1. Let be an IFNS. Then, a sequence is said to be(i)strongly intuitionistic fuzzy (Schauder) basis (for short, sif-basis) of if and only if for every there exists a unique sequence of scalars such that this means that for each there exists such that implies for all ,(ii)weak intuitionistic fuzzy (Schauder) basis (for short, wif-basis) of if and only if for every there exists a unique sequence of scalars such that this means, for each and , there exists such that implies

Proposition 3.2. Let be a sequence in the IFNS satisfying (1.2). Then(i) is a wif-basis of if and only if for every there exists a unique sequence of scalars such that for each , (ii) is a sif-basis of if and only if for every there exists a unique sequence of scalars such that where are α-norms of the intuitionistic fuzzy norms .

The proof of the above theorem is similar to Propositions 2.2.

Definition 3.3. By the notations in Definition 3.1., the mappings are called coordinate functionals and natural projections, respectively, associated to the sif(wif)-basis in .

Proposition 3.4. Let be a basis in wif-complete IFNS satisfying (1.2). Then each and is wif-continuous.

Proof. By Proposition 3.2, is also a Schauder basis in the Banach space for each . Thus are continuous. Therefore, the mappings are wif-continuous for each .

Remark 3.5. It is obvious that, if is an sif-basis of then it is wif-basis of , but not conversely. For the converse part, let us consider the following example.

Example 3.6. Let , the Banach space of all absolutely -summable sequences with the norm , and consider the intuitionistic fuzzy norm on . We can find α-norms of intuitionistic fuzzy norm since it satisfies condition (1.2). Thus This shows that Now, we show that the sequence is wif-basis but not sif-basis. Take any . Put then Hence and by Proposition 3.2 is wif-basis for . However, this convergence is not uniform in α since as .
However, if we put on , then is an IFNS satisfying (1.2), and is a sif-basis for since for each .

Remark 3.7. In finite-dimensional spaces, the definition of basis is independent of the intuitionistic fuzzy norm and hence coincides with the definition of a classical vector space basis (Hamel basis).

We know that every intuitionistic fuzzy normed space induces a topology such that for some if and only if for each there exists some and such that , where

Proposition 3.8. is a vector topology for ; that is, the vector space operations are continuous in this topology.

Proof. Since the family is a countable local basis at , is the first countable topology of . Hence it is sufficient to show that the vector space operations are sequentially continuous in . Suppose and in the topological space . This means , and as , for all . Now for all . Further, if in or , the scalar field of , then
Analogous to the classical results, we prove here that a normed linear space having a Schauder basis is separable.

Theorem 3.9. Let be an IFNS having wif-basis . Then the topological space is separable.

Proof. Let denotes the set of all finite linear combinations , where each is a (real or complex) rational number. Obviously, is countable and let us show that it is dense in . Suppose is arbitrary. There exists a unique sequence of scalars such that for each and , we can find some integer such that That is, for all , On the other hand, one can constitute a sequence of scalars converging to , for each . Hence the sequence converges to in by the continuity of vector space operations. This implies that every -centered -open sphere includes an element of .

Theorem 3.10. Let be a normed space and a basis in . Then is a wif-basis for IFNS , where

Proof. By the hypothesis, for each , there exists a unique sequence of scalars with in the norm topology as . Explicitly, for each , there exists an integer such that implies Now, for each and , take . So, there exists an integer such that implies if and only if

4. Intuitionistic Fuzzy Approximation Property

In this section, we define strong and weak intuitionistic fuzzy approximation property and prove some interesting results.

Definition 4.1. We say that sif-complete IFNS is said to have strong intuitionistic fuzzy approximation property (for short, sif-AP) if for every sif-compact set and there exists an operator of finite rank such that for all and .

Definition 4.2. A wif-complete IFNS is said to have weak intuitionistic fuzzy approximation property (for short, wif-AP) if for every wif-compact set and for each and there exists an operator of finite rank such that for all .

Remark 4.3. The operator in wif-AP depends both on and whereas it depends only on in sif-AP. depends on the set in both situations.

Proposition 4.4. (i) A wif-complete IFNS satisfying (1.2) has wif-AP if and only if for every wif-compact set and for each and there exists an operator of finite rank such that for all .
(ii) A sif-complete IFNS satisfying (1.2) has sif-AP if and only if for every sif-compact set and for each there exists an operator of finite rank such that for all .

The proof of the above theorem directly follows from Propositions 2.2.

Theorem 4.5. Let be an IFNS possessing a wif-basis . Then has the wif-AP.

Proof. Let be a wif-compact subset of . Let and be arbitrary. By the hypothesis, for some , there exists a unique sequence of scalars such that Then, there exists some such that for all . Further, each has a finite rank in the linear space since . Hence, each such that can be taken as a desired finite rank operator in the definition.

Remark 4.6. Theorem 4.5 can also proved for sif-basis.

Example 4.7. Let , the Banach space of all bounded sequence with sup-norm . Also, is another norm on . Define the function Then is an intuitionistic fuzzy norm on . We can find α-norms of intuitionistic fuzzy norm since it satisfies (1.2) condition. Thus IFNS cannot have a wif and hence a sif-basis since for and the Banach space is not separable. However, has sif-AP. Recall that the set of all partitions of natural numbers is a directed set by the relation which means that each is included in some . Now, for each where is the distinguished point in and is the characteristic function of for . Then is a projection on of finite rank. It is well known that the set converges to in . Let be sif-compact. Given and , then there exists a partition such that for But for since for every . That is, for , for all . Hence for some meets all requirements for sif-AP in Proposition 4.4.

Acknowledgment

This paper was completed when the first author (M. Mursaleen) visited the Yildiz Technical University, Istanbul, during June 20–July 19, 2010 under the project supported by TUBITAK BIDEB.

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