Abstract and Applied Analysis

Volume 2012 (2012), Article ID 969313, 11 pages

doi:10.1155/2012/969313

Research Article

The Intuitionistic Fuzzy Normed Space of Coefficients

B. T. Bilalov, S. M. Farahani, and F. A. Guliyeva

Institute of Mathematics and Mechanics of Azerbaijan, National Academy of Sciences, 1141 Baku, Azerbaijan

Received 25 January 2012; Accepted 19 March 2012

Academic Editor: Gabriel Turinici

Copyright © 2012 B. T. Bilalov 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.

Intuitionistic fuzzy normed space is defined using concepts of -norm and -conorm. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity, and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy norm and strong basicity of canonical system in this space are proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator.

1. Introduction

The fuzzy theory, dating back to Zadeh [1], has emerged as the most active area of research in many branches of mathematics and engineering. Fuzzy set theory is a powerful handset for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. The concept of fuzzy topology may have very important applications in quantum particle physics, particularly in connection with both string and theories introduced and studied by El Naschie [2–4] and further developed in [5]. So, further development in E-Infinity may lead to a set transitional resolution of quantum entanglement [6]. A large number of research works are appearing these days which deal with the concept of fuzzy set numbers, and the fuzzification of many classical theories has also been made. The concept of Schauder basis in intuitionistic fuzzy normed space and some results related to this concept have recently been studied in [7–9]. These works introduced the concepts of strongly and weakly intuitionistic fuzzy (Schauder) basis in intuitionistic fuzzy Banach spaces (IFBS in short). Some of their properties are revealed. The concepts of strongly and weakly intuitionistic fuzzy approximation properties (sif-AP and wif-AP in short, resp.) are also introduced in these works. It is proved that if the intuitionistic fuzzy space has a sif-basis, then it has a sif-AP. All the results in these works are obtained on condition that IFBS admits equivalent topology using the family of norms generated by -norm and -conorm (we will define them later).

In our work, we define the basic concepts of classical basis theory in intuitionistic fuzzy normed spaces (IFNS in short). Concepts of weakly and strongly fuzzy spaces of coefficients are introduced. Strong completeness of these spaces with regard to fuzzy norm and strong basicity of canonical system in them is proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator.

In Section 2, we recall some notations and concepts. In Section 3, we state our main results. We first define the fuzzy space of coefficients and then introduce the corresponding fuzzy norms. We prove that for nondegenerate system the corresponding fuzzy space of coefficients is strongly fuzzy complete. Moreover, we show that the canonical system forms a strong basis for this space.

2. Some Preliminary Notations and Concepts

We will use the usual notations: will denote the set of all positive integers, will be the set of all real numbers, will be the set of complex numbers, and will denote a field of scalars (, or ), . We state some concepts and facts from IFNS theory to be used later.

One of the most important problems in fuzzy topology is to obtain an appropriate concept of intuitionistic fuzzy normed space. This problem has been investigated by Park [10]. He has introduced and studied a notion of intuitionistic fuzzy metric space. We recall it.

Definition 2.1. A binary operation is a continuous -norm if it satisfies the following conditions:

(a) is associative and commutative,(b) is continuous,(c), ,(d) whenever and , , , , .

Example 2.2. Two typical examples of continuous -norm are and .

Definition 2.3. A binary operation is a continuous -conorm if it satisfies the following conditions:

is associative and commutative, is continuous,, , whenever and , .

Example 2.4. Two typical examples of continuous -conorm are and .

Definition 2.5. Let be a linear space over a field . Functions are called fuzzy norms on if they hold the following conditions:

(1), , ,(2), ,(3), ,(4) is a nondecreasing function of for and , ,(5), , ,(6), , ,(7), ,(8), ,(9) is a nonincreasing function of for and , ,(10), , ,(11), , .

Then the 5-tuple is said to be an intuitionistic fuzzy normed space (shortly IFNS).

Example 2.6. Let be a normed space. Denote and , for, and define and as follows:

Then is an IFNS.

The above concepts allow to introduce the following kinds of convergence (or topology) in IFNS.

Definition 2.7. Let be a fuzzy normed space, and let be some sequence, then it is said to be strongly intuitionistic fuzzy convergent to (denoted by , or in short) if and only if for , , , , .

Definition 2.8. Let be a fuzzy normed space, and let be some sequence, then it is said to be weakly intuitionistic fuzzy convergent to (denoted by , , or in short) if and only if for , , , , . More details on these concepts can be found in [10–19].

Let be an IFNS, and let be some set. By , we denote the linear span of in . The weakly (strongly) intuitionistic fuzzy convergent closure of will be denoted by (). If is complete with respect to the weakly (strongly) intuitionistic fuzzy convergence, then we will call it intuitionistic fuzzy weakly (strongly) Banach space ( or ( or ) in short). Let be an (). We denote by () the linear space of linear and continuous in () functionals over the same field .

Now, we define the corresponding concepts of basis theory for IFNS. Let be some system.

Definition 2.9. System is called -complete (-complete) in (in ) if .

Definition 2.10. System is called -biorthogonal (-biorthogonal) to the system if , , where is the Kronecker symbol.

Definition 2.11. System is called -linearly (-linearly) independent in if in (in ) implies , .

Definition 2.12. System is called -basis ( -basis) for (for ) if , in (in ).

We will also need the following concept.

Definition 2.13. System is called nondegenerate if , .

3. Main Results

3.1. Space of Coefficients

Let be an IFNS, and let be some system.

Assume that

It is not difficult to see that and are linear spaces with regard to component-specific summation and component-specific multiplication by a scalar. Take , and assume that

Let us show that and satisfy the conditions (1)–(11).

(1)It is clear that , .(2)Let , . Hence, , , . Suppose that the system is nondegenerate. It follows from the above-stated relations that, for , we have , . Hence, . Continuing this way, we get at the end of this process that, , that is, .(3)The validity of relation , is beyond any doubt.(4) As is a nondecreasing function on it is not difficult to see that has the same property. Let us show that . Take . Let and . It is clear that . Then it follows from the definition of - that,. Property (4) implies

As a result, we get

As is a nondecreasing function of, it follows from (3.4) that

We have

where. As for, we have,. Let, then it is clear that

It follows from (3.5) and (3.6) that

Let. Hence, we obtain from (3.6) and (3.7) that

Thus,,.(5) Let and . We have

(6) As , , it is clear that , , .(7) Let the system be nondegenerate. Assume that , , then,,. For, we have,. Continuing this process, we get, .(8) Clearly, , .(9) It follows from the property (9) that is a nonincreasing function on . Therefore, is a nonincreasing function on . Let us show that . Let and . Take . It is clear that . Then it follows from the definition of - that , . We have

As is a nonincreasing function, it is clear that

We have

As for , we have , . Let . It is clear that , . It follows from (3.12) that , . Let , then it is clear that , .(10) Let and . We have

(11)Consider the following:

Thus, we have proved the validity of the following.

Theorem 3.1. Let be a fuzzy normed space, and let be a nondegenerate system, then the space of coefficients is also strongly fuzzy normed space.

The following theorem is proved in absolutely the same way.

Theorem 3.2. Let be a fuzzy normed space, and let be a nondegenerate system, then the space of coefficients is also weakly fuzzy normed space.

3.2. Completeness of the Space of Coefficients

Subsequently, we assume that is IFBS. Let us show that is a strongly fuzzy complete normed space. First, we prove the following.

Lemma 3.3. Let , , and let be some sequence. If , that is, , , , , and , then , .

Proof. As , it is clear that . We have for . Assume that the relation is not true, then and , . It is clear that uniformly in . On the other hand, for , we have . So we came upon a contradiction which proves the lemma.

Further, we assume that the following condition is also fulfilled.

(12) The functions , are continuous for .

Take-fundamental sequence,. Then uniformly in, that is,

uniformly in . Take and fix it. We have

Then from property (5), we get

It follows directly from this relation that uniformly in . As , Lemma 3.3 implies , that is, the sequence is fundamental in . Let , as . Denote . Let us show that uniformly in . Take . It is clear that , , , . Consequently,

Hence,

As shown above, uniformly in . Now let us take into account the fact that , . Indeed, if , then , , and clearly, for . If , then for sufficiently large values of we have , and as a result,

Passage to the limit in the inequality (3.20) as yields

We have

As , it is clear that , :

We have

It follows that the series is strongly fuzzy convergent, that is, -. Consequently,, and the relation (3.22) implies that uniformly in, . It can be proved in a similar way that uniformly in . As a result, we obtain that the space is strongly fuzzy complete. Thus, we have proved the following.

Theorem 3.4. Let be a fuzzy Banach space with condition (12), and let be a nondegenerate system, then the space of coefficients is a strongly fuzzy complete normed space.

Consider operator defined by

Let in , where . We have

It follows directly that -, that is, the operator is strongly fuzzy continuous. Let , that is, , where . It is clear that if the system is -linearly independent, then , , and as a result, . In this case, ImT. If, in addition, ImT is -closed in , then is also continuous.

Denote by a canonical system in, where . Obviously, , . Let us prove that forms an -basis for . Take and show that the series is strongly fuzzy convergent in . In fact, the existence of - in implies that , ,

We have

It follows that the series is strongly fuzzy convergent in . Moreover,

Consequently, -, that is, . Consider the functionals , . Let us show that they are -continuous. Let -, where . As established in the proof of Theorem 3.4, we have as , that is, as for . Thus, is -continuous in for . On the other hand, it is easy to see that , , that is, is -biorthogonal to . As a result, we obtain that the system forms an -basis for . So we get the validity of the following.

Theorem 3.5. Let be a fuzzy Banach space with condition (12), and let be a nondegenerate system. Then the corresponding space of coefficients is strongly fuzzy complete with canonical -basis .

Suppose that the system is-linearly independent and ImT is closed, then it is easily seen that forms an -basis for ImT, and in case of its -completeness in , it forms an -basis for . In this case, and are isomorphic, and is an isomorphism between them. The opposite of it is also true, that is, if the above-defined operator is an isomorphism between and , then the system forms an -basis for . We will call a coefficient operator. Thus, the following theorem holds.

Theorem 3.6. Let be a fuzzy Banach space with condition (12), let be a nondegenerate system, let be a corresponding strongly fuzzy complete normed space, and let be a coefficient operator. System forms an -basis for if and only if the operator is an isomorphism between and .

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