Fuzzy Convergence Sequence and Fuzzy Compact Operators on Standard Fuzzy Normed Spaces

The main purpose of this work is to introduce some types of fuzzy convergence sequences of operators defined on a standard fuzzy normed space (SFN-spaces) and investigate some properties and relationships between these concepts. Firstly, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional is given. Then the notions of weakly and strongly fuzzy convergence sequences of operators (Γn ) are introduced and essential theorems related to these concepts are proved. In particular, if (Γn) is a strongly fuzzy convergent sequence with a limit Γ where Γ linear operator from complete standard fuzzy normed space (U, NU,⊛) into a standard fuzzy normed space (V, NV,⊛) then Γ belongs to the set of all fuzzy bounded linear operators FB(U, V). Furthermore, the concept of a fuzzy compact linear operator in a standard fuzzy normed space is introduced. Also, several fundamental theorems of fuzzy compact linear operators are studied in the same space. More accurately, every fuzzy compact linear operator Γ: U → V is proved to be fuzzy bounded where (U, NU,⊛) and (V, NV,⊛) are two standard fuzzy normed spaces.


Introduction:
Fuzzy set theory was initiated by Zadeh in 1965(1), then numerous mathematicians have studied this concept and obtained different main results from various points of view (2,3). The definition of the fuzzy norm was firstly introduced by Katrasas (4) who defined the fuzzy norm in linear space. Felbin (5) presented another different type of fuzzy norm defined on a vector space. Thought of a fuzzy normed linear space and the fuzzy linear operator was introduced by Shih Chuan and John Mordeson (6). Bag and Samanta (7) studied the relation between fuzzy boundedness and fuzzy continuity and gave a notion of boundedness of a linear operator in fuzzy normed spaces. They also debated the notions of convergent sequence and Cauchy sequence in the same space. In (8) Sadeqi and Salehi introduced the concept of the fuzzy compact operator and proved the equivalence of the concepts of fuzzy boundedness and fuzzy norm continuity of linear operators. Moreover, different approaches for the space of fuzzy norm can be found in (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). In this paper, the first main goal is to present the definition of weak and strong fuzzy convergence sequence of operators in a standard fuzzy normed space (briefly SFN-space) where is an arbitrary continuous tnorm. Then basic properties related to these concepts are given. The second goal is to introduce the notion of fuzzy compact linear operators and discussed some important properties of them. Structurally, this paper involves the following: the basic properties of a standard fuzzy normed space are given, then, in Section 3, the definition of strongly and weakly fuzzy convergence sequence of operators in an SFN-space are presented. In Section 4, the concept of a fuzzy compact linear operator from an SFN-space into another SFN-space is given. Besides, different properties of the fuzzy compact linear operator will be proved, for example, the product of a fuzzy compact linear operator Γ and fuzzy bounded linear operator Λ on a standard fuzzy normed space is proved to be fuzzy compact and in case U is finite dimension then the linear operator Γ is proved to be fuzzy compact. Finally, the paper finished with a conclusion section.

Preliminaries
This section presents some definitions of standard fuzzy normed spaces and basic properties related to these concepts.

Definition 1(23)
Suppose that a binary operation. Then is called continuous t-norm if satisfies the following properties for each : The following definition of the standard fuzzy normed space considered in (24) is given.

Definition 2 (24)
A Standard fuzzy normed space (briefly, SFNspace) is a triple where is a linear space over field , is a continuous t-norm and is a fuzzy set on satisfying the conditions: The next proposition shows that an ordinary normed space with some conditions will be standard fuzzy normed space induced by the norm ‖.‖ as in (24).

Proposition 1(24)
Suppose that ‖ ‖ be an ordinary normed space with ‖ ‖ is an integer for each and for each . Define is standard fuzzy normed space deduced from the norm ‖ ‖. The following group of definitions are presenting the notions of the open ball, open and closed set convergent and Cauchy sequence, and the closure of a set respectively that given in (24).

Definition 3 (24)
Let be an SFN-space. Put . Then is called an open ball centered at with radius .

Definition 4 (24)
Suppose that is an SFN-space and . Then (1) is open set if for each there is an with .
is open, that is .

Definition 5 (24)
In an SFN-space , for any a sequence in is said to be: 1-convergent to a point if there is a natural number with ) for each . 2-Cauchy if there is a natural number with .

Definition 6 (24)
Let be a subset of an SFN-space . Then ̅ is called the closure of which is the intersection of all closed sets containing: ̅ ⋂ .

Definition 7 (24)
Suppose that is an SFN-space and . Then is called fuzzy bounded if for each there is a real number say r, with .

Definition 8 (24)
Suppose that and be two SFN-spaces and : D( ) → be a linear operator, where D( ) . The operator is said to be fuzzy bounded if there is a real number , 0 < < 1 such that for all D( ), (T ) ≥ (1-) ( ) . The set of all fuzzy bounded operator is denoted by such that | (25).

Theorem 1(25)
Suppose that and are two SFN-spaces. Then ( is SFN-space such that for each .

Definition 9 (25)
A linear functional from an SFN-space into the SFN-space is called fuzzy bounded if there exists , with for each . Furthermore, the standard fuzzy norm of is and .

Theorem 2 (25)
Let be SFN-space. Then is compact if and only if each sequence of elements in has a subsequence converging to an element in .

Fuzzy Convergence Sequence and Fuzzy Convergence Sequence of Operators
This section is devoted to introduce the notations of strongly and weakly fuzzy convergent sequence of operators and prove some properties in the standard fuzzy normed space. It is well known that if is an SFN-space, then the set of all fuzzy bounded functional , is fuzzy bounded linear functional} with a fuzzy norm which is called the fuzzy dual space of (25).
First, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional on a standard fuzzy normed space is introduce as follows.

Definition 10
Let be an SFN-space. A sequence in U is called weakly fuzzy convergent if there exists an element such that ) for every

Theorem 3
Suppose that be a sequence in an SFNspace . If fuzzy convergent sequence then weak fuzzy convergent with the same limit. Proof: let be a sequence in an SFN-space . Since fuzzy converges to then there exists an element and a positive integer with for all n . Now for each . Put for some , then there is , with . Hence and this shows that is a weakly fuzzy convergent. To prove that a sequence in an SFN-space is weak fuzzy convergence, the following definition is introduced.

Definition 11
Suppose that be an SFN-space and . Then is said to be fuzzy total set if ̅̅̅̅̅̅̅̅̅̅ .

Theorem 4
Let be an SFN-space and be a sequence in . If the sequence is a fuzzy bounded and the sequence fuzzy converges to for each element of a fuzzy total subset , then the sequence is a weakly fuzzy convergent to .

Proof:
Since is a fuzzy bounded sequence then for all and for some To prove that the sequence is a weakly fuzzy convergent to , suppose that for every , the sequence should converge to . Since is fuzzy total in , for each , there is a sequence in span such that fuzzy converges to . Hence for any given obtain . Moreover, since , there is an such that for all and . Hence for n> N, obtain Hence there exists such that and . This shows that the sequence fuzzy converges weakly to .

Definition 12
Suppose that is an SFN-space and . Then is called : (ii) Meager in if represents the countable union of rare sets. The complement of a meager set is called residual.
It is well known that in an SFN-space every convergent sequence is Cauchy. Also, an SFN-space is called complete if each Cauchy sequence in converges to an element in . One of the essential theorems in functional analysis is Baire's theorem (26). Now, Fuzzy Baire's 4024 theorem in the standard fuzzy normed spaces is proved.

Theorem 5
If an SFN-space is complete, it is non-meager in itself. Hence if is complete and ⋃ ( closed) then there is at least one contains an open subset which is not equal to .
Proof: Assume the complete SFN-space was meager in itself. Then with each nowhere in U. Cauchy sequence ( will be constructed whose limit is is in no this contradicts eq. (1)

Fuzzy Compact Linear Operators on SFN-space
In this section, the notion of fuzzy compact linear operators on a standard fuzzy normed space will be given. Moreover, the main general properties of this notion will be proved.

Definition 15
Let and be an SFN-spaces and a linear operator between and . Then is called fuzzy compact operator if for each bounded subset of , is relatively compact in , that is, the closure ̅̅̅̅̅̅̅ is compact.

Theorem 8
Let and be two SFNspaces. The linear operator is fuzzy compact if and only if the sequence has a convergent subsequence in for every fuzzy bounded sequence in .
Proof: Consider be a linear operator from an SFNspace into an SFN-space and let be fuzzy compact. Assume that be a fuzzy bounded sequence in . Then by Definition15, is relatively compact, this means that the closure of is compact in . Hence by Theorem 2, the sequence contains a convergent subsequence. For the converse, assume that each fuzzy bounded sequence in contains a subsequence, say with converges in . Suppose that be any fuzzy bounded subset and consider be any sequence in . Then for and the sequence is fuzzy bounded since is fuzzy bounded. The sequence by the assumption contains a convergent subsequence, hence the closure ̅̅̅̅̅̅̅ is compact. Thus is fuzzy compact. Relying on the above theorem, the next proposition discusses the relationship between the fuzzy compact linear operator and the fuzzy bounded operator.

Proposition 2
Every fuzzy compact linear operator is fuzzy bounded where and are two SFN-spaces.
Proof: let be a fuzzy compact linear operator the operator have to proved fuzzy bounded. The set is fuzzy bounded for each . Now consider an arbitrary sequence in that can be written as for some in . Since is fuzzy bounded and by assumption, is fuzzy compact operator, hence by Theorem 8, the sequence contains a convergent subsequence. Now suppose that is unbounded, this means that contains unbounded sequence say and this sequence has no convergent subsequence. But since each sequence in has convergent subsequence therefore must be fuzzy bounded.

Proposition 3
Let and be an SFN-spaces and be a linear operator. If is fuzzy bounded and the dimension of is finite then is a fuzzy compact operator. Proof: Let be any fuzzy bounded sequence in . Then by using the inequality (see (24), Remark2.2) it is clear that the sequence is fuzzy bounded. Thus is relatively compact since the dimension of is finite that follows, the sequence has a convergent subsequence. Because was an arbitrary fuzzy bounded sequence in , hence by Theorem 2, is fuzzy compact.
For a finite-dimensional SFN-space , each linear operator on is fuzzy bounded (see (25), Theorem2.8.14). In the following, a linear operator over a finite dimension SFN-space is proved to be a fuzzy compact.

Proposition 4
Let and be an SFN-spaces and be a linear operator. If the dimension of is finite then is a fuzzy compact operator.
Proof: Because the dimension of is finite, hence is fuzzy bounded operator by Theorem 2.8.14 (25) and since therefore by Proposition 3, is a fuzzy compact operator. Let and are two SFN-spaces, then the set of all fuzzy compact linear operators is denoted by such that | . Now by Theorem 8, the following result is provided.

Theorem 9
Let be an SFN-space and complete SFN-space. If and is a fuzzy convergent sequence of operators then the limit operator is fuzzy compact.  (23). Therefore, obtain which proves that ( ) is Cauchy. Since is complete so ( ) converges. Notice that is a subsequence of the sequence and using Theorem 8; therefore, is a fuzzy compact.
The final result explains the relationship between the product of two linear operators on standard fuzzy normed spaces.

Proposition 5
Let be a fuzzy compact linear operator and be a fuzzy bounded linear operator on SFN-space . Then the product and are fuzzy compact. Proof: Let be a fuzzy compact operator and be a fuzzy bounded operator on . To show that is a fuzzy compact operator, take a fuzzy bounded set in since is fuzzy bounded operator so is fuzzy bounded set and is relatively compact because is a fuzzy compact operator. Hence is a fuzzy compact operator. Now to prove that is fuzzy compact, assume that be any fuzzy bounded sequence in then the sequence has a convergent subsequence say, by Theorem 8 and the sequence converges by Corollary (2.7)(i) (24). Thus is fuzzy compact.

Conclusion:
In this paper, the convergence of the sequence of operators in a standard fuzzy normed space is studied then some of their essential properties are discussed. The concept of fuzzy compact linear operators between standard fuzzy normed spaces is given. Finally, several properties of fuzzy compact linear operators in a standard fuzzy normed space (SFN-spaces) have been studied.

Author's declaration:
-Conflicts of Interest: None. -Ethical Clearance: The project was approved by the local ethical committee in University of Technology.