Finite Dimensional Intuitionistic Fuzzy Normed Linear Space

Introduction : The authors T. Bag and S. K. Samanta [5] introduced the definition of fuzzy norm over a linear space following the definition S. C. Cheng and J. N. Moordeson [4] and they have studied finite dimensional fuzzy normed linear spaces. Also the definition of intuitionistic fuzzy n-normed linear space was introduced in the paper [3] and established a sufficient condition for an intuitionistic fuzzy n-normed linear space to be complete. In this paper, following the definition of intuitionistic fuzzy n-norm [3] , the definition of intuitionistic fuzzy norm ( in short IFN ) is defined over a linear space. There after a sufficient condition is given for an intuitionistic fuzzy normed linear space to be complete and also it is proved that a finite dimensional intuitionistic fuzzy norm linear space is complete. In such spaces, it is established that a necessary and sufficient condition for a subset to be compact. Thereafter following the definition of fuzzy continuous mapping [6], the definition of intuitionistic fuzzy continuity, strongly intuitionistic fuzzy continuity and sequentially intuitionistic fuzzy continuity are defined and proved that the concept of intuitionistic fuzzy continuity and sequentially intuitionistic fuzzy continuity are equivalent. There after it is shown that intuitionistic fuzzy continuous image of a compact set is again a compact set.

( b ) For any r 5 ε ( 0 , 1 ) , there exist r 6 , r 7 ε ( 0 , 1 ) such that r 6 * r 6 ≥ r 5 and r 7 ⋄ r 7 ≤ r 5 .

Let E be any set. An intuitionistic fuzzy set A of E is an object of the form A = { ( x , µ A ( x ) , ν A ( x ) ) : x ε E } , where the functions µ A : E -→ [ 0 , 1 ] and ν A : E -→ [ 0 , 1 ] denotes the degree of membership and the non -membership of the element x ε E respectively and for every x ε E ,

If A and B are two intuitionistic fuzzy sets of a non -empty set E , then A ⊆ B if and only if for all x ε E ,

Definition 5. Let * be a continuous t -norm , ⋄ be a continuous t -co -norm and V be a linear space over the field F ( = R or C ) . An intuitionistic fuzzy norm or in short IFN on V is an object of the form A = { ( ( x , t ) , N ( x , t ) , M ( x , t ) ) : ( x , t ) ε V × R + } , where N , M are fuzzy sets on V × R + , N denotes the degree of membership and M denotes the degree of non -membership ( x , t ) ε V × R + satisfying the following conditions :

( vii ) M ( x , t ) > 0 ;

( viii ) M ( x , t ) = 0 if and only if x = 0 ;

( ix )

where k > 0 . We now consider

Proof. Obviously follows from the calculation of the example 3.2 [ 3 ] . Definition 6. If A is an IFN on V ( a linear space over the field F ( = R or C )) then ( V , A ) is called an intuitionistic fuzzy normed linear space or in short IFNLS.

Proof. The proof directly follows from the proof of the theorem 3.4 [3] .

Taking limit, we have

Thus, we see that lim

Proof. Obvious.

Theorem 5. In an IFNLS ( V , A ), every subsequence of a convergent sequence converges to the limit of the sequence .

Proof. Obvious.

Theorem 6. In an IFNLS ( V , A ) , every convergent sequence is a Cauchy sequence.

Proof. Let { x n } n be a convergent sequence in the IFNLS ( V , A ) with lim

Note 1. The converse of the above theorem is not necessarily true . It is verified by the following example . where k > 0 . It is easy to see that

Theorem 7. Let ( V , A ) be an IFNLS , such that every Cauchy sequence in ( V , A ) has a convergent subsequence. Then ( V , A ) is complete .

Proof. Let { x n } n be a Cauchy sequence in ( V , A ) and { x n k } k be a subsequence of { x n } n that converges to x ε V and t > 0 . Since { x n } n is a Cauchy sequence in ( V , A ) , we have

M ( x nx , t ) = 0 Thus, { x n } n converges to x in ( V , A ) and hence ( V , A ) is complete . Theorem 8. Let ( V , A ) be an IFNLS , we further assume that ,

α : α ε ( 0 , 1 ) } are ascending family of norms on V . We call these norms as α -norm on V corresponding to the IFN A on V .

Proof. Let α ε ( 0 , 1 ) . To prove x 1 α is a norm on V , we will prove the followings :

( 1 )

The proof of ( 1) , ( 2) and ( 3 ) directly follows from the proof of the theorem 2.1 [5] . So, we now prove ( 4 ) .

} is an ascending family of norms on V . Now we shall prove that { x 2 α : α ε ( 0 , 1 ) } is also an ascending family of norms on V . Let α ε ( 0 , 1 ) and x , y ε V . It is obvious that

} is an ascending family of norms on V . Lemma 1. [5] Let ( V , A ) be an IFNLS satisfying the condition ( Xiii ) and { x 1 , x 2 , • • • , x n } be a finite set of linearly independent vectors of V . Then for each α ε ( 0 , 1 ) there exists a constant C α > 0 such that for any scalars

α is defined in the previous theorem.

Theorem 9. Every finite dimensional IFNLS satisfying the conditions ( Xii ) and ( Xiii ) is complete .

Proof. Let ( V , A ) be a finite dimensional IFNLS satisfying the conditions ( Xii ) and ( Xiii ). Also, let dim V = k and { e 1 , e 2 , • • • , e k } be a basis of V . Consider { x n } n as an arbitrary Cau

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