Abstract
The purpose of this study was to introduce the inner product over fuzzy matrices. By virtue of this definition, -norm is defined and the parallelogram law is proved. Again the relative fuzzy norm with respect to the inner product over fuzzy matrices is defined. Moreover Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle are established. Some equivalent conditions are also proved.
1. Introduction
The concept of fuzzy set was introduced by Zadeh [1] in 1965. Biswas [2] and El-Abyad and El-Hamouly [3] first tried to give a meaningful definition of fuzzy inner product space and associated fuzzy norm function. Also those definitions are restricted to the real linear space only. Felbin [4], Ragab, and Emam [5] introduced fuzzy 2-normed linear spaces. Meenakshi and Cokilavany [6], Bag and Samanta [7], and so forth have given different definitions of fuzzy normed linear spaces. Gani and Kalyani [8] introduced the concept of binormed sequences in fuzzy matrices. Gani and Kalyani [9] introduced the properties of fuzzy -norm matrices. Gani and Kalyani Introduced the definition of fuzzy equivalence relation. Recently Majumdar and Samanta [10] and Hasankhani et al. [11] have introduced a definition of fuzzy inner product space whose associated norm is of Felbin type. Somasundaram and Beaula [12] defined the 2-fuzzy 2-normed linear space. Some standard results in fuzzy 2-normed linear spaces were extended. The organization of the paper is as follows. Section 2 provides some preliminary results which are used in this paper. In Section 3, we study some results on inner product over fuzzy matrices and the parallelogram law is proved. Section 4 is devoted to establishing Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle theorem in fuzzy setting.
In this paper, we have introduced the new concept of inner product over fuzzy matrices on , the set of all fuzzy sets of . We introduce the notion of -norm on an inner product over fuzzy matrices. With the help of it, the standard parallelogram law is proved. Some important results on inner product over fuzzy matrices are proved. Moreover Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle are established. Some equivalent conditions are also proved.
2. Preliminaries
We consider the fuzzy algebra with operation and the standard order “” where and for all , in . is a commutative semiring with additive and multiplicative identities 0 and 1, respectively. Let denote the set of all fuzzy matrices over . In short, is the set of all fuzzy matrices of order . Define “+” and scalar multiplication in as , where and , and , where is in ; with these operations forms a linear space.
3. -Norm and Binorm Fuzzy Matrices
Definition 1. Let be the set of all fuzzy matrices over . For every in define -norm of denoted by as
Definition 2. Let be in and let be in such that . Then the pair is called a fuzzy point in and it is denoted by . The dual fuzzy point for is the point with -norm () denoted by .
Definition 3. The set of all fuzzy points in is given by .
In we follow the usual order relation; correspondingly we define an order relation in as follows.
Definition 4. We define if and only if and if and only if (then automatically ).
Definition 5. A binorm with respect to the -norm in is a real valued function defined on to satisfying the following conditions: (i) if and only if .(ii).(iii).(iv).Then is called a fuzzy binormed linear space with respect to the -norm.
[ is defined for as 0 (by ()). Therefore is defined for ].(i)When both the fuzzy points coincide. Therefore . Conversely both fuzzy points should coincide .(ii). Therefore .(iii)Case (i): let ; From (2) and (3), Case (ii): let ; From (5) and (3),(iv)Case (i): let ; Case (ii): let ; From (9) and (11), Therefore . Since , [in fact ].Thus is a fuzzy binorm with respect to the -norm in and hence is a fuzzy binormed linear space with respect to the -norm.
Definition 6. Let be a linear space over (the field is real or complex number). A fuzzy subset of is called a fuzzy norm of if and only if, for all and in .For all with , .For all with , .For all with , , if .For all , , . is a nondecreasing function of and as .The pair will be referred to as a fuzzy normed linear space.
Theorem 7. Let be nonempty and be the set of all fuzzy sets in . If and and is a bounded function for .
Let be the space of real numbers; then is a linear space over the field ; where the addition and scalar multiplication are defined by
Proof. is said to be fuzzy normed space if to every there is an associated nonnegative real number called the -norm of in such a way that
(i) therefore if and only if ;
(ii) one has (iii) one has ForTherefore is a fuzzy normed linear space.
4. Inner Product over Fuzzy Matrices
Definition 8. Let be a linear space over the filed . The fuzzy subset defined a mapping from to such that, for all , :(i)For , .(ii)For , .(iii)For , .(iv)For , for all .(v) for all .(vi) for all if and only if .(vii) is a monotonic nondecreasing function of and as .Then is said to be 2-fuzzy inner product (2-FIP) on and the pair is called a 2-fuzzy inner product space (2-FIPS).
Example 9. Iftherefore . ConsiderTherefore . One has the following:(i) For (ii)For ,(iii) For ,(iv) For , For
Theorem 10. If a 2-fuzzy inner product (2-FIP) on is strictly convex, and if , then are 2-fuzzy inner product on .
Proof. (i) For and , we haveTherefore .
(ii) One has (iii) One has (iv) One has(v) One has(vi) One has(vii) Since and is a monotonic nondecreasing function of and , this implies that has also the property. Thus is a 2-fuzzy inner product on .
Definition 11. Let be a 2-FIP satisfying the condition implying that . Then, for all in , define as a crisp norm on , called the -norm on generated by . Now using these definitions let one define fuzzy norm on and verify the conditions as follows.
Theorem 12. Let be a 2-FIP on ; then defined byis a fuzzy -norm on .
Proof. (i) for all in and .
(ii) For all , if and only if .
It follows that if and only if .
(iii) For all and ,(iv) One has
Theorem 13 (Cauchy Schwarz inequality). Let be a 2-fuzzy inner product on , , and let be a -norm generated from 2-FIP on ; then , for all .
Proof. We can assume that We can find a fuzzy sequence Then, taking limit into the above,Therefore .
Theorem 14 (parallelogram law). Let be a 2-fuzzy inner product on , , and let be a -norm generated from 2-FIP on ; then .
Proof. Considerwhere is the fuzzy -norm induced from .
One has the following:Therefore, .
Theorem 15 (Pythagoras). Let be a 2-fuzzy inner product on , , and let be a -norm generated from 2-FIP on ; then .
Proof. Consider the following:AlsoFrom and ,
Definition 16. A sequence in a fuzzy -normed linear space is called a Cauchy sequence with respect to 2-norm if as , .
Definition 17. A sequence in a fuzzy -norm linear space is called a convergent sequence with respect to 2-norm if there exists , such that .
Definition 18. A fuzzy 2-normed linear space is said to be complete if every Cauchy sequence converges.
Definition 19. A complete fuzzy 2-normed linear space is called 2-fuzzy Banach space.
Definition 20. A complex 2-fuzzy Banach space is said to be 2-fuzzy Hilbert space if its norm is induced by the 2-fuzzy inner product.
Theorem 21 (Fundamental Minimum Principle). A nonempty closed convex fuzzy subset of a fuzzy Hilbert space has a unique element of smallest -norm.
Proof. If is a 2-fuzzy convex set.
It contains whenever it contains and ; we know that And so,Let where .
Let , be a closed convex fuzzy set , inside a fuzzy Hilbert space, so that both and are within of the of the -norm in . Then there exists a sequence in such that .
By parallelogram law,Since is convex, is a 2-fuzzy Cauchy sequence in , since is complete and is closed convex. Thus such that ; also , which implies that with smallest -norm. Now let us prove the uniqueness part. Suppose that there exists a other than with the same -norm .
Again, by parallelogram law,Which is a contradiction to the definition of :Hence there exists a unique element of smallest -norm.
Theorem 22. The 2-FIP is a continuous function on .
Proof. Let be a 2-FIP and let and be sequence in such that and ; then,Since and converge,
Theorem 23. Let be a 2-FIP and is indeed a -norm on . It also satisfies (i) for every nonzero ,(ii), for all and .
Proof. ConsiderFor ,Also, (i)Let be a nonzero element in ; then and hence in .(ii)Let ; Since is a fuzzy norm,The square roots exist and are unique in .
Therefore .
Theorem 24 (polarisation identity). If are element of 2-FIPS , then
Proof. ConsiderTherefore,
5. Concluding Remarks
In this paper the concept of inner product over fuzzy matrices has been discussed. We plan to extend our research work to (1) orthogonal of fuzzy matrices and (2) the Moore Penrose inverse and spectral inverse of fuzzy matrices.
Conflict of Interests
The authors declare that they do not have any conflict of interests regarding the publication of this paper.