Research Article | Open Access
J. J. Font, J. Galindo, S. Macario, M. Sanchis, "A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces", Abstract and Applied Analysis, vol. 2014, Article ID 624920, 4 pages, 2014. https://doi.org/10.1155/2014/624920
A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces
We introduce fuzzy norm-preserving maps, which generalize the concept of fuzzy isometry. Based on the ideas from Vogt, 1973, and Väisälä, 2003, we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: let , be fuzzy normed spaces and let be a fuzzy norm-preserving surjection satisfying . Then is additive.
Studies on fuzzy normed spaces are relatively recent in the field of fuzzy functional analysis. It was Katsaras who in 1984 , while studying topological vector spaces, was the first to introduce the idea of fuzzy norm on a linear space. Eight years later, Felbin  offered an alternative definition. With this definition, the induced fuzzy metric is of Kaleva and Seikkala type . In 1994, Cheng and Mordeson  defined another type of fuzzy norm on a linear space whose associated fuzzy metric is of Kramosil and Michalek type . Finally, in  (see also ), Bag and Samanta redefined the concept of fuzzy norm given in  as follows.
Definition 1. Let be a real linear space. A function is said to be a fuzzy norm on if, for all and all , it satisfies the following:(N1) for ;(N2) if and only if for all ;(N3) if ;(N4);(N5).
The pair is called a fuzzy normed space.
We point out that classical normed spaces are strictly included in the class of fuzzy normed spaces (see ) and that (N2) and (N4) imply that, for a fixed , the function is nondecreasing. It is a well-known fact that every fuzzy norm on a real linear space induces a topology on defined as follows: a subbase for the neighborhood system at a point consists of the sets for all and . It is straightforward to verify that the filter of neighborhoods of the origin generated by the family satisfies the properties which make a Hausdorff topological vector space.
The theory of isometric mappings on classical normed spaces has its roots in the seminal paper by Mazur and Ulam (; see also ), who proved that every bijective isometry between two real normed spaces is affine. It is known that the surjective assumption is essential in this result and that it is not true for complex normed spaces. The Mazur-Ulam theorem has been extended in many directions. For example, Baker  proved that the result remains true if we consider an (not necessarily onto) isometry between a real normed space and a strictly convex real normed space. Another direction was provided by Vogt , who replaced isometries by the more general notion of equality of distance preserving maps (see also ).
In this paper, following the ideas of Vogt, we introduce a generalization of the concept of fuzzy isometry as follows.
Definition 2. Let and be two fuzzy normed spaces. One says that is a fuzzy norm-preserving mapping if given , then, for all ,
Let us recall here the definition of a fuzzy isometry.
Definition 3. Let and be two fuzzy normed spaces. It is said that is a fuzzy isometry if , for all and .
We provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: let , be fuzzy normed spaces and let be a fuzzy norm-preserving surjection satisfying . Then, is additive. As a corollary, we deduce that if such an is a fuzzy isometry, then is affine.
2. The Results
Let be a fuzzy normed space. Fix and define a function as . It is apparent that is bijective; indeed, , where stands for the identity map on . Furthermore, is a fuzzy isometry since , for all and .
Let . It is clear that and that .
Lemma 4. Every fuzzy isometry from onto fixes .
Proof. Let . This is a nonempty set since the identity map belongs to it. Fix and let us define, for each ,
If , then we define , which is in . Then
Moreover, for all , we have which yields . That is, , which leads us to the following equalities:
On the other hand, given , we have since .
Let us suppose that there exists such that . Then, there exists such that In addition, for each , there exists such that . Hence, by (6) and (7), we have that but a contradiction which completes the proof.
Remark 5. It can be checked that a fuzzy norm-preserving mapping is associated, for each , with a function such that
It is then apparent that fuzzy isometries are fuzzy norm-preserving mappings taking , .
Theorem 6. Let and be two fuzzy normed spaces and let be a fuzzy norm-preserving surjection satisfying . Then is additive.
Proof. Fix and let
We know that .
We now define map as for a fixed and .
Claim 1. is a fuzzy isometry from to which does not depend on the choice of .
Let us first check that does not depend on the choice of . To this end, suppose that . Then, which is to say that .
Let us next prove that is a fuzzy isometry. Suppose that and , with and . Then,
Finally, let us check that maps onto . Fix and let such that . From the definition of , we know that Hence, Furthermore, As a consequence, we deduce that . Since it is a routine matter to verify that , we infer that , and the claim is proved.
Thanks to Claim 1, we can apply Lemma 4 and conclude that fixes . Hence, and, then, for all , which is to say that
Next, let us define, for a fixed , the following map: for all .
It is clear that and is surjective since is assumed to be also surjective. Furthermore, for all and , which is to say that is also a fuzzy norm-preserving map. Hence, by (20), we infer that , for all . Then, for all , we have which yields that is, is additive.
Let us recall here that additivity yields -linearity, which, in presence of continuity, implies linearity. Hence, as a straightforward corollary of Theorem 6, we obtain a fuzzy version of the Mazur-Ulam theorem.
Corollary 7. Let and be two fuzzy normed spaces and let be a surjective fuzzy isometry. Then, is affine.
Proof. It suffices to apply Theorem 6 to , .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Spanish Ministry of Education and Science (Grant no. MTM2011-23118) and by Bancaixa (Project P1-1B2011-30).
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