A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces

Abstract

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 𝑋, 𝑌 be fuzzy normed spaces and let 𝑓∶𝑋→𝑌 be a fuzzy norm-preserving surjection satisfying 𝑓(0)=0. Then 𝑓 is additive.

1. Introduction

Studies on fuzzy normed spaces are relatively recent in the field of fuzzy functional analysis. It was Katsaras who in 1984 [1], while studying topological vector spaces, was the first to introduce the idea of fuzzy norm on a linear space. Eight years later, Felbin [2] offered an alternative definition. With this definition, the induced fuzzy metric is of Kaleva and Seikkala type [3]. In 1994, Cheng and Mordeson [4] defined another type of fuzzy norm on a linear space whose associated fuzzy metric is of Kramosil and Michalek type [5]. Finally, in [6] (see also [7]), Bag and Samanta redefined the concept of fuzzy norm given in [4] as follows.

Definition 1. Let 𝑋 be a real linear space. A function 𝑁∶𝑋×ℝ→[0,1] is said to be a fuzzy norm on 𝑋 if, for all 𝑥,𝑦∈𝑋 and all 𝑡,𝑠,𝑘∈ℝ, it satisfies the following:(N1)𝑁(𝑥,𝑡)=0 for 𝑡≤0;(N2)𝑥=0 if and only if 𝑁(𝑥,𝑡)=1 for all 𝑡>0;(N3)𝑁(𝑘𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑘|) if 𝑘≠0;(N4)𝑁(𝑥+𝑦,𝑡+𝑠)≥min{𝑁(𝑥,𝑡),𝑁(𝑦,𝑠)};(N5)lim_𝑡→∞𝑁(𝑥,𝑡)=1.
The pair (𝑋,𝑁) is called a fuzzy normed space.

We point out that classical normed spaces are strictly included in the class of fuzzy normed spaces (see [6]) and that (N2) and (N4) imply that, for a fixed 𝑥∈𝑋, the function 𝑁(𝑥,⋅) is nondecreasing. It is a well-known fact that every fuzzy norm on a real linear space 𝑋 induces a topology on 𝑋 defined as follows: a subbase for the neighborhood system at a point 𝑥∈𝑋 consists of the sets 𝐵_𝑁(𝑥,𝜀,𝑡)=𝑦∈𝑋∶𝑁𝑥−𝑦,𝑡>1−𝜀 (1) for all 0<𝜀<1 and 𝑡>0. It is straightforward to verify that the filter of neighborhoods of the origin generated by the family {𝐵_𝑁(0,𝜀,𝑡)∶0<𝜀<1,𝑡>0} satisfies the properties which make (𝑋,𝑁) 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 ([8]; see also [9]), 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 [10] 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 [11], who replaced isometries by the more general notion of equality of distance preserving maps (see also [12]).

In this paper, following the ideas of Vogt, we introduce a generalization of the concept of fuzzy isometry as follows.

Definition 2. Let (𝑋,𝑁) and (𝑌,𝑁^′) be two fuzzy normed spaces. One says that 𝑓∶𝑋→𝑌 is a fuzzy norm-preserving mapping if given 𝑥,𝑦,𝑥^′,𝑦^′∈𝑋, then, for all 𝑡>0, 𝑁𝑥−𝑦,𝑡=𝑁𝑥^′−𝑦^′,𝑡⟹𝑁^′𝑓(𝑥)−𝑓𝑦,𝑡=𝑁^′𝑓𝑥^′−𝑓𝑦^′,𝑡. (2)
Let us recall here the definition of a fuzzy isometry.

Definition 3. Let (𝑋,𝑁) and (𝑌,𝑁^′) be two fuzzy normed spaces. It is said that 𝑓∶𝑋→𝑌 is a fuzzy isometry if 𝑁^′(𝑓(𝑥)−𝑓(𝑦),𝑡)=𝑁(𝑥−𝑦,𝑡), for all 𝑥,𝑦∈𝑋 and 𝑡>0.
We provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: let 𝑋, 𝑌 be fuzzy normed spaces and let 𝑓∶𝑋→𝑌 be a fuzzy norm-preserving surjection satisfying 𝑓(0)=0. Then, 𝑓 is additive. As a corollary, we deduce that if such an 𝑓 is a fuzzy isometry, then 𝑓 is affine.

2. The Results

Let (𝑋,𝑁) be a fuzzy normed space. Fix 𝑐∈𝑋 and define a function 𝜙_𝑐∶𝑋→𝑋 as 𝜙_𝑐(𝑥)∶=2𝑐−𝑥. It is apparent that 𝜙_𝑐 is bijective; indeed, 𝜙_𝑐∘𝜙_𝑐=𝐼𝑑, where 𝐼𝑑 stands for the identity map on 𝑋. Furthermore, 𝜙_𝑐 is a fuzzy isometry since 𝑁(𝜙_𝑐(𝑥)−𝜙_𝑐(𝑦),𝑡)=𝑁(2𝑐−𝑥−2𝑐+𝑦,𝑡)=𝑁(𝑥−𝑦,𝑡), for all 𝑥,𝑦∈𝑋 and 𝑡>0.

Let 𝐿(𝑐,𝑁)∶={𝑥∈𝑋∶𝑁(𝑥,𝑡)=𝑁(2𝑐−𝑥,𝑡)≥𝑁(𝑐,𝑡/2)𝑓𝑜𝑟𝑎𝑙𝑙𝑡>0}. It is clear that 𝑐∈𝐿(𝑐,𝑁)≠0/ and that 𝜙_𝑐(𝐿(𝑐,𝑁))⊆𝐿(𝑐,𝑁).

Lemma 4. Every fuzzy isometry from 𝐿(𝑐,𝑁) onto 𝐿(𝑐,𝑁) fixes 𝑐.

Proof. Let 𝐼∶={𝑔∶𝐿(𝑐,𝑁)→𝐿(𝑐,𝑁)∶𝑔isanontofuzzyisometry}. This is a nonempty set since the identity map belongs to it. Fix 𝑡>0 and let us define, for each 𝑖=1,2,3,…, 𝜆^𝑡_𝑖𝑁𝑔(𝑐)𝑡∶=inf−𝑐,𝑖∶𝑔∈𝐼. (3) If 𝑔∈𝐼, then we define 𝑔^′∶=𝑔⁻¹∘𝜙_𝑐∘𝑔, which is in 𝐼. Then 𝑁𝑔^′(𝑐)−𝑐,𝑡=𝑁𝑔⁻¹∘𝜙_𝑐(𝑐)𝜙∘𝑔−𝑐,𝑡=𝑁_𝑐𝑔(𝑐)(𝑐)2𝑔(𝑐)𝑔(𝑐)𝑡−𝑔,𝑡=𝑁−𝑐,𝑡=𝑁−𝑐,2. (4) Consequently, 𝑁(𝑔(𝑐)−𝑐,𝑡/2)≥𝜆^𝑡₁.
Moreover, for all 𝑔∈𝐼, we have 𝑁𝑔(𝑐)𝑔(𝑐)𝑡−𝑐,𝑡≥𝑁−𝑐,2≥𝜆^𝑡₁, (5) which yields 𝜆^𝑡₁≥𝜆^𝑡₂≥𝜆^𝑡₁. That is, 𝜆^𝑡₁=𝜆^𝑡₂, which leads us to the following equalities: 𝜆^𝑡₁=𝜆^𝑡₂=⋯=𝜆^𝑡_𝑛=⋯. (6)
On the other hand, given 𝑔∈𝐼, we have 𝑁𝑔(𝑐)𝑁𝑔(𝑐),𝑡−𝑐,𝑡≥min2𝑡,𝑁𝑐,2𝑡=𝑁𝑐,2, (7) since 𝑔(𝑐)∈𝐿(𝑐,𝑁).
Let us suppose that there exists 𝑔₀∈𝐼 such that 𝑔₀(𝑐)≠𝑐. Then, there exists 𝑡₀>0 such that 𝑁𝑔₀(𝑐)−𝑐,𝑡₀=𝑘<1. (8) In addition, for each 𝑡>0, there exists 𝑛_𝑡 such that 𝑁(𝑔₀(𝑐)−𝑐,𝑡/𝑛_𝑡)≤𝑁(𝑔₀(𝑐)−𝑐,𝑡₀). Hence, by (6) and (7), we have that 𝑁𝑡𝑐,2≤𝜆^𝑡_𝑛𝑡≤𝑁𝑔₀(𝑐)𝑡−𝑐,𝑛_𝑡≤𝑁𝑔₀(𝑐)−𝑐,𝑡₀=𝑘<1, (9) but 1=lim_𝑡→∞𝑁𝑡𝑐,2≤𝑘<1, (10) a contradiction which completes the proof.

Remark 5. It can be checked that a fuzzy norm-preserving mapping is associated, for each 𝑡>0, with a function 𝜌_𝑡∶𝐷_𝑡⊆[0,1]→[0,1] such that 𝑁^′𝑓(𝑥)−𝑓𝑦,𝑡=𝜌_𝑡𝑁𝑥−𝑦,𝑡. (11)
It is then apparent that fuzzy isometries are fuzzy norm-preserving mappings taking 𝜌_𝑡=𝐼𝑑, 𝑡>0.

Theorem 6. Let (𝑋,𝑁) and (𝑌,𝑁^′) be two fuzzy normed spaces and let 𝑓∶𝑋→𝑌 be a fuzzy norm-preserving surjection satisfying 𝑓(0)=0. Then 𝑓 is additive.

Proof. Fix 𝑥₀∈𝑋 and let 𝐿𝑓𝑥₀,𝑁^′∶=𝑦∈𝑌∶𝑁^′𝑦,𝑡=𝑁^′2𝑓𝑥₀−𝑦,𝑡≥𝑁^′𝑓𝑥₀,𝑡2,𝑡>0. (12)
We know that 𝑓(𝑥₀)∈𝐿(𝑓(𝑥₀),𝑁^′)≠0/.
We now define map ℎ∶𝐿(𝑓(𝑥₀),𝑁^′)→𝐿(𝑓(𝑥₀),𝑁^′) as ℎ(𝑦)∶=𝑓(𝑥₁−𝑥^′) for a fixed 𝑥₁∈𝑓⁻¹(2𝑓(𝑥₀)) and 𝑥^′∈𝑓⁻¹(𝑦).
Claim 1. ℎ is a fuzzy isometry from 𝐿(𝑓(𝑥₀),𝑁^′) to 𝐿(𝑓(𝑥₀),𝑁^′) which does not depend on the choice of 𝑥^′∈𝑓⁻¹(𝑦).
Let us first check that ℎ does not depend on the choice of 𝑥^′∈𝑓⁻¹(𝑦). To this end, suppose that {𝑥^′,𝑥^′′}⊆𝑓⁻¹(𝑦). Then, 𝑁^′𝑓𝑥₁−𝑥^′−𝑓𝑥₁−𝑥^′′,𝑡=𝜌_𝑡𝑁𝑥^′′−𝑥^′,𝑡=𝑁^′𝑓𝑥^′′−𝑓𝑥^′,𝑡=𝑁^′(0,𝑡)=1, (13) which is to say that 𝑓(𝑥₁−𝑥^′)=𝑓(𝑥₁−𝑥^′′).
Let us next prove that ℎ is a fuzzy isometry. Suppose that 𝑓(𝑥^′)=𝑦^′ and 𝑓(𝑥^′′)=𝑦^′′, with 𝑥^′,𝑥^′′∈𝑋 and 𝑦^′,𝑦^′′∈𝑌. Then, 𝑁^′ℎ𝑦^′−ℎ𝑦^′′,𝑡=𝑁^′𝑓𝑥₁−𝑥^′−𝑓𝑥₁−𝑥^′′,𝑡=𝜌_𝑡𝑁𝑥^′′−𝑥^′,𝑡=𝑁^′𝑓𝑥^′′−𝑓𝑥^′,𝑡=𝑁^′𝑦^′′−𝑦^′,𝑡. (14)
Finally, let us check that ℎ maps 𝐿(𝑓(𝑥₀),𝑁^′) onto 𝐿(𝑓(𝑥₀),𝑁^′). Fix 𝑦^′∈𝐿(𝑓(𝑥₀),𝑁^′) and let 𝑥^′∈𝑋 such that 𝑓(𝑥^′)=𝑦^′. From the definition of 𝐿(𝑓(𝑥₀),𝑁^′), we know that 𝑁^′𝑦^′,𝑡=𝑁^′2𝑓𝑥₀−𝑦^′,𝑡≥𝑁^′𝑓𝑥₀,𝑡2. (15) Hence, 𝑁^′ℎ𝑦^′,𝑡=𝑁^′𝑓𝑥₁−𝑥^′−𝑓(0),𝑡=𝜌_𝑡𝑁𝑥₁−𝑥^′−0,𝑡=𝑁^′𝑓𝑥₁−𝑓𝑥^′,𝑡=𝑁^′2𝑓𝑥₀−𝑦^′,𝑡=𝑁^′𝑦^′,𝑡≥𝑁^′𝑓𝑥₀,𝑡2. (16) Furthermore, 𝑁^′2𝑓𝑥₀−ℎ𝑦^′,𝑡=𝑁^′2𝑓𝑥₀−𝑓𝑥₁−𝑥^′,𝑡=𝑁^′𝑓𝑥₁−𝑓𝑥₁−𝑥^′,𝑡=𝜌_𝑡𝑁𝑥₁−𝑥₁+𝑥^′−0,𝑡=𝑁^′𝑓𝑥^′−𝑓(0),𝑡=𝑁^′𝑦^′,𝑡=𝑁^′ℎ𝑦^′,𝑡. (17) As a consequence, we deduce that ℎ(𝐿(𝑓(𝑥₀),𝑁^′))⊆𝐿(𝑓(𝑥₀),𝑁^′). Since it is a routine matter to verify that ℎ⁻¹=ℎ, we infer that ℎ(𝐿(𝑓(𝑥₀),𝑁^′))=𝐿(𝑓(𝑥₀),𝑁^′), and the claim is proved.
Thanks to Claim 1, we can apply Lemma 4 and conclude that ℎ fixes 𝑓(𝑥₀). Hence, 𝑓𝑥₀=ℎ𝑓𝑥₀=𝑓𝑥₁−𝑥₀ (18) and, then, 1=𝑁^′𝑓𝑥₀−𝑓𝑥₁−𝑥₀,𝑡=𝜌_𝑡𝑁2𝑥₀−𝑥₁,𝑡=𝑁^′𝑓2𝑥₀−𝑓𝑥₁,𝑡=𝑁^′𝑓2𝑥₀−2𝑓𝑥₀,𝑡 (19) for all 𝑡>0, which is to say that 𝑓2𝑥₀=2𝑓𝑥₀. (20)
Next, let us define, for a fixed 𝑧∈𝑋, the following map: 𝑓_𝑧(𝑥)∶=𝑓(𝑥+𝑧)−𝑓(𝑧), (21) for all 𝑥∈𝑋.
It is clear that 𝑓_𝑧(0)=0 and 𝑓_𝑧 is surjective since 𝑓 is assumed to be also surjective. Furthermore, for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝑁^′𝑓_𝑧(𝑥)−𝑓_𝑧𝑦,𝑡=𝑁^′𝑓(𝑥+𝑧)−𝑓𝑦+𝑧,𝑡=𝜌_𝑡𝑁𝑥−𝑦,𝑡, (22) which is to say that 𝑓_𝑧 is also a fuzzy norm-preserving map. Hence, by (20), we infer that 𝑓_𝑧(2𝑥)=2𝑓_𝑧(𝑥), for all 𝑥∈𝑋. Then, for all 𝑥,𝑦∈𝑋, we have 𝑓𝑥−𝑦+𝑦−𝑓𝑦=𝑓_𝑦𝑥−𝑦=2𝑓_𝑦𝑥−𝑦2=2𝑓𝑥−𝑦2+𝑦−𝑓𝑦, (23) which yields 𝑓(𝑥)+𝑓𝑦=2𝑓𝑥+𝑦2=𝑓𝑥+𝑦; (24) that is, 𝑓 is additive.

Let us recall here that additivity yields ℚ-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 (𝑋,𝑁) and (𝑌,𝑁^′) be two fuzzy normed spaces and let 𝑓∶𝑋→𝑌 be a surjective fuzzy isometry. Then, 𝑓 is affine.

Proof. It suffices to apply Theorem 6 to 𝑔(𝑥)∶=𝑓(𝑥)−𝑓(0), 𝑥∈𝑋.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Spanish Ministry of Education and Science (Grant no. MTM2011-23118) and by Bancaixa (Project P1-1B2011-30).