Corrigendum to "On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces"
aa r X i v : . [ m a t h . F A ] F e b Filomat xx (yyyy), zzz–zzzDOI (will be added later)
Published by Faculty of Sciences and Mathematics,University of Niˇs, SerbiaAvailable at:
Corrigendum to “On the Mazur-Ulam theorem in non-Archimedeanfuzzy anti-2-normed spaces”
Javier Cabello S´anchez a , Jos´e Navarro Garmendia b a Departamento de Matem´aticas, Universidad de Extremadura, Avenida de Elvas s / n, 06006; Badajoz. Spain b Departamento de Matem´aticas, Universidad de Extremadura, Avenida de Elvas s / n, 06006; Badajoz. Spain Abstract.
In this note we correct a paper by D. Kang (“On the Mazur-Ulam theorem in non-Archimedeanfuzzy anti-2-normed spaces”, Filomat, 2017).The research in that paper applies to what the author calls strictly convex spaces. Nevertheless, weprove that this notion is void: there is no single space that satisfies the definition.
1. Introduction
In [2, Theorem 3.5], we can see the following non-Archimedean fuzzy version of the classical Mazur-Ulam theorem:
Theorem 1.1.
Let X, Y be non-Archimedean fuzzy anti-2-normed spaces over a certain type of non-Archimedeanfield K . If both X and Y are strictly convex, then any centred fuzzy 2-isometry f : X → Y is an additive map.
The class of strictly convex fuzzy anti-2-normed spaces was introduced in that paper (cf. [2, Defini-tion 2.5]), although there appeared no examples there. In this note, we prove:
Proposition 1.2.
There are no such strictly convex spaces at all.
As a consequence, the statement of the above Theorem is void.Let us also point out that the situation is similar in the di ff erent non-Archimedean versions of theMazur-Ulam theorem that have appeared in recent years (cf. [1] and references in [2]).
2. Voidness of the notion of strictly convex fuzzy space
We reflect the following definition just for the sake of completeness.
Definition 2.1. A non-Archimedean fuzzy anti-2-normed space is a linear space X over a non-Archimedeanfield ( K , | · | ) together with a fuzzy anti-2-norm; that is to say, with a function N : X × R → [0 ,
1] such that,for all x , y ∈ X and all s , t ∈ R , Mathematics Subject Classification . Primary 46S10; Secondary 26E30
Keywords . fuzzy normed spaces; Mazur-Ulam Theorem; non-Archimedean normed spaces; strict convexityReceived: dd Month yyyy; Revised: dd Month yyyy; Accepted: dd Month yyyyCommunicated by (name of the Editor, mandatory)Research supported in part by DGICYT project PID2019-103961GB-C21 (Spain), ERDF funds and Junta de Extremadura programsIB-16056 and IB-18087
Email addresses: [email protected] (Javier Cabello S´anchez), [email protected] (Jos´e Navarro Garmendia) . Author, S. Author / Filomat xx (yyyy), zzz–zzz t ≤ , then N ( x , y , t ) = t >
0, then N ( x , y , t ) = x and y are linearly dependent,(A2N-3) N ( x , y , t ) = N ( y , x , t ) , (A2N-4) N ( x , cy , t ) = N ( x , y , t / | c | ) for any non-zero c ∈ K (A2N-5) N ( x , y + z , max { s , t } ) ≤ max { N ( x , y , s ) , N ( x , z , t ) } ,(A2N-6) N ( x , y , ∗ ) is a non-increasing function of R and lim t →∞ N ( x , y , t ) = Definition 2.2. [2, Definition 2.5]
A non-Archimedean fuzzy anti-2-normed space ( X , N ) is strictly convex if N ( x , y , s ) = N ( x , z , t ) = N ( x , y + z , max { s , t } ) ⇒ y = z and s = t . (1) Proposition 2.3.
There are no strictly convex spaces at all –in the sense of the above Definition.Proof.