Open Access

An introduction to 2-fuzzy n-normed linear spaces and a new perspective to the Mazur-Ulam problem

Journal of Inequalities and Applications20122012:14

DOI: 10.1186/1029-242X-2012-14

Received: 24 May 2011

Accepted: 19 January 2012

Published: 19 January 2012

Abstract

The purpose of this article is to introduce the concept of 2-fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set. We define the concepts of n-isometry, n-collinearity n-Lipschitz mapping in this space. Also, we generalize the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or (X) is a fuzzy n-normed linear space, the Mazur-Ulam theorem holds. Moreover, it is shown that each n-isometry in 2-fuzzy n-normed linear spaces is affine.

Mathematics Subject Classification (2010): 03E72; 46B20; 51M25; 46B04; 46S40.

Keywords

Mazur-Ulam theorem α-n-norm 2-fuzzy n-normed linear spaces n-isometry n-Lipschitz mapping.

1. Introduction

A satisfactory theory of 2-norms and n-norms on a linear space has been introduced and developed by Gähler [1, 2]. Following Misiak [3], Kim and Cho [4], and Malčeski [5] developed the theory of n-normed space. In [6], Gunawan and Mashadi gave a simple way to derive an (n - 1)-norm from the n-norms and realized that any n-normed space is an (n - 1)-normed space. Different authors introduced the definitions of fuzzy norms on a linear space. Cheng and Mordeson [7] and Bag and Samanta [8] introduced a concept of fuzzy norm on a linear space. The concept of fuzzy n-normed linear spaces has been studied by many authors (see [4, 9]).

Recently, Somasundaram and Beaula [10] introduced the concept of 2-fuzzy 2-normed linear space or fuzzy 2-normed linear space of the set of all fuzzy sets of a set. The authors gave the notion of α-2-norm on a linear space corresponding to the 2-fuzzy 2-norm by using some ideas of Bag and Samanta [8] and also gave some fundamental properties of this space.

In 1932, Mazur and Ulam [11] proved the following theorem.

Mazur-Ulam Theorem. Every isometry of a real normed linear space onto a real normed linear space is a linear mapping up to translation.

Baker [12] showed an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian [13] investigated the generalizations of the Mazur-Ulam theorem in F*-spaces. Rassias and Wagner [14] described all volume preserving mappings from a real finite dimensional vector space into itself and Väisälä [15] gave a short and simple proof of the Mazur-Ulam theorem. Chu [16] proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al. [17] generalized the Mazur-Ulam theorem when X is a linear n-normed space, that is, the Mazur-Ulam theorem holds, when the n-isometry mapped to a linear n-normed space is affine. They also obtain extensions of Rassias and Šemrl's theorem [18]. Moslehian and Sadeghi [19] investigated the Mazur-Ulam theorem in non-archimedean spaces. Choy et al. [20] proved the Mazur-Ulam theorem for the interior preserving mappings in linear 2-normed spaces. They also proved the theorem on non-Archimedean 2-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption. Choy and Ku [21] proved that the barycenter of triangle carries the barycenter of corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean 2-normed spaces using the above statement. Xiaoyun and Meimei [22] introduced the concept of weak n-isometry and then they got under some conditions, a weak n-isometry is also an n-isometry. Cobzaş [23] gave some results of the Mazur-Ulam theorem for the probabilistic normed spaces as defined by Alsina et al. [24]. Cho et al. [25] investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Alaca [26] introduced the concepts of 2-isometry, collinearity, 2-Lipschitz mapping in 2-fuzzy 2-normed linear spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy 2-normed linear space or (X) is a fuzzy 2-normed linear space. Kang et al. [27] proved that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space. Kubzdela [28] gave some new results for isometries, Mazur-Ulam theorem and Aleksandrov problem in the framework of non-Archimedean normed spaces. The Mazur-Ulam theorem has been extensively studied by many authors (see [29, 30]).

In the present article, we introduce the concept of 2-fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set. We define the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this space. Also, we generalize the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or (X) is a fuzzy n-normed linear space, the Mazur-Ulam theorem holds. It is moreover shown that each n-isometry in 2-fuzzy n-normed linear spaces is affine.

2. Preliminaries

Definition 2.1([31]) Let n and let X be a real vector space of dimension dn. (Here we allow d to be infinite.) A real-valued function , ..., on X × × X n satisfying the following properties
  1. (1)

    x1, x2, ..., xn = 0 if and only if x1, x2, ..., x n are linearly dependent,

     
  2. (2)

    x1, x2, ..., x n is invariant under any permutation,

     
  3. (3)

    x1, x2, ..., αx n = |α| x1, x2, ..., xn for any α ,

     
  4. (4)

    x1, x2, ..., xn-1, y + zx1, x2, ..., xn-1, y + x1, x2, ..., xn-1, z, is called an n-norm on X and the pair (X, , ..., ) is called an n-normed linear space.

     

Definition 2.2 [9] Let X be a linear space over S (field of real or complex numbers). A fuzzy subset N of X n × (, the set of real numbers) is called a fuzzy n-norm on X if and only if:

(N1) For all t with t ≤ 0, N(x1, x2, ..., x n , t) = 0,

(N2) For all t with t > 0, N(x1, x2, ..., x n , t) = 1 if and only if x1, x2, ..., x n are linearly dependent,

(N3) N(x1, x2, ..., x n , t) is invariant under any permutation of x1, x2, ..., x n ,

(N4) For all t with t > 0, N ( x 1 , x 2 , , λ x n , t ) = N x 1 , x 2 , , x n , t λ , if λ ≠ 0, λ S,

(N5) For all s, t
N ( x 1 , x 2 , , x n + x n , s + t ) min { N ( x 1 , x 2 , , x n , s ) , N ( x 1 , x 2 , , x n , t ) } ,

(N6) N(x1, x2, ..., x n , t) is a non-decreasing function of t and lim t N ( x 1 , x 2 , , x n , t ) = 1 .

Then (X, N) is called a fuzzy n-normed linear space or in short f-n-NLS.

Theorem 2.1 [9] Let (X, N) be an f-n-NLS. Assume that

(N7) N(x1, x2, ..., x n ,t) > 0 for all t > 0 implies that x1, x2, ..., x n are linearly dependent.

Define
| | x 1 , x 2 , , x n | | α = inf { t : N ( x 1 , x 2 , x n , t ) α , α ( 0 , 1 ) } .

Then {, , ..., α : α (0, 1)} is an ascending family of n-norms on X.

We call these n-norms as α-n-norms on X corresponding to the fuzzy n-norm on X.

Definition 2.3 Let X be any non-empty set and (X) the set of all fuzzy sets on X. For U, V (X) and λ S the field of real numbers, define
U + V = { ( x + y , ν μ ) : ( x , ν ) U , ( y , μ ) V }

and λU = {(λx, ν): (x, ν) U}.

Definition 2.4 A fuzzy linear space X ^ = X × ( 0 , 1 ] over the number field S, where the addition and scalar multiplication operation on X are defined by (x, ν) + (y, μ) = (x + y, νμ), λ(x, ν) = (λx, ν) is a fuzzy normed space if to every ( x , ν ) X ^ there is associated a non-negative real number, (x, ν), called the fuzzy norm of (x, ν), in such away that
  1. (i)

    (x, ν) = 0 iff x = 0 the zero element of X, ν (0, 1],

     
  2. (ii)

    λ(x, ν) = |λ| (x, ν) for all ( x , ν ) X ^ and all λ S,

     
  3. (iii)

    (x, ν) + (y, μ) || ≤ (x, ν μ) + (y, ν μ) for all ( x , ν ) , ( y , μ ) X ^ ,

     
  4. (iv)

    (x, t ν t ) = t (x, ν t ) for all ν t (0, 1].

     

3. 2-fuzzy n-normed linear spaces

In this section, we define the concepts of 2-fuzzy n-normed linear spaces and α-n-norms on the set of all fuzzy sets of a non-empty set.

Definition 3.1 Let X be a non-empty and (X) be the set of all fuzzy sets in X. If f (X) then f = {(x, μ): x X and μ (0, 1]}. Clearly f is bounded function for |f(x)| ≤ 1. Let S be the space of real numbers, then (X) is a linear space over the field S where the addition and scalar multiplication are defined by
f + g = { ( x , μ ) + ( y , η ) } = { ( x + y , μ η ) : ( x , μ ) f and ( y , η ) g }
and
λ f = { ( λ x , μ ) : ( x , μ ) f }

where λ S.

The linear space (X) is said to be normed linear space if, for every f (X), there exists an associated non-negative real number f (called the norm of f) which satisfies
  1. (i)
    f = 0 if and only if f = 0. For
    | | f | | = 0 { | | ( x , μ ) | | : ( x , μ ) f } = 0 x = 0 , μ ( 0 , 1 ] f = 0 .
     
  2. (ii)
    λf = |λ| f, λ S. For
    | | λ f | | = { | | λ ( x , μ ) | | : ( x , μ ) f , λ S } = { | λ | | | ( x , μ ) | | : ( x , μ ) f } = | λ | | | f | | .
     
  3. (iii)
    f + gf + g for every f, g (X). For
    | | f + g | | = { | | ( x , μ ) + ( y , η ) | | : x , y X , μ , η ( 0 , 1 ] } = { | | ( x + y ) , ( μ η ) | | : x , y X , μ , η ( 0 , 1 ] } = { | | ( x , μ η ) | | + | | ( y , μ η ) | | : ( x , μ ) f , ( y , η ) g } = | | f | | + | | g | | .
     

Then ((X),) is a normed linear space.

Definition 3.2 A 2-fuzzy set on X is a fuzzy set on (X).

Definition 3.3 Let X be a real vector space of dimension dn (n ) and (X) be the set of all fuzzy sets in X. Here we allow d to be infinite. Assume that a [0, 1]-valued function , ..., on ( X ) × × ( X ) n satisfies the following properties
  1. (1)

    f1, f2, ..., f n = 0 if and only if f1, f2, ..., f n are linearly dependent,

     
  2. (2)

    f1, f2, ..., f n is invariant under any permutation,

     
  3. (3)

    f1, f2, ..., λf n = |λ| f1, f2, ..., f n for any λ S,

     
  4. (4)

    f1, f2, ..., fn-1, y + zf1, f2, ..., fn-1, y + f1, f2, ..., fn-1, z.

     

Then ((X),,...,) is an n-normed linear space or (X, , ..., ) is a 2-n-normed linear space.

Definition 3.4 Let (X) be a linear space over the real field S. A fuzzy subset N of ( X ) × × ( X ) n × is called a 2-fuzzy n-norm on X (or fuzzy n-norm on (X)) if and only if

(2-N1) for all t with t ≤ 0, N(f1, f2, ..., f n , t) = 0,

(2-N2) for all t with t > 0, N(f1, f2, ..., f n , t) = 1 if and only if f1, f2, ..., f n are linearly dependent,

(2-N3) N(f1, f2, ..., f n , t) is invariant under any permutation of f1, f2, ..., f n ,

(2-N4) for all t with t > 0, N(f1, f2, ..., λf n , t) = N(f1, f2, ..., f n , t/|λ|), if λ ≠ 0, λ S,

(2-N5) for all s, t ,
N ( f 1 , f 2 , , f n + f n , s + t ) min { N ( f 1 , f 2 , , f n , s ) , N ( f 1 , f 2 , , f n , t ) } ,

(2-N6) N(f1, f2, ..., f n , ·): (0, ∞) → [0, 1] is continuous,

(2-N7) lim t N ( f 1 , f 2 , , f n , t ) = 1 .

Then (X), N) is a fuzzy n-normed linear space or (X, N) is a 2-fuzzy n-normed linear space.

Remark 3.1 In a 2-fuzzy n-normed linear space (X, N), N(f1, f2, ..., f n , ·) is a non-decreasing function of for all f1, f2,...,f n (X).

Remark 3.2 From (2-N4) and (2-N5), it follows that in a 2-fuzzy n-normed linear space,

(2-N4) for all t with t > 0, N ( f 1 , f 2 , , λ f i , , f n , t ) = N f 1 , f 2 , , f i , , f n , t | λ | , if λ ≠ 0, λ S,

(2-N5) for all s, t ,
N ( f 1 , f 2 , , f i + f i , , f n , s + t ) min { N ( f 1 , f 2 , , f i , , f n , s ) , N ( f 1 , f 2 , , f i , , f n , t ) } .

The following example agrees with our notion of 2-fuzzy n-normed linear space.

Example 3.1 Let ((X),,,...,) be an n-normed linear space as in Definition 3.3. Define
N ( f 1 , f 2 , , f n , t ) = t t + | | f 1 , f 2 , , f n | | i f t > 0 , t , 0 i f t 0

for all ( f 1 , f 2 , , f n ) ( X ) × × ( X ) n . Then (X, N) is a 2-fuzzy n-normed linear space.

Solution. (2-N1) For all t with t ≤ 0, by definition, we have N(f1, f2, ..., f n , t) = 0.

(2-N2) For all t with t > 0,
N ( f 1 , f 2 , , f n , t ) = 1 t t + | | f 1 , f 2 , , f n | | = 1 t = t + | | f 1 , f 2 , , f n | | | | f 1 , f 2 , , f n | | = 0 f 1 , f 2 , , f n  are linearly dependent .
(2-N3) For all t with t > 0,
N ( f 1 , f 2 , , f n , t ) = t t + | | f 1 , f 2 , , f n | | = t t + | | f 1 , f 2 , , f n , f n - 1 | | = N ( f 1 , f 2 , , f n , f n - 1 , t ) = .
(2-N4) For all t with t > 0 and λ F, λ ≠ 0,
N ( f 1 , f 2 , , f n , t / | λ | ) = t / | λ | t / | λ | + | | f 1 , f 2 , , f n | | = t / | λ | ( t + | λ | | | f 1 , f 2 , , f n | | ) / | λ | = t t + | λ | | | f 1 , f 2 , , f n | | = t ( t + | | f 1 , f 2 , , λ f n | | = N ( f 1 , f 2 , , λ f n , t ) .
(2-N5) We have to prove
N ( f 1 , f 2 , , f n + f n , s + t ) min { f ( x 1 , f 2 , , f n , s ) , N ( f 1 , f 2 , , f n , t ) } .
  1. (i)

    s + t < 0,

     
  2. (ii)

    s = t = 0,

     
  3. (iii)

    s + t > 0; s > 0, t < 0; s < 0, t > 0, then the above relation is obvious. If

     
  4. (iv)
    s > 0, t > 0, s + t > 0, then
    N ( f 1 , f 2 , , f n + f n , s + t = s + t s + t + | | f 1 , f 2 , , f n + f n | | .
     
If
s s + | | f 1 , f 2 , , f n | | t t + | | f 1 , f 2 , , f n | | | | f 1 , f 2 , , f n | | s | | x 1 , x 2 , , x n | | t | | f 1 , f 2 , , f n | | s + | | f 1 , f 2 , , f n | | s | | f 1 , f 2 , , f n | | t + | | f 1 , f 2 , , f n | | s | | f 1 , f 2 , , f n + f n | | s s + t s t | | f 1 , f 2 , , f n | | | | f 1 , f 2 , , f n + f n | | s + t | | f 1 , f 2 , , f n | | t s + t + | | f 1 , f 2 , , f n + f n | | s + t t + | | f 1 , f 2 , , f n | | t s + t s + t + | | f 1 , f 2 , , f n + f n | | t t + | | f 1 , f 2 , , f n | | N ( f 1 , f 2 , , f n + f n , s + t ) N ( f 1 , f 2 , , f n , t ) .
Similarly, if
t t + | | f 1 , f 2 , , f n | | s s + | | f 1 , f 2 , , f n | | N ( f 1 , f 2 , , f n + f n , s + t ) N ( f 1 , f 2 , , f n , t ) .
Thus
N ( f 1 , f 2 , , f n + f n , s + t ) min { N ( f 1 , f 2 , , f n , s ) , N ( f 1 , f 2 , , f n , t ) } .

(2-N6) It is clear that N(f1, f2, ..., f n , ·): (0, ∞) → [0, 1] is continuous.

(2-N7) For all t with t > 0,
lim t N ( f 1 , f 2 , , f n , t ) = lim t t t + | | f 1 , f 2 , , f n | | = lim t t t ( 1 + ( 1 / t ) | | f 1 , f 2 , , f n | | ) = 1 ,

as desired.

As a consequence of Theorem 3.2 in [10], we introduce an interesting notion of ascending family of α-n-norms corresponding to the fuzzy n-norms in the following theorem.

Theorem 3.1 Let ((X), N) is a fuzzy n-normed linear space. Assume that

(2-N8) N(f1, f2, ..., f n , t) > 0 for all t > 0 implies f1, f2, ..., f n are linearly dependent.

Define
| | f 1 , f 2 , , f n | | α = inf ( t : N ( f 1 , f 2 , , f n , t ) α , α ( 0 , 1 ) } .

Then {, , ..., α : α (0, 1)} is an ascending family of n-norms on (X).

These n-norms are called α-n-norms on (X) corresponding to the 2-fuzzy n-norm on X.

Proof. (i) Let f1, ..., f n α = 0. This implies that inf {t : N(f1, ..., f n , t) ≥ α}. Then, N(f1, f2, ..., f n , t) ≥ α > 0, for all t > 0, α (0, 1), which implies that f1, f2, ..., f n are linearly dependent, by (2-N8).

Conversely, assume f1, f2, ..., f n are linearly dependent. This implies that N(f1, f2, ..., f n , t) = 1 for all t > 0. For all α (0, 1), inf {t : N(f1, f2, ..., f n , t) ≥ α}, which implies that f1, f2, ..., f n α = 0.
  1. (ii)

    Since N(f1, f2, ..., f n , t) is invariant under any permutation, f1, f2, ..., f n α = 0 under any permutation.

     
  2. (iii)
    If λ ≠ 0, then
    | | f 1 , f 2 , , λ f n | | α = inf { s : N ( f 1 , f 2 , , f n , s ) α } = inf { s : N ( f 1 , f 2 , , f n , s | λ | α } .
     
Let t = s | λ | , then
| | f 1 , f 2 , , λ f n | | α = inf { | λ | t : N ( f 1 , f 2 , , f n , t ) α } = | λ | inf { t : N ( f 1 , f 2 , , f n , t ) α } = | λ | | | f 1 , f 2 , , f n | | α .
If λ = 0, then
| | f 1 , f 2 , , λ f n | | α = | | f 1 , f 2 , , 0 | | α = 0 = 0 | | f 1 , f 2 , , f n | | α = | λ | | | f 1 , f 2 , , f n | | α , λ S ( field ) .
(iv)
f 1 , f 2 , , f n α + f 1 , f 2 , , f n α = inf { t : N ( f 1 , f 2 , , f n , t ) α } + inf ( s : N ( f 1 , f 2 , , f n , s ) α } = inf { t + s : N ( f 1 , f 2 , , f n , t ) α , N ( f 1 , f 2 , , f n , s ) α } inf { t + s : N ( f 1 , f 2 , , f n + f n , t + s ) α } , inf { r : N ( f 1 , f 2 , , f n + f n , r ) α } , r = t + s = f 1 , f 2 , , f n + f n α .
Hence
| | f 1 , f 2 , , f n + f n | | α | | f 1 , f 2 , , f n | | α + | | f 1 , f 2 , , f n | | α .

Thus {, , ..., α : α (0, 1)} is an α-n-norm on X.

Let 0 < α1 < α2. Then,
f 1 , f 2 , , f n α 1 = inf { t : N ( f 1 , f 2 , , f n , t ) α 1 } , f 1 , f 2 , , f n α 2 = inf { t : N ( f 1 , f 2 , , f n , t ) α 2 } .
As α1 < α2,
{ t : N ( f 1 , f 2 , , f n , t ) α 2 } { t : N ( f 1 , f 2 , , f n , t ) α 1 }
implies that
inf { t : N ( f 1 , f 2 , , f n , t ) α 2 } inf { t : N ( f 1 , f 2 , , f n , t ) α 1 }
which implies that
| | f 1 , f 2 , , f n | | α 2 | | f 1 , f 2 , , f n | | α 1 .

Hence {, , ..., α : α (0, 1)} is an ascending family of α-n-norms on x corresponding to the 2-fuzzy n-norm on X.

4. On the Mazur-Ulam problem

In this section, we give a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy n-normed linear space or (X) is a fuzzy n-normed linear space. Hereafter, we use the notion of fuzzy n-normed linear space on (X) instead of 2-fuzzy n-normed linear space on X.

Definition 4.1 Let (X) and (X) be fuzzy n-normed linear spaces and Ψ : (X) → (Y) a mapping. We call Ψ an n-isometry if
f 1 - f 0 , , f n - f 0 α = Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) β

for all f0, f1, f2,...,f n (X) and α, β (0, 1).

For a mapping Ψ, consider the following condition which is called the n-distance one preserving property (n DOPP).
( n DOPP ) Let f 0 , f 1 , f 2 , , f n ( X ) with f 1 - f 0 , , f n - f 0 α = 1 .

Then Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0) β = 1.

Lemma 4.1 Let f1, f2,...,f n (X), α (0, 1) and ħ . Then,
| | f 1 , , f i , , f j , , f n | | α = | | f 1 , , f i , , f j + f i , , f n | | α

for all 1 ≤ ijn.

Proof. It is obviously true.

Lemma 4.2 For f 0 , f 0 ( X ) , if f0 and f 0 are linearly dependent with some direction, that is, f 0 = t f 0 for some t > 0, then
| | f 0 + f 0 , f 1 , , f n | | α = | | f 0 , f 1 , , f n | | α + | | f 0 , f 1 , , f n | | α

for all f1, f2,...,f n (X) and α (0, 1).

Proof. Let f 0 = t f 0 for some t > 0. Then we have
| | f 0 + f 0 , f 1 , , f n | | α = | | f 0 + t f 0 , f 1 , , f n | | α = ( 1 + t ) | | f 0 , f 1 , , f n | | α = | | f 0 , f 1 , , f n | | α + t | | f 0 , f 1 , , f n | | α = | | f 0 , f 1 , , f n | | α + | | f 0 , f 1 , , f n | | α

for all f1, f2,...,f n (X) and α (0, 1).

Definition 4.2 The elements f0, f1, f2, ..., f n of (X) are said to be n-collinear if for every i, {f j - f i : 0 ≤ jin} is linearly dependent.

Remark 4.1 The elements f0, f1, and f2 are said to be 2-collinear if and only if f2 - f0 = r(f1 - f0) for some real number r.

Now we define the concept of n-Lipschitz mapping.

Definition 4.3 We call Ψ an n-Lipschitz mapping if there is a κ ≥ 0 such that
| | Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β k | | f 1 - f 0 , , f n - f 0 | | α

for all f0, f1, f2,...,f n (X) and α, β (0, 1). The smallest such κ is called the n-Lipschitz constant.

Lemma 4.3 Assume that if f0, f1, and f2 are 2 -collinear then Ψ(f0), Ψ(f1) and Ψ(f2) are 2-collinear, and that Ψ satisfies (n DOPP). Then Ψ preserves the n-distance k for each k .

Proof. Suppose that there exist f0, f1 (X) with f0f1 such that Ψ(f0) = Ψ(f1). Since dim(X) ≥ n, there are f2,...,f n (X) such that f1 - f0, f2 - f0, ..., f n - f0 are linearly independent. Since f1 - f0, f2 - f0, ..., f n - f0 α ≠ 0, we can set
z 2 = f 0 + f 2 - f 0 | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α .
Then we have
| | f 1 - f 0 , z 2 - f 0 , f 3 - f 0 , , f n - f 0 | | α = f 1 - f 0 , f 2 - f 0 | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α , f 3 - f 0 , , f n - f 0 α = 1 .
Since Ψ preserves the unit n-distance,
| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( z 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = 1 .
But it follows from Ψ (f0) = Ψ (f1) that
| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( z 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = 0 ,

which is a contradiction. Hence, Ψ is injective.

Let f0, f1, f2, ..., f n be elements of (X), k and
| | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α = k .
We put
g i = f 0 + i k ( f 1 - f 0 ) , i = 0 , 1 , , k .
Then
| | g i + 1 - g i , f 2 - f 0 , , f n - f 0 | | α = f 0 + i + 1 k ( f 1 - f 0 ) - f 0 + i k ( f 1 - f 0 ) , f 2 - f 0 , , f n - f 0 α = 1 k ( f 1 - f 0 ) , f 2 - f 0 , , f n - f 0 α = i k | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α = k k = 1
for all i = 0, 1, ..., k - 1. Since Ψ satisfies (n DOPP),
| | Ψ ( g i + 1 ) - Ψ ( g i ) , Ψ ( f 2 ) - Ψ ( f 0 ) , Ψ ( f n ) - Ψ ( f 0 ) | | β = 1
(4.1)
for all i = 0, 1, ..., k - 1. Since g0, g1, and g2 are 2-collinear, Ψ (g0), Ψ(g1) and Ψ(g2) are also 2-collinear. Thus there is a real number r0 such that Ψ(g2) - Ψ(g1) = r0 (Ψ(g1) - Ψ(g0)). It follows from (4.1) that
| | Ψ ( g 1 ) - Ψ ( g 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | Ψ ( g 2 ) - Ψ ( g 1 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | r 0 ( Ψ ( g 1 ) - Ψ ( g 2 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | r 0 | | | ( Ψ ( g 1 ) - Ψ ( g 0 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β .
Thus, we have r0 = 1 or -1. If r0 = -1, Ψ(g2) - Ψ(g1) = -Ψ(g1) + Ψ(g0), that is, Ψ(g2) = Ψ(g0). Since Ψ is injective, g2 = g0, which is a contradiction. Thus r0 = 1. Then we have Ψ(g2) - Ψ(g1) = Ψ(g1) - Ψ(g0). Similarly, one can obtain that Ψ(gi+1) - Ψ(g i ) = Ψ(g i ) - Ψ(gi-1) for all i = 0, 1, ..., k - 1. Thus Ψ(gi+1) - Ψ(g i ) = Ψ(g1) - Ψ(g0) for all i = 0, 1, ..., k - 1. Hence
Ψ ( f 1 ) - Ψ ( f 0 ) = Ψ ( g k ) - Ψ ( g 0 ) = Ψ ( g k ) - Ψ ( g k - 1 ) + Ψ ( g k - 1 ) - Ψ ( g k - 2 ) + + Ψ ( g 1 ) - Ψ ( g 0 ) = k ( Ψ ( g 1 ) - Ψ ( g 0 ) ) .
Hence
| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | k ( Ψ ( g 1 ) - Ψ ( g 0 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | k Ψ ( g 1 ) - Ψ ( g 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = k .

This completes the proof.

Lemma 4.4 Let h, f0, f1, ..., f n be elements of (X) and let h, f0, f1 be 2-collinear. Then
| | f 1 - h , f 2 - h , , f n - h | | α = | | f 1 - h , f 2 - f 0 , , f n - f 0 | | α .
Proof. Since h, f0, f1 are 2-collinear, there exists a real number r such that f1 - h = r(f0 - h). It follows from Lemma 4.1 that
| | f 1 - h , f 2 - f 0 , , f n - f 0 | | α = | | r ( f 0 - h ) , f 2 - f 0 , , f n - f 0 | | α = | r | | | f 0 - h , f 2 - f 0 , , f n - f 0 | | α = | r | | | f 0 - h , f 2 - h , , f n - h | | α = | | r ( f 0 - h ) , f 2 - h , , f n - h | | α = | | f 1 - h , f 2 - h , , f n - h | | α .

This completes the proof.

Theorem 4.1 Let Ψ be an n-Lipschitz mapping with the n-Lipschitz constant κ ≤ 1. Assume that if f0, f1, ..., f n are m-collinear then Ψ(f0), Ψ(f1), ..., Ψ(f m ) are m-collinear, m = 2, n, and that Ψ satisfies (n DOPP), then Ψ is an n-isometry.

Proof. It follows from Lemma 4.3 that Ψ preserves n-distance k for all k . For f0, f1, ..., f n X, there are two cases depending upon whether f1 - f0, ..., f n - f0 α = 0 or not. In the case f1 - f0, ..., f n - f0 α = 0, f1 - f0, ..., f n - f0 are linearly dependent, that is, n-collinear. Thus f1 - f0, ..., f n - f0 are linearly dependent. Thus Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0) β = 0.

In the case f1 - f0, ..., f n - f0 α > 0, there exists an n0 such that
n 0 > | | f 1 - f 0 , , f n - f 0 | | α .
Assume that
| | Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β < | | f 1 - f 0 , , f n - f 0 | | α .
We can set
h = f 0 + n 0 | | f 1 - f 0 , , f n - f 0 | | α ( f 1 - f 0 ) .
Then we get
| | h - f 0 , , f n - f 0 | | α = f 0 + n 0 | | f 1 - f 0 , , f n - f 0 | | α ( f 1 - f 0 ) - f 0 , , f n - f 0 α = n 0 | | f 1 - f 0 , , f n - f 0 | | α | | f 1 - f 0 , , f n - f 0 | | α = n 0 .
It follows from Lemma 4.3 that
| | Ψ ( h ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = n 0 .
By the definition of h,
h - f 1 = n 0 | | f 1 - f 0 , , f n - f 0 | | α - 1 ( f 1 - f 0 ) .
Since
n 0 | | f 1 - f 0 , , f n - f 0 | | α > 1 ,
h - f1 and f1 - f0 have the same direction. It follows from Lemma 4.2 that
| | h - f 0 , f 2 - f 0 , , f n - f 0 | | α = | | h - f 1 , f 2 - f 0 , , f n - f 0 | | α + | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α .
Since Ψ(h), Ψ(f1), Ψ(f2) are 2-collinear, we have
| | Ψ ( h ) - Ψ ( f 1 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | Ψ ( f 1 ) - Ψ ( h ) , Ψ ( f 2 ) - Ψ ( h ) , , Ψ ( f n ) - Ψ ( h ) | | β | | f 1 - h , f 2 - h , , f n - h | | α = | | f 1 - h , f 2 - f 0 , , f n - f 0 | | α = n 0 - | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α
by Lemma 4.4. By the assumption,
n 0 = | | Ψ ( h ) - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β | | Ψ ( h ) - Ψ ( f 1 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β + | | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β < n 0 - | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α + | | f 1 - f 0 , f 2 - f 0 , , f n - f 0 | | α = n 0 ,

which is a contradiction. Hence Ψ is an n-isometry.

Lemma 4.5 Let g0, g1 be elements of (X). Then v = g 0 + g 1 2 is the unique element of (X) satisfying
1 2 | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | | g 1 - v , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | | g 0 - g n , g 0 - v , g 2 - g n , , g n - 1 - g n | | α

for some g2,...,g n (X) with g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n α ≠ 0 and v, g0, g1 2-collinear.

Proof. Let g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n α ≠ 0 and | | f 0 - f n , f 1 - f n , f 2 - f n , , f n - 1 - f n | | α 0 . .

Then v, g0, g1 are 2-collinear. It follows from Lemma 4.1 and g n - g 0 = g1 - g0 - (g1 - g n ) that
| | g 1 - v , g 1 - g n , g 2 - g n , g n - 1 - g n | | α = g 1 g 0 + g 1 2 , g 1 - g n , g 2 - g n , , g n - 1 - g n α = 1 2 | | g 1 - g 0 , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = 1 2 | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α
and similarly
| | g 0 - g n , g 0 - v , g 2 - g n , , g n - 1 - g n | | α = 1 2 | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α .

Now we prove the uniqueness.

Let u be an element of (X) satisfying the above properties. Since u, g0, g1 are 2-collinear, there exists a real number t such that u = tg0 + (1 - t)g1. It follows from Lemma 4.1 that
1 2 | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | | g 1 - u , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | | g 1 - ( t g 0 + ( 1 - t ) g 1 ) , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | t | | | g 1 - g 0 , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | t | | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α
and
1 2 | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α = | | g 0 - g n , g 0 - u , g 2 - g n , , g n - 1 - g n | | α = | | g 0 - g n , g 0 - ( t g 0 + ( 1 - t ) g 1 ) , g 2 - g n , , g n - 1 - g n | | α = | 1 - t | | | g 0 - g n , g 0 - g 1 , g 2 - g n , , g n - 1 - g n | | α = | 1 - t | | | g 0 - g n , g 1 - g n , g 2 - g n , , g n - 1 - g n | | α .

Since g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n α ≠ 0, we have 1 2 = | 1 - t | = | t | . Therefore, we get t = 1 2 and hence v = u.

Lemma 4.6 If Ψ is an n-isometry and f0, f1, f2 are 2-collinear then Ψ(f0), Ψ(f1), Ψ(f2) are 2-collinear.

Proof. Since dim(X) ≥ n, for any f0 (X), there exist g1,...,g n (X) such that g1 - f0, ..., g n - f0 are linearly independent. Then
| | g 1 - f 0 , , g n - f 0 | | α = | | Ψ ( g 1 ) - Ψ ( f 0 ) , , Ψ ( g n ) - Ψ ( f 0 ) | | β 0

and hence, the set A = {Ψ(f) -Ψ(f0) : f (X)} contains n linearly independent vectors.

Assume that f0, f1, f2 are 2-collinear. Then, for any f3,...,f n (X),
| | f 1 - f 0 , , f n - f 0 | | α = | | Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n ) - Ψ ( f 0 ) | | β = 0 ,

i.e. Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0) are linearly dependent.

If there exist f3, ..., fn-1such that Ψ(f1) - Ψ(f0), ..., Ψ(fn- 1) - Ψ(f0) are linearly independent, then
A = { Ψ ( f n ) - Ψ ( f 0 ) : f n ( X ) } span  { Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n - 1 ) - Ψ ( f 0 ) } ,

which contradicts the fact that A contains n linearly independent vectors.

Then, for any f3, ..., fn-1, Ψ(f1) - Ψ(f0), ..., Ψ(fn-1) - Ψ(f0) are linearly dependent.

If there exist f3, ..., fn-2such that Ψ(f1) - Ψ(f0), ..., Ψ(fn-2) - Ψ(f0) are linearly independent, then
A = { Ψ ( f n - 1 ) - Ψ ( f 0 ) : f n - 1 ( X ) } span  { Ψ ( f 1 ) - Ψ ( f 0 ) , , Ψ ( f n - 2 ) - Ψ ( f 0 ) } ,

which contradicts the fact that A contains n linearly independent vectors.

And so on, Ψ(f1) - Ψ(f0), Ψ(f2) - Ψ(f0) are linearly dependent. Thus Ψ(f0), Ψ(f1), and Ψ(f2) are 2-collinear.

Theorem 4.2 Every n-isometry mapping is affine.

Proof. Let Ψ be an n-isometry and Φ(f) = Ψ(f) - Ψ(0). Then Φ is an n-isometry and Φ(0) = 0. Thus we may assume that Ψ(0) = 0. Hence it suffices to show that Ψ is linear.

Let f0, f1 (X) with f0f1. Since dim(X) ≥ n, there exist f2,...,f n (X) such that
| | f 0 - f n , f 1 - f n , f 2 - f n , , f n - 1 - f n | | α 0 .
Since Ψ is an n-isometry, we have
| | Ψ ( f 0 ) - Ψ ( f n ) , Ψ ( f 1 ) - Ψ ( f n ) , Ψ ( f 2 ) - Ψ ( f n ) , , Ψ ( f n - 1 ) - Ψ ( f n ) | | β 0 .
It follows from Lemma 4.1 that
Ψ ( f 0 ) - Ψ ( f n ) , Ψ ( f 0 ) - Ψ f 0 + f 1 2 , Ψ ( f 2 ) - Ψ ( f n ) , , Ψ ( f n - 1 ) - Ψ ( f n ) β = Ψ ( f n ) - Ψ ( f 0 ) , Ψ f 0 + f 1 2 - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n - 1 ) - Ψ ( f 0 ) β = f n - f 0 , f 0 + f 1 2 - f 0 , f 2 - f 0 , , f n - 1 - f 0 α = 1 2 | | f n - f 0 , f 1 - f 0 , f 2 - f 0 , , f n - 1 - f 0 | | α = 1 2 | | Ψ ( f n ) - Ψ ( f 0 ) , Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , , Ψ ( f n - 1 ) - Ψ ( f 0 ) | | β = 1 2 | | Ψ ( f 0 ) - Ψ ( f n ) , Ψ ( f 1 ) - Ψ ( f n ) , Ψ ( f 2 ) - Ψ ( f n ) , , Ψ ( f n - 1 ) - Ψ ( f n ) | | β .
And we get
Ψ ( f 1 ) - Ψ f 0 + f 1 2 , Ψ ( f 1 ) - Ψ ( f n ) , Ψ ( f 2 ) - Ψ ( f n ) , , Ψ ( f n - 1 ) - Ψ ( f n ) β = Ψ f 0 + f 1 2 - Ψ ( f 1 ) , Ψ ( f n ) - Ψ ( f 1 ) , Ψ ( f 2 ) - Ψ ( f 1 ) , , Ψ ( f n - 1 ) - Ψ ( f 1 ) β = f 0 + f 1 2 - f 1 , f n - f 1 , f 2 - f 1 , , f n - 1 - f 1 α = 1 2 | | f 0 - f 1 , f n - f 1 , f 2 - f 1 , , f n - 1 - f 1 | | α = 1 2 | | Ψ ( f 0 ) - Ψ ( f 1 ) , Ψ ( f n ) - Ψ ( f 1 ) , Ψ ( f 2 ) - Ψ ( f 1 ) , , Ψ ( f n - 1 ) - Ψ ( f 1 ) | | β = 1 2 | | Ψ ( f 0 ) - Ψ ( f n ) , Ψ ( f 1 ) - Ψ ( f n ) , Ψ ( f 2 ) - Ψ ( f n ) , , Ψ ( f n - 1 ) - Ψ ( f n ) | | β .

By Lemma 4.6, we obtain that Ψ f 0 + f 1 2 , Ψ(f0), and Ψ(f1) are 2-collinear. By Lemma 4.5, we get Ψ f 0 + f 1 2 = Ψ ( f 0 ) + Ψ ( f 1 ) 2 for all f, g (X) and α, β (0, 1). Since Ψ(0) = 0, we can easily show that Ψ is additive. It follows that Ψ is - linear -linear.

Let r + with r ≠ 1 and f (X). By Lemma 4.6, Ψ(0), Ψ(f) and Ψ(rf) are also 2-collinear. It follows from Ψ(0) = 0 that there exists a real number k such that Ψ(rf) = k Ψ(f). Since dim(X) ≥ n, there exist f1, ..., fn-1 (X) such that f, f1, f2, ..., fn-1 α ≠ 0. Since Ψ(0) = 0, for every f0, f1, f2, ..., fn-1 (X),
| | f 0 , f 1 , f 2 , , f n - 1 | | α = | | f 0 - 0 , f 1 - 0 , f 2 - 0 , , f n - 1 - 0 | | α = | | Ψ ( f 0 ) - Ψ ( 0 ) , Ψ ( f 1 ) - Ψ ( 0 ) , Ψ ( f 2 ) - Ψ ( 0 ) , , Ψ ( f n - 1 ) - Ψ ( 0 ) | | β = | | Ψ ( f 0 ) , Ψ ( f 1 ) , Ψ ( f 2 ) , , Ψ ( f n - 1 ) | | β .
Thus we have
r | | f , f 1 , f 2 , , f n - 1 | | α = | | r f , f 1 , f 2 , , f n - 1 | | α = | | Ψ ( r f ) , Ψ ( f 1 ) , Ψ ( f 2 ) , , Ψ ( f n - 1 ) | | β = | | k Ψ ( f ) , Ψ ( f 1 ) , Ψ ( f 2 ) , , Ψ ( f n - 1 ) | | β = | k | | | Ψ ( f ) , Ψ ( f 1 ) , Ψ ( f 2 ) , , Ψ ( f n - 1 ) | | β = | k | | | f , f 1 , f 2 , , f n - 1 | | α .
Since f, f1, f2, ..., fn-1 α ≠ 0, |k| = r. Then Ψ(rf) = r Ψ(f) or Ψ(rf) = -r Ψ(f). First of all, assume that k = -r, that is, Ψ(rf) = -r Ψ(f). Then there exist positive rational numbers q1, q2 such that o < q1 < r < q2. Since dim(X) ≥ n, there exist h1, ..., hn-1 (X) such that
| | r f - q 2 f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α 0 .
Then we have
( q 2 + r ) | | Ψ ( f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = | | ( q 2 + r ) Ψ ( f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = | | q 2 Ψ ( f ) - ( - r Ψ ( f ) ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = | | Ψ ( r f ) - Ψ ( q 2 f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = | | r f - q 2 f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α = ( q 2 - r ) | | f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α ( q 2 - q 1 ) | | f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α = | | q 1 f - q 2 f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α = | | Ψ ( q 1 f ) - Ψ ( q 2 f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β .
And also we have
| | r f - q 2 f , h 1 - q 2 f , h 2 - q 2 f , , h n - 1 - q 2 f | | α = | | Ψ ( r f ) - Ψ ( q 2 f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = | | - r Ψ ( f ) - q 2 Ψ ( f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β = ( r + q 2 ) | | Ψ ( f ) , , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β .
Since rf - q2f, h1 - q2f, h2 - q2f, ..., hn-1- q2f α ≠ 0,
| | Ψ ( f ) , Ψ ( h 1 ) - Ψ ( q 2 f ) , Ψ ( h 2 ) - Ψ ( q 2 f ) , , Ψ ( h n - 1 ) - Ψ ( q 2 f ) | | β 0 .

Thus we have r + q2 < q2 - q1, which is a contradiction. Hence k = r, that is, Ψ(rf) = r Ψ(f) for all positive real numbers r. Therefore Ψ is -linear, as desired.

We get the following corollary from Theorems 4.1 and 4.2.

Corollary 4.1 Let Ψ be an n-Lipschitz mapping with the n-Lipschitz constant κ ≤ 1. Suppose that if f, g, h are 2-collinear, then Ψ(f), Ψ(g), Ψ(h) are 2-collinear. If Ψ satisfies (n DOPP), then Ψ is an affine n-isometry.

5. Conclusion

In this article, the concept of 2-fuzzy n-normed linear space is defined and the concepts of n-isometry, n-collinearity, n-Lipschitz mapping are given. Also, the Mazur-Ulam theorem is generalized into 2-fuzzy n-normed linear spaces.

Declarations

Acknowledgements

The authors would like to thank the referees and area editor Professor Mohamed A. El-Gebeily for giving useful suggestions and comments for the improvement of this article.

Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(2)
Department of Mathematics, Faculty of Science and Arts, Celal Bayar University

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