## Journal of Inequalities and Applications

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# 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

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 $\underset{n}{\underset{⏟}{X×\cdots ×X}}$ 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, $N\left({x}_{1},{x}_{2},\dots ,\lambda {x}_{n},t\right)=N\left({x}_{1},{x}_{2},\dots ,{x}_{n},\frac{t}{\lambda }\right)$, if λ ≠ 0, λ S,

(N5) For all s, t
$N\left({x}_{1},{x}_{2},\dots ,{x}_{n}+{x}_{n}^{\prime },s+t\right)\ge \text{min}\left\{N\left({x}_{1},{x}_{2},\dots ,{x}_{n},s\right),N\left({x}_{1},{x}_{2},\dots ,\underset{n}{\overset{\prime }{x}},t\right)\right\},$

(N6) N(x1, x2, ..., x n , t) is a non-decreasing function of t and $\underset{t\to \infty }{\text{lim}}N\left({x}_{1},{x}_{2},\dots ,{x}_{n},t\right)=1$.

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
$||{x}_{1},{x}_{2},\dots ,{x}_{n}|{|}_{\alpha }=\text{inf}\left\{t:N\left({x}_{1},{x}_{2},\dots {x}_{n},t\right)\ge \alpha ,\alpha \in \left(0,1\right)\right\}.$

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
$U+V=\left\{\left(x+y,\nu \wedge \mu \right):\left(x,\nu \right)\in U,\left(y,\mu \right)\in V\right\}$

and λU = {(λx, ν): (x, ν) U}.

Definition 2.4 A fuzzy linear space $\stackrel{^}{X}=X×\left(0,1\right]$ 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 $\left(x,\nu \right)\in \stackrel{^}{X}$ 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 $\left(x,\nu \right)\in \stackrel{^}{X}$ and all λ S,

3. (iii)

(x, ν) + (y, μ) || ≤ (x, ν μ) + (y, ν μ) for all $\left(x,\nu \right),\left(y,\mu \right)\in \stackrel{^}{X}$,

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
$f+g=\left\{\left(x,\mu \right)+\left(y,\eta \right)\right\}=\left\{\left(x+y,\mu \wedge \eta \right):\left(x,\mu \right)\in f\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\left(y,\eta \right)\in g\right\}$
and
$\lambda f=\left\{\left(\lambda x,\mu \right):\left(x,\mu \right)\in f\right\}$

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
$\begin{array}{l}\hfill ||f||=0\\ ⇔\left\{||\left(x,\mu \right)||:\left(x,\mu \right)\in f\right\}=0\phantom{\rule{2em}{0ex}}\\ ⇔x=0,\mu \in \left(0,1\right]⇔f=0.\phantom{\rule{2em}{0ex}}\end{array}$

2. (ii)
λf = |λ| f, λ S. For
$\begin{array}{ll}\hfill ||\lambda f||& =\left\{||\lambda \left(x,\mu \right)||:\left(x,\mu \right)\in f,\lambda \in S\right\}\phantom{\rule{2em}{0ex}}\\ =\left\{|\lambda |||\left(x,\mu \right)||:\left(x,\mu \right)\in f\right\}=|\lambda |||f||.\phantom{\rule{2em}{0ex}}\end{array}$

3. (iii)
f + gf + g for every f, g (X). For
$\begin{array}{ll}\hfill ||f+g||& =\left\{||\left(x,\mu \right)+\left(y,\eta \right)||:x,y\in X,\mu ,\eta \in \left(0,1\right]\right\}\phantom{\rule{2em}{0ex}}\\ =\left\{||\left(x+y\right),\left(\mu \wedge \eta \right)||:x,y\in X,\mu ,\eta \in \left(0,1\right]\right\}\phantom{\rule{2em}{0ex}}\\ =\left\{||\left(x,\mu \wedge \eta \right)||+||\left(y,\mu \wedge \eta \right)||:\left(x,\mu \right)\in f,\left(y,\eta \right)\in g\right\}\phantom{\rule{2em}{0ex}}\\ =||f||+||g||.\phantom{\rule{2em}{0ex}}\end{array}$

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 $\underset{n}{\underset{⏟}{\Im \left(X\right)×\cdots ×\Im \left(X\right)}}$ 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 $\underset{n}{\underset{⏟}{\Im \left(X\right)×\cdots ×\Im \left(X\right)}}×ℝ$ 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 ,
$N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{f}_{n}^{\prime },s+t\right)\ge \text{min}\left\{N\left({f}_{1},{f}_{2},\dots ,{f}_{n},s\right),N\left({f}_{1},{f}_{2},\dots ,\underset{n}{\overset{\prime }{f}},t\right)\right\},$

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

(2-N7) $\underset{t\to \infty }{\text{lim}}N\left({f}_{1},{f}_{2},\dots ,{f}_{n,}t\right)=1$.

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, $N\left({f}_{1},{f}_{2},\dots ,\lambda {f}_{i},\dots ,{f}_{n},t\right)=N\left({f}_{1},{f}_{2},\dots ,{f}_{i},\dots ,{f}_{n},\frac{t}{|\lambda |}\right),$ if λ ≠ 0, λ S,

(2-N5) for all s, t ,
$\begin{array}{c}N\left({f}_{1},{f}_{2},\dots ,{f}_{i}+{{f}^{\prime }}_{i},\dots ,{f}_{n},s+t\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\ge \text{min}\left\{N\left({f}_{1},{f}_{2},\dots ,{f}_{i},\dots ,{f}_{n},s\right),N\left({f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{i},\dots ,{f}_{n},t\right)\right\}.\end{array}$

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\left({f}_{1},{f}_{2},\dots ,{f}_{n,}t\right)=\left\{\begin{array}{c}\frac{t}{t+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}if\phantom{\rule{2.77695pt}{0ex}}t>0,t\in ℝ,\hfill \\ 0\hfill & if\phantom{\rule{1em}{0ex}}t\le 0\hfill \end{array}\right\$

for all $\left({f}_{1},{f}_{2},\dots ,{f}_{n}\right)\in \underset{n}{\underset{⏟}{\Im \left(X\right)×\cdots ×\Im \left(X\right)}}$. 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,
(2-N3) For all t with t > 0,
$\begin{array}{ll}\hfill N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)& =\frac{t}{t+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}=\frac{t}{t+||{f}_{1},{f}_{2},\dots ,{f}_{n},{f}_{n-1}||}\phantom{\rule{2em}{0ex}}\\ =N\left({f}_{1},{f}_{2},\dots ,{f}_{n},{f}_{n-1},t\right)=\cdots \phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}$
(2-N4) For all t with t > 0 and λ F, λ ≠ 0,
$\begin{array}{l}N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t/|\lambda |\right)=\frac{t/|\lambda |}{t/|\lambda |+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}=\frac{t/|\lambda |}{\left(t+|\lambda |\phantom{\rule{0.1em}{0ex}}||{f}_{1},{f}_{2},\dots ,{f}_{n}||\right)/|\lambda |}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}=\frac{t}{t+|\lambda |\phantom{\rule{0.1em}{0ex}}||{f}_{1},{f}_{2},\dots ,{f}_{n}||}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}=\frac{t}{\left(t+||{f}_{1},{f}_{2},\dots ,\lambda {f}_{n}||}=N\left({f}_{1},{f}_{2},\dots ,\lambda {f}_{n},t\right).\end{array}$
(2-N5) We have to prove
$N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{f}_{n}^{\prime },s+t\right)\ge \text{min}\left\{f\left({x}_{1},{f}_{2},\dots ,{f}_{n},s\right),N\left({f}_{1},{f}_{2},\dots ,\underset{n}{\overset{\prime }{f}},t\right)\right\}.$
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
$N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n},s+t=\frac{s+t}{s+t+||{f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n}||}.$

If
$\begin{array}{ll}\hfill \frac{s}{s+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}& \ge \frac{t}{t+||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}⇒\frac{||{f}_{1},{f}_{2},\dots ,{f}_{n}||}{s}\le \frac{||{x}_{1},{x}_{2},\dots ,{{x}^{\prime }}_{n}||}{t}\phantom{\rule{2em}{0ex}}\\ ⇒\frac{||{f}_{1},{f}_{2},\dots ,{f}_{n}||}{s}+\frac{||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}{s}\le \frac{||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}{t}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\frac{||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}{s}\phantom{\rule{2em}{0ex}}\\ ⇒\frac{||{f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n}||}{s}\le \left(\frac{s+t}{s\cdot t}\right)||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||\phantom{\rule{2em}{0ex}}\\ ⇒\frac{||{f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n}||}{s+t}\le \frac{||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}{t}\phantom{\rule{2em}{0ex}}\\ ⇒\frac{s+t+||{f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n}||}{s+t}\le \frac{t+||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}{t}\phantom{\rule{2em}{0ex}}\\ ⇒\frac{s+t}{s+t+||{f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n}||}\ge \frac{t}{t+||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}\phantom{\rule{2em}{0ex}}\\ ⇒N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n},s+t\right)\ge N\left({f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n},t\right).\phantom{\rule{2em}{0ex}}\end{array}$
Similarly, if
$\begin{array}{c}\frac{t}{t+||{f}_{1},{f}_{2},\dots ,{{f}^{\prime }}_{n}||}\ge \frac{s}{s+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}⇒N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{{f}^{\prime }}_{n},s+t\right)\ge N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right).\end{array}$
Thus
$N\left({f}_{1},{f}_{2},\dots ,{f}_{n}+{f}_{n}^{\prime },s+t\right)\ge \text{min}\left\{N\left({f}_{1},{f}_{2},\dots ,{f}_{n},s\right),N\left({f}_{1},{f}_{2},\dots ,\underset{n}{\overset{\prime }{f}},t\right)\right\}.$

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

(2-N7) For all t with t > 0,
$\begin{array}{ll}\hfill \underset{t\to \infty }{\text{lim}}N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)& =\underset{t\to \infty }{\text{lim}}\frac{t}{t+||{f}_{1},{f}_{2},\dots ,{f}_{n}||}\phantom{\rule{2em}{0ex}}\\ =\underset{t\to \infty }{\text{lim}}\frac{t}{t\left(1+\left(1/t\right)||{f}_{1},{f}_{2},\dots ,{f}_{n}||\right)}=1,\phantom{\rule{2em}{0ex}}\end{array}$

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
$||{f}_{1},{f}_{2},\dots ,{f}_{n}|{|}_{\alpha }=\text{inf}\left(t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge \alpha ,\alpha \in \left(0,1\right)\right\}.$

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
$\begin{array}{l}||{f}_{1},{f}_{2},\dots ,\lambda {f}_{n}|{|}_{\alpha }=\mathrm{inf}\left\{s:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},s\right)\ge \alpha \right\}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}=\mathrm{inf}\left\{s:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},\frac{s}{|\lambda |}\ge \alpha \right\}.\end{array}$

Let $t=\frac{s}{|\lambda |}$, then
$\begin{array}{c}||{f}_{1},{f}_{2},\dots ,\lambda {f}_{n}|{|}_{\alpha }=\text{inf}\left\{|\lambda |t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge \alpha \right\}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=|\lambda |\text{inf}\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge \alpha \right\}=|\lambda |\phantom{\rule{2.77695pt}{0ex}}||{f}_{1},{f}_{2},\dots ,{f}_{n}||\alpha .\end{array}$
If λ = 0, then
$\begin{array}{l}||{f}_{1},{f}_{2},\dots ,\lambda {f}_{n}|{|}_{\alpha }=||{f}_{1},{f}_{2},\dots ,0|{|}_{\alpha }=0=0||{f}_{1},{f}_{2},\dots ,{f}_{n}||\alpha \\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}=|\lambda |\phantom{\rule{0.1em}{0ex}}||{f}_{1},{f}_{2},\dots ,{f}_{n}||\alpha ,\forall \lambda \in S\left(\text{field}\right).\end{array}$
(iv)
Hence
$||{f}_{1},{f}_{2},\dots ,{f}_{n}+{f}_{n}^{\prime }|{|}_{\alpha }\le ||{f}_{1},{f}_{2},\dots ,{f}_{n}|{|}_{\alpha }+||{f}_{1},{f}_{2},\dots ,{f}_{n}^{\prime }|{|}_{\alpha }.$

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

Let 0 < α1 < α2. Then,
$\begin{array}{c}{∥{f}_{1},{f}_{2},\dots ,{f}_{n}∥}_{{\alpha }_{1}}=\text{inf}\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{1}\right\},\\ {∥{f}_{1},{f}_{2},\dots ,{f}_{n}∥}_{{\alpha }_{2}}=\text{inf}\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{2}\right\}.\end{array}$
As α1 < α2,
$\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{2}\right\}\subset \left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{1}\right\}$
implies that
$\text{inf}\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{2}\right\}\ge \text{inf}\left\{t:N\left({f}_{1},{f}_{2},\dots ,{f}_{n},t\right)\ge {\alpha }_{1}\right\}$
which implies that
$||{f}_{1},{f}_{2},\dots ,{f}_{n}|{|}_{{\alpha }_{2}}\ge ||{f}_{1},{f}_{2},\dots ,{f}_{n}|{|}_{{\alpha }_{1}}.$

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
${∥{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }={∥\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)∥}_{\beta }$

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).
$\left(n\text{DOPP}\right)\phantom{\rule{2.77695pt}{0ex}}\text{Let}\phantom{\rule{2.77695pt}{0ex}}{f}_{0},{f}_{1},{f}_{2},\dots ,{f}_{n}\in \Im \left(X\right)\phantom{\rule{2.77695pt}{0ex}}\text{with}\phantom{\rule{2.77695pt}{0ex}}{∥{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }=1.$

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,
$||{f}_{1},\dots ,{f}_{i},\dots ,{f}_{j},\dots ,{f}_{n}|{|}_{\alpha }=\phantom{\rule{2.77695pt}{0ex}}||{f}_{1},\dots ,{f}_{i},\dots ,{f}_{j}+\hslash {f}_{i},\dots ,{f}_{n}|{|}_{\alpha }$

for all 1 ≤ ijn.

Proof. It is obviously true.

Lemma 4.2 For ${f}_{0},{f}_{0}^{\prime }\in \Im \left(X\right)$, if f0 and ${f}_{0}^{\prime }$ are linearly dependent with some direction, that is, ${f}_{0}^{\prime }=t{f}_{0}$ for some t > 0, then
$||{f}_{0}+{f}_{0}^{\prime },{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }=\phantom{\rule{2.77695pt}{0ex}}||{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }+||{f}_{0}^{\prime },{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }$

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

Proof. Let ${f}_{0}^{\prime }=t{f}_{0}$ for some t > 0. Then we have
$\begin{array}{ll}\hfill ||{f}_{0}+{{f}^{\prime }}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }& =||{f}_{0}+t{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =\left(1+t\right)||{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =||{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }+t||{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =||{f}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }+||{{f}^{\prime }}_{0},{f}_{1},\dots ,{f}_{n}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\end{array}$

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
$||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\le k||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }$

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
${z}_{2}={f}_{0}+\frac{{f}_{2}-{f}_{0}}{||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}.$
Then we have
$\begin{array}{c}||{f}_{1}-{f}_{0},{z}_{2}-{f}_{0},{f}_{3}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={∥{f}_{1}-{f}_{0},\frac{{f}_{2}-{f}_{0}}{||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }},{f}_{3}-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }=1.\end{array}$
Since Ψ preserves the unit n-distance,
$||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\Psi \left({z}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }=1.$
But it follows from Ψ (f0) = Ψ (f1) that
$||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\Psi \left({z}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }=0,$

which is a contradiction. Hence, Ψ is injective.

Let f0, f1, f2, ..., f n be elements of (X), k and
$||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }=k.$
We put
${g}_{i}={f}_{0}+\frac{i}{k}\left({f}_{1}-{f}_{0}\right),\phantom{\rule{1em}{0ex}}i=0,1,\dots ,k.$
Then
$\begin{array}{c}||{g}_{i+1}-{g}_{i},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={∥{f}_{0}+\frac{i+1}{k}\left({f}_{1}-{f}_{0}\right)-\left({f}_{0}+\frac{i}{k}\left({f}_{1}-{f}_{0}\right)\right),{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={∥\frac{1}{k}\left({f}_{1}-{f}_{0}\right),{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=\frac{i}{k}||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }=\frac{k}{k}=1\end{array}$
for all i = 0, 1, ..., k - 1. Since Ψ satisfies (n DOPP),
$||\Psi \left({g}_{i+1}\right)-\Psi \left({g}_{i}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right)\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }=1$
(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
$\begin{array}{c}||\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left({g}_{2}\right)-\Psi \left({g}_{1}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||{r}_{0}\left(\Psi \left({g}_{1}\right)-\Psi \left({g}_{2}\right)\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=|{r}_{0}|\phantom{\rule{2.77695pt}{0ex}}||\left(\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right)\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }.\end{array}$
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
$\begin{array}{c}\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right)=\Psi \left({g}_{k}\right)-\Psi \left({g}_{0}\right)\\ \phantom{\rule{1em}{0ex}}=\Psi \left({g}_{k}\right)-\Psi \left({g}_{k-1}\right)+\Psi \left({g}_{k-1}\right)-\Psi \left({g}_{k-2}\right)+\cdots +\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right)\\ \phantom{\rule{1em}{0ex}}=k\left(\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right)\right).\end{array}$
Hence
$\begin{array}{c}||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||k\left(\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right)\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||k\Psi \left({g}_{1}\right)-\Psi \left({g}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }=k.\end{array}$

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
$||{f}_{1}-h,{f}_{2}-h,\dots ,{f}_{n}-h|{|}_{\alpha }=\phantom{\rule{2.77695pt}{0ex}}||{f}_{1}-h,{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }.$
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
$\begin{array}{ll}\hfill ||{f}_{1}-h,{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }& =\phantom{\rule{2.77695pt}{0ex}}||r\left({f}_{0}-h\right),{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =|r|\phantom{\rule{2.77695pt}{0ex}}||{f}_{0}-h,{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =|r|\phantom{\rule{2.77695pt}{0ex}}||{f}_{0}-h,{f}_{2}-h,\dots ,{f}_{n}-h|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =||r\left({f}_{0}-h\right),{f}_{2}-h,\dots ,{f}_{n}-h|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =||{f}_{1}-h,{f}_{2}-h,\dots ,{f}_{n}-h|{|}_{\alpha }.\phantom{\rule{2em}{0ex}}\end{array}$

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
${n}_{0}>||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }.$
Assume that
$||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }<||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }.$
We can set
$h={f}_{0}+\frac{{n}_{0}}{||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}\left({f}_{1}-{f}_{0}\right).$
Then we get
$\begin{array}{c}||h-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={∥{f}_{0}+\frac{{n}_{0}}{||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}\left({f}_{1}-{f}_{0}\right)-{f}_{0},\dots ,{f}_{n}-{f}_{0}∥}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=\frac{{n}_{0}}{||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }={n}_{0}.\end{array}$
It follows from Lemma 4.3 that
$||\Psi \left(h\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }={n}_{0}.$
By the definition of h,
$h-{f}_{1}=\left(\frac{{n}_{0}}{||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}-1\right)\left({f}_{1}-{f}_{0}\right).$
Since
$\frac{{n}_{0}}{||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }}>1,$
h - f1 and f1 - f0 have the same direction. It follows from Lemma 4.2 that
$\begin{array}{c}||h-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||h-{f}_{1},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }+||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }.\end{array}$
Since Ψ(h), Ψ(f1), Ψ(f2) are 2-collinear, we have
$\begin{array}{c}||\Psi \left(h\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left({f}_{1}\right)-\Psi \left(h\right),\Psi \left({f}_{2}\right)-\Psi \left(h\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left(h\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}\le ||{f}_{1}-h,{f}_{2}-h,\dots ,{f}_{n}-h|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{f}_{1}-h,{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={n}_{0}-||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\end{array}$
by Lemma 4.4. By the assumption,
$\begin{array}{ll}\hfill {n}_{0}& =||\Psi \left(h\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ \le ||\Psi \left(h\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ <{n}_{0}-||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+||{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n}-{f}_{0}||\alpha \phantom{\rule{2em}{0ex}}\\ ={n}_{0},\phantom{\rule{2em}{0ex}}\end{array}$

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

Lemma 4.5 Let g0, g1 be elements of (X). Then $v=\frac{{g}_{0}+{g}_{1}}{2}$ is the unique element of (X) satisfying
$\begin{array}{c}\frac{1}{2}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ =||{g}_{1}-v,{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ =||{g}_{0}-{g}_{n},{g}_{0}-v,{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\end{array}$

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 $||{f}_{0}-{f}_{n},{f}_{1}-{f}_{n},{f}_{2}-{f}_{n},\dots ,{f}_{n-1}-{f}_{n}|{|}_{\alpha }\ne 0.$.

Then v, g0, g1 are 2-collinear. It follows from Lemma 4.1 and g n - g 0 = g1 - g0 - (g1 - g n ) that
$\begin{array}{c}||{g}_{1}-v,{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots {g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}={∥{g}_{1}\frac{{g}_{0}+{g}_{1}}{2},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}∥}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=\frac{1}{2}||{g}_{1}-{g}_{0},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=\frac{1}{2}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}||\alpha \end{array}$
and similarly
$||{g}_{0}-{g}_{n},{g}_{0}-v,{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }=\frac{1}{2}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }.$

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
$\begin{array}{c}\frac{1}{2}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{g}_{1}-u,{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{g}_{1}-\left(t{g}_{0}+\left(1-t\right){g}_{1}\right),{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=|t|\phantom{\rule{2.77695pt}{0ex}}||{g}_{1}-{g}_{0},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=|t|\phantom{\rule{2.77695pt}{0ex}}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\end{array}$
and
$\begin{array}{c}\frac{1}{2}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{g}_{0}-{g}_{n},{g}_{0}-u,{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{g}_{0}-{g}_{n},{g}_{0}-\left(t{g}_{0}+\left(1-t\right){g}_{1}\right),{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=|1-t|\phantom{\rule{2.77695pt}{0ex}}||{g}_{0}-{g}_{n},{g}_{0}-{g}_{1},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=|1-t|\phantom{\rule{2.77695pt}{0ex}}||{g}_{0}-{g}_{n},{g}_{1}-{g}_{n},{g}_{2}-{g}_{n},\dots ,{g}_{n-1}-{g}_{n}|{|}_{\alpha }.\end{array}$

Since g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n α ≠ 0, we have $\frac{1}{2}=|1-t|=|t|$. Therefore, we get $t=\frac{1}{2}$ 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
$||{g}_{1}-{f}_{0},\dots ,{g}_{n}-{f}_{0}|{|}_{\alpha }=\phantom{\rule{2.77695pt}{0ex}}||\Psi \left({g}_{1}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({g}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\ne 0$

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),
$||{f}_{1}-{f}_{0},\dots ,{f}_{n}-{f}_{0}|{|}_{\alpha }=\phantom{\rule{2.77695pt}{0ex}}||\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }=0,$

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

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

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
$||{f}_{0}-{f}_{n},{f}_{1}-{f}_{n},{f}_{2}-{f}_{n},\dots ,{f}_{n-1}-{f}_{n}|{|}_{\alpha }\ne 0.$
Since Ψ is an n-isometry, we have
$||\Psi \left({f}_{0}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{1}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{n}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{n}\right)|{|}_{\beta }\ne 0.$
It follows from Lemma 4.1 that
$\begin{array}{c}{∥\Psi \left({f}_{0}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{0}\right)-\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{n}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{n}\right)∥}_{\beta }\\ ={∥\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right),\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{0}\right)∥}_{\beta }\\ ={∥{f}_{n}-{f}_{0},\frac{{f}_{0}+{f}_{1}}{2}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n-1}-{f}_{0}∥}_{\alpha }\\ =\frac{1}{2}||{f}_{n}-{f}_{0},{f}_{1}-{f}_{0},{f}_{2}-{f}_{0},\dots ,{f}_{n-1}-{f}_{0}|{|}_{\alpha }\\ =\frac{1}{2}||\Psi \left({f}_{n}\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{1}\right)-\Psi \left({f}_{0}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{0}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{0}\right)|{|}_{\beta }\\ =\frac{1}{2}||\Psi \left({f}_{0}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{1}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{n}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{n}\right)|{|}_{\beta }.\end{array}$
And we get
$\begin{array}{c}{∥\Psi \left({f}_{1}\right)-\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right),\Psi \left({f}_{1}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{n}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{n}\right)∥}_{\beta }\\ ={∥\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{n}\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{1}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{1}\right)∥}_{\beta }\\ ={∥\frac{{f}_{0}+{f}_{1}}{2}-{f}_{1},{f}_{n}-{f}_{1},{f}_{2}-{f}_{1},\dots ,{f}_{n-1}-{f}_{1}∥}_{\alpha }\\ =\frac{1}{2}||{f}_{0}-{f}_{1},{f}_{n}-{f}_{1},{f}_{2}-{f}_{1},\dots ,{f}_{n-1}-{f}_{1}|{|}_{\alpha }\\ =\frac{1}{2}||\Psi \left({f}_{0}\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{n}\right)-\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{1}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{1}\right)|{|}_{\beta }\\ =\frac{1}{2}||\Psi \left({f}_{0}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{1}\right)-\Psi \left({f}_{n}\right),\Psi \left({f}_{2}\right)-\Psi \left({f}_{n}\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left({f}_{n}\right)|{|}_{\beta }.\end{array}$

By Lemma 4.6, we obtain that $\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right)$, Ψ(f0), and Ψ(f1) are 2-collinear. By Lemma 4.5, we get $\Psi \left(\frac{{f}_{0}+{f}_{1}}{2}\right)=\frac{\Psi \left({f}_{0}\right)+\Psi \left({f}_{1}\right)}{2}$ 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 $ℚ-\text{linear}$-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),
$\begin{array}{c}||{f}_{0},{f}_{1},{f}_{2},\dots ,{f}_{n-1}|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{f}_{0}-0,{f}_{1}-0,{f}_{2}-0,\dots ,{f}_{n-1}-0|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left({f}_{0}\right)-\Psi \left(0\right),\Psi \left({f}_{1}\right)-\Psi \left(0\right),\Psi \left({f}_{2}\right)-\Psi \left(0\right),\dots ,\Psi \left({f}_{n-1}\right)-\Psi \left(0\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left({f}_{0}\right),\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right),\dots ,\Psi \left({f}_{n-1}\right)|{|}_{\beta }.\end{array}$
Thus we have
$\begin{array}{ll}\hfill r||f,{f}_{1},{f}_{2},\dots ,{f}_{n-1}|{|}_{\alpha }& =||rf,{f}_{1},{f}_{2},\dots ,{f}_{n-1}|{|}_{\alpha }\phantom{\rule{2em}{0ex}}\\ =||\Psi \left(rf\right),\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right),\dots ,\Psi \left({f}_{n-1}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ =||k\Psi \left(f\right),\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right),\dots ,\Psi \left({f}_{n-1}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ =|k|\phantom{\rule{2.77695pt}{0ex}}||\Psi \left(f\right),\Psi \left({f}_{1}\right),\Psi \left({f}_{2}\right),\dots ,\Psi \left({f}_{n-1}\right)|{|}_{\beta }\phantom{\rule{2em}{0ex}}\\ =|k|\phantom{\rule{2.77695pt}{0ex}}||f,{f}_{1},{f}_{2},\dots ,{f}_{n-1}|{|}_{\alpha }.\phantom{\rule{2em}{0ex}}\end{array}$
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
$||rf-{q}_{2}f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\ne 0.$
Then we have
$\begin{array}{c}\left({q}_{2}+r\right)||\Psi \left(f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||\left({q}_{2}+r\right)\Psi \left(f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||{q}_{2}\Psi \left(f\right)-\left(-r\Psi \left(f\right)\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left(rf\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||rf-{q}_{2}f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=\left({q}_{2}-r\right)||f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}\le \left({q}_{2}-{q}_{1}\right)||f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||{q}_{1}f-{q}_{2}f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left({q}_{1}f\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }.\end{array}$
And also we have
$\begin{array}{c}||rf-{q}_{2}f,{h}_{1}-{q}_{2}f,{h}_{2}-{q}_{2}f,\dots ,{h}_{n-1}-{q}_{2}f|{|}_{\alpha }\\ \phantom{\rule{1em}{0ex}}=||\Psi \left(rf\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=||-r\Psi \left(f\right)-{q}_{2}\Psi \left(f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\\ \phantom{\rule{1em}{0ex}}=\left(r+{q}_{2}\right)||\Psi \left(f\right),,\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }.\end{array}$
Since rf - q2f, h1 - q2f, h2 - q2f, ..., hn-1- q2f α ≠ 0,
$||\Psi \left(f\right),\Psi \left({h}_{1}\right)-\Psi \left({q}_{2}f\right),\Psi \left({h}_{2}\right)-\Psi \left({q}_{2}f\right),\dots ,\Psi \left({h}_{n-1}\right)-\Psi \left({q}_{2}f\right)|{|}_{\beta }\ne 0.$

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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