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

Choonkil Park1 and Cihangir Alaca2*

Author Affiliations

1 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

2 Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, 45140 Manisa, Turkey

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Journal of Inequalities and Applications 2012, 2012:14  doi:10.1186/1029-242X-2012-14

 Received: 24 May 2011 Accepted: 19 January 2012 Published: 19 January 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 ℑ(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 ℑ(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 ℑ(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 ∈ ℕ and let X be a real vector space of dimension d n. (Here we allow d to be infinite.) A real-valued function ∥●, ..., ●∥ on X × × X n satisfying the following properties

(1) ∥x1, x2, ..., xn∥ = 0 if and only if x1, x2, ..., xn are linearly dependent,

(2) ∥x1, x2, ..., xn∥ is invariant under any permutation,

(3) ∥x1, x2, ..., αxn∥ = |α| ∥x1, x2, ..., xn∥ for any α ∈ ℝ,

(4) ∥x1, x2, ..., xn-1, y + z∥ ≤ ∥x1, 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 Xn × ℝ (ℝ, the set of real numbers) is called a fuzzy n-norm on X if and only if:

(N1) For all t ∈ ℝ with t ≤ 0, N(x1, x2, ..., xn, t) = 0,

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

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

(N4) For all t ∈ ℝ 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 ∈ ℝ

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, ..., xn, t) is a non-decreasing function of t ∈ ℝ 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, ..., xn,t) > 0 for all t > 0 implies that x1, x2, ..., xn 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 ℑ(X) the set of all fuzzy sets on X. For U, V ∈ ℑ(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

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

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

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

(iv) ∥(x, ∨tνt)∥ = ∧t ∥(x, νt)∥ for all νt ∈ (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 ℑ(X) be the set of all fuzzy sets in X. If f ∈ ℑ(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 ℑ(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 ℑ(X) is said to be normed linear space if, for every f ∈ ℑ(X), there exists an associated non-negative real number ∥f∥ (called the norm of f) which satisfies

(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 .

(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 | | .

(iii) ∥f + g∥ ≤ ∥f∥ + ∥g∥ for every f, g ∈ ℑ(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 (ℑ(X),∥●∥) is a normed linear space.

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

Definition 3.3 Let X be a real vector space of dimension d n (n ∈ ℕ) and ℑ(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) ∥f1, f2, ..., fn∥ = 0 if and only if f1, f2, ..., fn are linearly dependent,

(2) ∥f1, f2, ..., fn∥ is invariant under any permutation,

(3) ∥f1, f2, ..., λfn∥ = |λ| ∥f1, f2, ..., fn∥ for any λ S,

(4) ∥f1, f2, ..., fn-1, y + z∥ ≤ ∥f1, f2, ..., fn-1, y∥ + ∥f1, f2, ..., fn-1, z∥.

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

Definition 3.4 Let ℑ(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 ℑ(X)) if and only if

(2-N1) for all t ∈ ℝ with t ≤ 0, N(f1, f2, ..., fn, t) = 0,

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

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

(2-N4) for all t ∈ ℝ with t > 0, N(f1, f2, ..., λfn, t) = N(f1, f2, ..., fn, t/|λ|), if λ ≠ 0, λ S,

(2-N5) for all s, t ∈ ℝ,

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, ..., fn, ·): (0, ∞) → [0, 1] is continuous,

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

Then ℑ(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, ..., fn, ·) is a non-decreasing function of ℝ for all f1, f2,...,fn ∈ ℑ(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 ∈ ℝ 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 ∈ ℝ,

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 (ℑ(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 ∈ ℝ with t ≤ 0, by definition, we have N(f1, f2, ..., fn, t) = 0.

(2-N2) For all t ∈ ℝ 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 ∈ ℝ 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 ∈ ℝ 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 ) } .

(i) s + t < 0,

(ii) s = t = 0,

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

(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, ..., fn, ·): (0, ∞) → [0, 1] is continuous.

(2-N7) For all t ∈ ℝ 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 (ℑ(X), N) is a fuzzy n-normed linear space. Assume that

(2-N8) N(f1, f2, ..., fn, t) > 0 for all t > 0 implies f1, f2, ..., fn 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 ℑ(X).

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

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

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

(ii) Since N(f1, f2, ..., fn, t) is invariant under any permutation, ∥f1, f2, ..., fnα = 0 under any permutation.

(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 ℑ(X) is a fuzzy n-normed linear space. Hereafter, we use the notion of fuzzy n-normed linear space on ℑ(X) instead of 2-fuzzy n-normed linear space on X.

Definition 4.1 Let ℑ(X) and ℑ(X) be fuzzy n-normed linear spaces and Ψ : ℑ(X) → ℑ(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,...,fn ∈ ℑ(X) and α, β ∈ (0, 1).

For a mapping Ψ, consider the following condition which is called the n-distance one preserving property (nDOPP).

( 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(fn) - Ψ(f0)∥β = 1.

Lemma 4.1 Let f1, f2,...,fn ∈ ℑ(X), α ∈ (0, 1) and ħ ∈ ℝ. Then,

| | f 1 , , f i , , f j , , f n | | α = | | f 1 , , f i , , f j + f i , , f n | | α

for all 1 ≤ i j n.

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,...,fn ∈ ℑ(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,...,fn ∈ ℑ(X) and α ∈ (0, 1).

Definition 4.2 The elements f0, f1, f2, ..., fn of ℑ(X) are said to be n-collinear if for every i, {fj - fi : 0 ≤ j i n} 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,...,fn ∈ ℑ(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 (nDOPP). Then Ψ preserves the n-distance k for each k ∈ ℕ.

Proof. Suppose that there exist f0, f1 ∈ ℑ(X) with f0 f1 such that Ψ(f0) = Ψ(f1). Since dimℑ(X) ≥ n, there are f2,...,fn ∈ ℑ(X) such that f1 - f0, f2 - f0, ..., fn - f0 are linearly independent. Since ∥f1 - f0, f2 - f0, ..., fn - f0α ≠ 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, ..., fn be elements of ℑ(X), k ∈ ℕ 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 (nDOPP),

| | 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(gi) = Ψ(gi) - Ψ(gi-1) for all i = 0, 1, ..., k - 1. Thus Ψ(gi+1) - Ψ(gi) = 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, ..., fn be elements of ℑ(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, ..., fn are m-collinear then Ψ(f0), Ψ(f1), ..., Ψ(fm) are m-collinear, m = 2, n, and that Ψ satisfies (nDOPP), then Ψ is an n-isometry.

Proof. It follows from Lemma 4.3 that Ψ preserves n-distance k for all k ∈ ℕ. For f0, f1, ..., fn X, there are two cases depending upon whether ∥f1 - f0, ..., fn - f0α = 0 or not. In the case ∥f1 - f0, ..., fn - f0α = 0, f1 - f0, ..., fn - f0 are linearly dependent, that is, n-collinear. Thus f1 - f0, ..., fn - f0 are linearly dependent. Thus ∥Ψ(f1) - Ψ(f0), ..., Ψ(fn) - Ψ(f0)∥β = 0.

In the case ∥f1 - f0, ..., fn - f0α > 0, there exists an n0 ∈ ℕ 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 ℑ(X). Then v = g 0 + g 1 2 is the unique element of ℑ(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,...,gn ∈ ℑ(X) with ∥g0 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 0 and v, g0, g1 2-collinear.

Proof. Let ∥g0 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 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 gn - g0 = g1 - g0 - (g1 - gn) 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 ℑ(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 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 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 dimℑ(X) ≥ n, for any f0 ∈ ℑ(X), there exist g1,...,gn ∈ ℑ(X) such that g1 - f0, ..., gn - 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 ∈ ℑ(X)} contains n linearly independent vectors.

Assume that f0, f1, f2 are 2-collinear. Then, for any f3,...,fn ∈ ℑ(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(fn) - Ψ(f0) are linearly dependent.

If there exist f3, ..., fn-1 such 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-2 such 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 ∈ ℑ(X) with f0 f1. Since dimℑ(X) ≥ n, there exist f2,...,fn ∈ ℑ(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 ∈ ℑ(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 ∈ ℝ+ with r ≠ 1 and f ∈ ℑ(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) = kΨ(f). Since dimℑ(X) ≥ n, there exist f1, ..., fn-1 ∈ ℑ(X) such that ∥f, f1, f2, ..., fn-1α ≠ 0. Since Ψ(0) = 0, for every f0, f1, f2, ..., fn-1 ∈ ℑ(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-1α ≠ 0, |k| = r. Then Ψ(rf) = rΨ(f) or Ψ(rf) = -rΨ(f). First of all, assume that k = -r, that is, Ψ(rf) = -rΨ(f). Then there exist positive rational numbers q1, q2 such that o < q1 < r < q2. Since dimℑ(X) ≥ n, there exist h1, ..., hn-1 ∈ ℑ(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 =