A Mazur-Ulam problem in non-Archimedean n-normed spaces

Let X and Y be metric spaces with metrics $d_X$ and $d_Y$, respectively. A map $f:X\to Y$ is called an isometry if $d_Y(f(x),f(y))=d_X(x,y)$ for every $x,y\in X$. Mazur and Ulam [1] treated the theory of isometry for the first time. They have proved the following theorem.

Mazur-Ulam theorem Let f be an isometric transformation from a real normed vector space X onto a real normed vector space Y with $f(0)=0$. Then f is linear.

It was natural to ask if the result holds without the onto assumption. Having asked this natural question, Baker [2] answered that every isometry of a real normed linear space into a strictly convex real normed linear space is affine. The Mazur-Ulam theorem has been widely studied by [3–12].

Chu et al. [13] have defined the notion of a 2-isometry which is suitable to represent the concept of an area-preserving mapping in linear 2-normed spaces. In [14], Chu proved that the Mazur-Ulam theorem holds in linear 2-normed spaces under the condition that a 2-isometry preserves collinearity. Chu et al. [15] discussed characteristics of 2-isometries. In [16], Amyari and Sadeghi proved the Mazur-Ulam theorem in non-Archimedean 2-normed spaces under the condition of strict convexity. Recently, Choy et al. [17] proved the theorem on non-Archimedean 2-normed spaces over linear ordered non-Archimedean fields without the strict convexity assumption.

Misiak [18, 19] defined the concept of an n-normed space and investigated the space. Park and Rassias [20] investigated the stability of linear n-isometries in a linear n-normed Banach module. Chu et al. [21], in linear n-normed spaces, defined the concept of an n-isometry that is suitable to represent the notion of a volume-preserving mapping. In [22], Chu et al. generalized the Mazur-Ulam theorem to n-normed spaces.

In this paper, without the condition of strict convexity, we prove the (additive) Mazur-Ulam theorem on non-Archimedean n-normed spaces. Firstly, we assert that an n-isometry f from a non-Archimedean space to a non-Archimedean space preserves the midpoint of a segment under some condition about the set of all elements of a valued field whose valuations are 1. Using the above result, we show the Mazur-Ulam theorem on non-Archimedean n-normed spaces over linear ordered non-Archimedean fields. In addition, we prove that the barycenter of a triangle in the non-Archimedean n-normed spaces is f-invariant under different conditions from those referred in previous statements. And then we also prove the (second type) Mazur-Ulam theorem in non-Archimedean n-normed spaces under some different conditions.

In this section, we introduce a non-Archimedean n-normed space which is a kind of generalization of a non-Archimedean 2-normed space, and we show the (additive) Mazur-Ulam theorem for an n-isometry f defined on a non-Archimedean n-normed space, that is, $f(x)-f(0)$ is additive. Firstly, we consider some definitions and lemmas which are needed to prove the theorem.

If every element of $\mathcal{K}$ carries a valuation, then a field $\mathcal{K}$ is called a valued field; for convenience, we simply call it a field. It is obvious that $|1|=|-1|=1$ and $|n|\le 1$ for all $n\in \mathbb{N}$. A trivial example of a non-Archimedean valuation is the map $|\cdot |$ taking everything but 0 into 1 and $|0|=0$ (see [23]).

Then we say $(\mathcal{X},\parallel \cdot \parallel )$ is a non-Archimedean space.

for all $\alpha \in \mathcal{K}$ and all $x,y,x_1,\dots ,x_n\in \mathcal{X}$. Then $(\mathcal{X},\parallel \cdot ,\dots ,\cdot \parallel )$ is called a non-Archimedean n-normed space.

From now on, let $\mathcal{X}$ and $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$.

for all $x_0,x_1,\dots ,x_n\in \mathcal{X}$.

Definition 2.3 The points $x_0,x_1,\dots ,x_n$ of a non-Archimedean n-normed space $\mathcal{X}$ are said to be n-collinear if for every i, $\{x_j-x_i\mid 0\le j\ne i\le n\}$ is linearly dependent.

The points $x_0$, $x_1$ and $x_2$ of a non-Archimedean n-normed space $\mathcal{X}$ are said to be 2-collinear if and only if $x_2-x_0=t(x_1-x_0)$ for some element t of the non-Archimedean field $\mathcal{K}$. We denote the set of all elements of $\mathcal{K}$ whose valuations are 1 by $\mathcal{C}$, that is, $\mathcal{C}=\{\gamma \in \mathcal{K}:|\gamma |=1\}$.

for all $1\le i\ne j\le n$.

Proof

One can easily prove the converse using similar methods. This completes the proof. □

Remark 2.5 Let $\mathcal{X}$, $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ and let $f:\mathcal{X}\to \mathcal{Y}$ be an n-isometry. One can show that the n-isometry f from $\mathcal{X}$ to $\mathcal{Y}$ preserves the 2-collinearity using a similar method to that in [[22], Lemma 3.2].

The midpoint of a segment with endpoints x and y in the non-Archimedean n-normed space $\mathcal{X}$ is defined by the point $\frac{x+y}{2}$.

Now, we prove the Mazur-Ulam theorem on non-Archimedean n-normed spaces. In the first step, we prove the following lemma. And then, using the lemma, we show that an n-isometry f from a non-Archimedean n-normed space $\mathcal{X}$ to a non-Archimedean n-normed space $\mathcal{Y}$ preserves the midpoint of a segment, i.e., the f-image of the midpoint of a segment in $\mathcal{X}$ is also the midpoint of a corresponding segment in $\mathcal{Y}$.

for some $x_2,\dots ,x_n\in \mathcal{X}$ with $\parallel x_0-x_1,x_0-x_2,\dots ,x_0-x_n\parallel \ne 0$ and u, $x_0$, $x_1$ are 2-collinear.

Since $\parallel x_0-x_1,x_0-x_2,\dots ,x_0-x_n\parallel \ne 0$, we have two equations $|1-t|=1$ and $|t|=1$. So, there are two integers $k_1$, $k_2$ such that $1-t=2^{k_1}$, $t=2^{k_2}$. Since $2^{k_1}+2^{k_2}=1$, $k_i<0$ for all $i=1,2$. Thus, we may assume that $1-t=2^{-n_1}$, $t=2^{-n_2}$ and $n_1\ge n_2\in \mathbb{N}$ without loss of generality. If $n_1⪈n_2$, then $1=2^{-n_1}+2^{-n_2}=2^{-n_1}(1+2^{n_1-n_2})$, that is, $2^{n_1}=1+2^{n_1-n_2}$. This is a contradiction because the left-hand side of the equation is a multiple of 2 but the right-hand side of the equation is not. Thus, $n_1=n_2=1$ and hence $v=\frac{1}{2}{x}_0+\frac{1}{2}{x}_1=u$. □

Theorem 2.7 Let $\mathcal{X}$, $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{2^n|n\in \mathbb{Z}\}$ and $f:\mathcal{X}\to \mathcal{Y}$ be an n-isometry. Then the midpoint of a segment is f-invariant, i.e., for every $x_0,x_1\in \mathcal{X}$ with $x_0\ne x_1$, $f(\frac{x_0+x_1}{2})$ is also the midpoint of a segment with endpoints $f(x_0)$ and $f(x_1)$ in $\mathcal{Y}$.

By Lemma 2.6, we obtain that $f(\frac{x_0+x_1}{2})=\frac{f(x_0)+f(x_1)}{2}$ for all $x_0,x_1\in \mathcal{X}$ with $x_0\ne x_1$. This completes the proof. □

for all $x_0,x_1\in \mathcal{X}$ with $x_0\ne x_1$.

Conversely, (i) trivially implies (ii). This completes the proof of this lemma. □

Remark 2.9 One can prove that the above lemma also holds in the case of linear n-normed spaces.

Theorem 2.10 Let $\mathcal{X}$ and $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{2^n|n\in \mathbb{Z}\}$. If $f:\mathcal{X}\to \mathcal{Y}$ is an n-isometry, then $f(x)-f(0)$ is additive.

Proof Let $g(x):=f(x)-f(0)$. Then it is clear that $g(0)=0$ and g is also an n-isometry.

Since $g(0)=0$, we obtain that g is additive which completes the proof. □

In this section, under different conditions from those previously referred in Theorem 2.10, we also prove the (second type) Mazur-Ulam theorem on a non-Archimedean n-normed space. Firstly, we show that an n-isometry f from a non-Archimedean n-normed space $\mathcal{X}$ to a non-Archimedean n-normed space $\mathcal{Y}$ preserves the barycenter of a triangle, i.e., the f-image of the barycenter of a triangle is also the barycenter of a corresponding triangle. Then, using Lemma 2.8, we also prove the Mazur-Ulam theorem (a non-Archimedean n-normed space version) under some different conditions.

for some $x_3,\dots ,x_n\in \mathcal{X}$ with $\parallel x_0-x_1,x_0-x_2,x_0-x_3,\dots ,x_0-x_n\parallel \ne 0$ and u is an interior point of ${\mathrm{△}}_{x_0{x}_1{x}_2}$.

and hence $|s_2|=1$ since $\parallel x_0-x_1,x_0-x_2,x_0-x_3,\dots ,x_0-x_n\parallel \ne 0$. Similarly, we obtain $|s_0|=|s_1|=1$. By the hypothesis of $\mathcal{C}$, there are integers $k_0$, $k_1$, $k_2$ such that $s_0=3^{k_1}$, $s_1=3^{k_2}$, $s_2=3^{k_2}$. Since $s_0+s_1+s_2=1$, every $k_i$ is less than 0. So, one may let $s_0=3^{-n_0}$, $s_1=3^{-n_1}$, $s_2=3^{-n_2}$ and $n_0\ge n_1\ge n_2\in \mathbb{N}$. Assume that one of the above inequalities holds. Then $1=s_0+s_1+s_2=3^{-n_0}(1+3^{n_0-n_1}+3^{n_0-n_2})$, i.e., $3^{n_0}=1+3^{n_0-n_1}+3^{n_0-n_2}$. This is a contradiction, because the left-hand side is a multiple of 3 whereas the right-hand side is not. Thus, $n_0=n_1=n_2$. Consequently, $s_0=s_1=s_2=\frac{1}{3}$. This means that u is unique. □

Theorem 3.2 Let $\mathcal{X}$, $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{3^n|n\in \mathbb{Z}\}$ and $f:\mathcal{X}\to \mathcal{Y}$ be an interior-preserving n-isometry. Then the barycenter of a triangle is f-invariant.

for all $x_0,x_1,x_2\in \mathcal{X}$ satisfying that $x_0$, $x_1$, $x_2$ are not 2-collinear. □

The next theorem is the Mazur-Ulam theorem II in non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$. The assumptions of this theorem are different from that of Theorem 2.10. In particular, in the proof of the theorem, we use the f-preserving property for the barycenter of a triangle.

Theorem 3.3 Let $\mathcal{X}$ and $\mathcal{Y}$ be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{3^n|n\in \mathbb{Z}\}$. If $f:\mathcal{X}\to \mathcal{Y}$ is an interior-preserving n-isometry, then $f(x)-f(0)$ is additive.

Proof Let $g(x):=f(x)-f(0)$. One can easily check that $g(0)=0$ and g is also an n-isometry. Using a similar method in [[17], Theorem 2.4], we can easily prove that g is also an interior-preserving mapping.

for any $x_0,x_1,x_2\in \mathcal{X}$ satisfying that $x_0$, $x_1$, $x_2$ are not 2-collinear. Using Lemma 2.8 and the property $g(0)=0$, we obtain that the interior-preserving n-isometry g is additive, which completes the proof. □