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A Mazur-Ulam problem in non-Archimedean n-normed spaces
Journal of Inequalities and Applications volume 2013, Article number: 34 (2013)
Abstract
In this article, we study the notions of n-isometries in non-Archimedean n-normed spaces over linear ordered non-Archimedean fields and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for n-isometries in non-Archimedean n-normed spaces.
MSC:46B20, 51M25, 46S10.
1 Introduction
Let X and Y be metric spaces with metrics and , respectively. A map is called an isometry if for every . 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 . 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.
2 The Mazur-Ulam theorem I in non-Archimedean n-normed spaces
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, is additive. Firstly, we consider some definitions and lemmas which are needed to prove the theorem.
Recall that a non-Archimedean (or ultrametric) valuation is given by a map from a field into such that for all ,
-
(i)
if and only if ;
-
(ii)
;
-
(iii)
.
If every element of carries a valuation, then a field is called a valued field; for convenience, we simply call it a field. It is obvious that and for all . A trivial example of a non-Archimedean valuation is the map taking everything but 0 into 1 and (see [23]).
Let be a vector space over a valued field . A non-Archimedean norm is a function such that for all and ,
-
(i)
if and only if ;
-
(ii)
;
-
(iii)
the strong triangle inequality
Then we say is a non-Archimedean space.
Definition 2.1 Let be a vector space with the dimension greater than over a valued field with a non-Archimedean valuation . A function is said to be a non-Archimedean n-norm if
-
(i)
are linearly dependent;
-
(ii)
for every permutation of ;
-
(iii)
;
-
(iv)
for all and all . Then is called a non-Archimedean n-normed space.
From now on, let and be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field .
Definition 2.2 Let and be non-Archimedean n-normed spaces and be a mapping. We call f an n-isometry if
for all .
Definition 2.3 The points of a non-Archimedean n-normed space are said to be n-collinear if for every i, is linearly dependent.
The points , and of a non-Archimedean n-normed space are said to be 2-collinear if and only if for some element t of the non-Archimedean field . We denote the set of all elements of whose valuations are 1 by , that is, .
Lemma 2.4 Let be an element of a non-Archimedean n-normed space for every and . Then
for all .
Proof
By Definition 2.1, we have
One can easily prove the converse using similar methods. This completes the proof. □
Remark 2.5 Let , be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field and let be an n-isometry. One can show that the n-isometry f from to 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 is defined by the point .
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 to a non-Archimedean n-normed space preserves the midpoint of a segment, i.e., the f-image of the midpoint of a segment in is also the midpoint of a corresponding segment in .
Lemma 2.6 Let be a non-Archimedean n-normed space over a linear ordered non-Archimedean field with and let with . Then is the unique member of satisfying
for some with and u, , are 2-collinear.
Proof Let . From the assumption for the dimension of , there exist elements in such that . One can easily prove that u satisfies the above equations and conditions. It suffices to show the uniqueness for u. Assume that there is another v satisfying
for some elements of with and v, , are 2-collinear. Since v, , are 2-collinear, for some . Then we have
Since , we have two equations and . So, there are two integers , such that , . Since , for all . Thus, we may assume that , and without loss of generality. If , then , that is, . 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, and hence . □
Theorem 2.7 Let , be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field with and be an n-isometry. Then the midpoint of a segment is f-invariant, i.e., for every with , is also the midpoint of a segment with endpoints and in .
Proof Let with . Since the dimension of is greater than , there exist elements of satisfying . Since , and their midpoint are 2-collinear in , , , are also 2-collinear in by Remark 2.5. Since f is an n-isometry, we have the following:
By Lemma 2.6, we obtain that for all with . This completes the proof. □
Lemma 2.8 Let and be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field and be an n-isometry. Then the following conditions are equivalent.
-
(i)
The n-isometry f preserves the midpoint of a segment in , i.e., for all with ;
-
(ii)
The n-isometry f preserves the barycenter of a triangle in , i.e., for all satisfying that , , are not 2-collinear.
Proof Assume that the n-isometry f preserves the barycenter of a triangle in . Let , be in with . Since the n-isometry f preserves the 2-collinearity, , , are 2-collinear. So,
for some element s of . By the hypothesis for the dimension of , we can choose the element of satisfying that , and are not 2-collinear. Since , , are 2-collinear, we have that , , are also 2-collinear by Remark 2.5. So, we obtain that
for some element t of the non-Archimedean field . By the equations (2.1), (2.2) and the barycenter-preserving property for the n-isometry f, we have
Thus, we get
So, we have the following equation:
By a calculation, we obtain
Since , , are not 2-collinear, , are linearly independent. Since , there are such that . Since f is an n-isometry,
So, and are linearly independent. Hence, from equation (2.3), we have and , i.e., we obtain , , which imply the equation
for all with .
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 and be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field with . If is an n-isometry, then is additive.
Proof Let . Then it is clear that and g is also an n-isometry.
From Theorem 2.7, for (), we have
Since , we obtain that g is additive which completes the proof. □
3 The Mazur-Ulam theorem II in non-Archimedean n-normed spaces
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 to a non-Archimedean n-normed space 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.
Lemma 3.1 Let be a non-Archimedean n-normed space over a linear ordered non-Archimedean field with and let , , be elements of such that , , are not 2-collinear. Then is the unique member of satisfying
for some with and u is an interior point of .
Proof Let . Thus u is an interior point of . Since , there are elements of such that . Applying Lemma 2.4, we have that
And we can also obtain that
For the proof of uniqueness, let v be another interior point of satisfying
with . Since v is an element of the set , there are elements , , of with , such that . Then we have
and hence since . Similarly, we obtain . By the hypothesis of , there are integers , , such that , , . Since , every is less than 0. So, one may let , , and . Assume that one of the above inequalities holds. Then , i.e., . This is a contradiction, because the left-hand side is a multiple of 3 whereas the right-hand side is not. Thus, . Consequently, . This means that u is unique. □
Theorem 3.2 Let , be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field with and be an interior-preserving n-isometry. Then the barycenter of a triangle is f-invariant.
Proof Let , and be elements of satisfying that , and are not 2-collinear. It is obvious that the barycenter of a triangle is an interior point of the triangle. By the assumption, is also the interior point of a triangle . Since , there exist elements in such that is not zero. Since f is an n-isometry, we have
Similarly, we obtain
From Lemma 3.1, we get
for all satisfying that , , 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 . 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 and be non-Archimedean n-normed spaces over a linear ordered non-Archimedean field with . If is an interior-preserving n-isometry, then is additive.
Proof Let . One can easily check that 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.
Now, let , , be elements of satisfying that , , are not 2-collinear. Since g is an interior-preserving n-isometry, by Theorem 3.2,
for any satisfying that , , are not 2-collinear. Using Lemma 2.8 and the property , we obtain that the interior-preserving n-isometry g is additive, which completes the proof. □
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Acknowledgements
The authors are deeply grateful to the referees whose helped to improve our manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013784).
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The authors conceived of the study, participated in design and coordination of the manuscript. They drafted the manuscript and participated in the sequence alignment. They also read and approved the final manuscript.
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Chu, HY., Ku, SH. A Mazur-Ulam problem in non-Archimedean n-normed spaces. J Inequal Appl 2013, 34 (2013). https://doi.org/10.1186/1029-242X-2013-34
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DOI: https://doi.org/10.1186/1029-242X-2013-34
Keywords
- Mazur-Ulam theorem
- n-isometry
- non-Archimedean n-normed space