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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 394216, 7 pages
On the Aleksandrov-Rassias Problems on Linear -Normed Spaces
Department of Mathematics, Dalian Nationality University, Dalian 116600, China
Received 11 May 2013; Accepted 17 July 2013
Academic Editor: Ji Gao
Copyright © 2013 Yumei Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper generalizes T. M. Rassias' results in 1993 to -normed spaces. If and are two real -normed spaces and is -strictly convex, a surjective mapping preserving unit distance in both directions and preserving any integer distance is an -isometry.
Let and be two metric spaces. A mapping is called an isometry if satisfies for all , where and denote the metrics in the spaces and , respectively. For some fixed number , suppose that preserves distance , that is, for all with , we have , then is called a conservative (or preserved) distance for the mapping . In particular, we denote DOPP as preserving the one distance property and SDOPP as preserving the strong one distance property and also for .
In 1970 , Aleksandrov posed the following problem. Examine whether the existence of a single conservative distance for some mapping implies that is an isometry. This question is of great significance for the Mazur-Ulam Theorem .
In 1993, T. M. Rassias and P. Šemrl proved the following.
Theorem 1 (see ). Let and be two real normed linear spaces such that one of them has a dimension greater than one. Assume also that one of them is strictly convex. Suppose that is a surjective mapping that satisfies SDOPP. Then, is an affine isometry (a linear isometry up to translation).
Theorem 2 (see ). Let and be two real normed linear spaces such that one of them has a dimension greater than one. Suppose that is a Lipschitz mapping. Assume also that is a surjective mapping satisfying (SDOPP). Then, is an isometry.
Definition 3 (see ). Let be a real linear space with and , a function, then is called a linear -normed space if for any and all : are linearly dependent, : for every permutation of , : , : . The function is called the -norm on .
Definition 4 (see ). Let and be two real linear -normed spaces. (i)A mapping is defined to be an -isometry if for all , (ii)A mapping is called the -distance one preserving property (-DOPP) if for , , it follows that .(iii)A mapping is called the -strong distance one preserving property (-SDOPP) if for , , it follows that and conversely.(iv)A mapping is called an -Lipschitz if for all ,
Definition 5 (see ). The points of are called -collinear if for every , is linearly dependent.
Definition 6. is said to be -strictly convex normed spaces if for any , and imply that and are linearly dependent.
C. Park and T. M. Rassias obtained the following.
Theorem 7 (see ). Let and be real linear -normed spaces. If a mapping satisfies the following conditions: (i) has the -DOPP,(ii) is -Lipschitz,(iii) preserves the -collinearity,(iv) preserves the -collinearity, then is an -isometry.
2. Main Results
The proof of the following lemma was presented in , to be published; the proof is given again for the convenience of readers.
Lemma 8. Let be an -normed space such that has dimension greater than and . Suppose that for . Then, there exists such that
Proof. Since are linearly independent and , then there exists with .
Set . For any , we have Let us define by then, we obtain Set Clearly, . And we have On the other hand, Thus, Define by It follows that
Obviously, is continuous on . Using the mean value theorem, there exists such that .
Set , , we have And from , we have
Lemma 9. Let and be two real linear -normed spaces whose dimensions are greater than , and let be -strictly convex normed space. Suppose that is a surjective mapping satisfying (-SDOPP) with preserving distance for any . Then, preserves distance for any .
Proof. Firstly, is injective. Suppose, on the contrary, that there are , , such that . As , it follows that there exist vectors such that are linearly independent. Then, .
Set Clearly, Then This implies that , which is a contradiction. Therefore, is a bijective mapping.
Let and satisfying
By Lemma 8, we can find with Set Clearly, we have It follows from the hypothesis of preserving any integer ; then, Clearly, we have We conclude that Otherwise, if for some , we have with or such that Suppose that . Then, Assume that Set Then, for , Since is bijective and preserves -SDOPP on both directions. Then, there exists with which satisfies that However, by (20), , and thus , are linear dependent. Then, This contradiction implies that This also contradicts with (26). Since is -strictly convex, then there exists such that Then, Since then . Thus, Similarly, Hence,
Lemma 10. Let and be real -normed spaces such that . If a mapping preserves the distance for each , then preserves the distance zero.
Proof. Choose such that ; that is, are linearly dependent. Assume that is a maximum linearly independent group of (. As , we can find a finite sequence of vectors such that are linearly independent. Hence, it holds that
We will prove that for every . Let . We can find a vector such that are linearly independent. Set for arbitrarily fixed . Then, Since , we get Since preserves the distance , we see that
For , we set for any . Then, we have for each . Since are linearly dependent, we get and hence, which together with (48) implies that for all . By a similar argument, we further obtain that
In view of (45), (50), and (51), we conclude that where denotes either or for .
Since preserves the distance for any , it follows from (52) that where is an arbitrary positive integer. Hence, we conclude that which implies that preserves the distance zero.
Remark 11. In Lemma 2.2 ( to be published), we give the same method under the condition of preserving 2-colinear.
Theorem 12. Let and be real -normed spaces such that and is -strictly convex. If a surjective mapping has the -SDOPP and preserves the distance for any , then is an affine -isometry.
Proof. Assume that for .
Take positive integers such that Set for , and Clearly, for , According to Lemma 8, there exists such that It follows from Lemma 9 that we have for .
On the other hand, Hence Suppose that For any , with find a positive integer satisfying .
Set . Clearly, , and .
It follows that and Then (63) is not valid. Hence,
Corollary 13. Let and be two real linear -normed spaces. Suppose that mapping preserves any positive integer -distance and Lipschitz condition. Then, is an -isometry.
This work is supported by the Fundamental Research Funds for the Central Universities in China and Education Department of Liaoning province in china.
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