拟凸n-赋范空间上的Aleksandrov问题和Mazur-Ulam定理
发布时间：2018-11-23 11:06
【摘要】：本文回顾总结了赋范线性空间以及m-赋范线性空间上(m = 2,n)的Aleksandrov问题、Mazur-Ulam定理及其Aleksandrov-Rassias问题的提出和研究现状..引入了拟凸n-赋范线性空间的定义,并在其上深入探究了上述三个方面的问题,得到一些结论.在第一章节中,我们主要回顾总结了 Aleksandrov问题、Mazur-Ulam定理在赋范线性空间及和m-赋范线性空间上(m=2,n)的提出和已经取得的成果,我们要特别关注的是H.Y.Chu等研究者在文献[13]和[14]中关于m-赋范线性空间上的Aleksandrov问题和Mazur-Ulam定理的研究和已经取得的结论.在第二章节中,我们主要讨论了在拟凸n-赋范线性空间上的Aleksandrov问题和Mazur-Ulam定理.我们证明了f仅在满足(nDOPP)和保共线的条件下即为n-等距映射以及在拟凸n-赋范线性空间上的等距映射即为仿射的结论.在第三章节中,我们主要研究的是拟凸n-赋范线性空间上的Aleksandrov-Rassias问题.我们在基于文献[33]和文献[44]中对赋范线性空间及n-赋范线性空间上获得的有关Aleksandrov-Rassias问题的已有结论的基础上,证明了在拟凸n-赋范线性空间上将条件:||x_1-y_1,x_2-y_2,…,x_n-y_n||≥1(?)||f(x_1)-f(y_1),f(x_2)-f(y_2),…,f(x_n)-f(y_n)||≥_1替换成:||x_1-y_1,x_2-y_2,…,x_n-y_n||≤1(?)||f(x_1)-f(y_1),f(x_2)-f(y_2),…,f(x_n)-f(y_n)||≤_1其相关结论仍然成立.
[Abstract]:In this paper, the Aleksandrov problem, Mazur-Ulam theorem and Aleksandrov-Rassias problem on normed linear space and m normed linear space (m = 2n) are reviewed and summarized. The definition of quasi convex n- normed linear space is introduced, on which the above three problems are deeply discussed, and some conclusions are obtained. In the first chapter, we review and summarize the Aleksandrov problem, the Mazur-Ulam theorem in normed linear space and m- normed linear space (mG2n) and the results obtained. We should pay special attention to the study of Aleksandrov problem and Mazur-Ulam theorem on m-normed linear spaces by H.Y.Chu and other researchers in [13] and [14]. In the second chapter, we mainly discuss the Aleksandrov problem and Mazur-Ulam theorem on quasiconvex n-normed linear space. We prove that f is an affine only if it satisfies (nDOPP) and collinear preserving conditions, that is, n-isometric mapping and isometric mapping on quasi-convex n-normed linear space. In the third chapter, we mainly study the Aleksandrov-Rassias problem on quasiconvex n-normed linear spaces. On the basis of the existing conclusions on Aleksandrov-Rassias problems on normed linear spaces and n- normed linear spaces in references [33] and [44], In this paper, we prove that the condition:\;\ , x_n-y_n 鈮，
本文编号：2351369
[Abstract]:In this paper, the Aleksandrov problem, Mazur-Ulam theorem and Aleksandrov-Rassias problem on normed linear space and m normed linear space (m = 2n) are reviewed and summarized. The definition of quasi convex n- normed linear space is introduced, on which the above three problems are deeply discussed, and some conclusions are obtained. In the first chapter, we review and summarize the Aleksandrov problem, the Mazur-Ulam theorem in normed linear space and m- normed linear space (mG2n) and the results obtained. We should pay special attention to the study of Aleksandrov problem and Mazur-Ulam theorem on m-normed linear spaces by H.Y.Chu and other researchers in [13] and [14]. In the second chapter, we mainly discuss the Aleksandrov problem and Mazur-Ulam theorem on quasiconvex n-normed linear space. We prove that f is an affine only if it satisfies (nDOPP) and collinear preserving conditions, that is, n-isometric mapping and isometric mapping on quasi-convex n-normed linear space. In the third chapter, we mainly study the Aleksandrov-Rassias problem on quasiconvex n-normed linear spaces. On the basis of the existing conclusions on Aleksandrov-Rassias problems on normed linear spaces and n- normed linear spaces in references [33] and [44], In this paper, we prove that the condition:\;\ , x_n-y_n 鈮，
本文编号：2351369
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