The Linear Normed Space Alexdrov According To Mazur-ulam Theorem

In 1932 Mazur and Ulam posed the theory of isometry and then defined the concept of conservative(or preserved) distance. The basic problem of conservative distance is whether the existence of a single conservative distance for some f implies that f is an isometry of metric space X into metric space Y. This is called the Alexandrov problem. The problem was posed in 1970 by Alexandrov. In recent years, Hahng-Yun Chu defined the concept of 2-isometry in linear 2-normed space and proved the theorem of Mazur-Ulam when X and Y were linear 2-normed space. After that, Chu et.al provided the concept of n-isometry in linear n-normed space and solved the Alexandrov problem in linear n-normed space. In 2009, Gao Jinmei introduced linear(2,p)-normed space and solved the Alexandrov problem in linear (2, p)-normed space.This dissertation consists of three chapters.In the first chapter, we introduce some characterizations of n-isometry and related results and theorems in linear n-normed space. In [14], the author introduced some characterizations of 2-isometry and related results and theorems. the characterizations of n-isometry in this chapter is generated on the basic of 2-isometry. Take the general-ization from weak 2-isometry to weak n-isometry for example:Let X and Y be linear 2-normed spaces and f :Xâ†'Y be a mapping. Then f is called a weak 2-isometry if for everyε> 0, there existsδ> 0 such that impliesThen next is the generalized weak n-isometry:Let X and Y be linear n-normed spaces and f :Xâ†'Y be a mapping. Then f is called a weak n-isometry if for x0, x1,…,xn∈X, everyε> 0, there existsδ> 0 such that impliesIn addition, we also prove that the Riesz theorem holds in linear n-normed space.In the second chapter, we consider the Alexandrov problem of distance preserving mappings. We first give some generalizations of earlier results. Then we introduce the concept of linear (n,p)-normed spaces on the basic of linear (2,p)- normed spaces. Here is the conceptLet X be a real linear space with dim X≥n and be a function. Suppose 0< p< 1. Then (X,‖·,…,·‖) is called a linear (n,p)-normed space if are linearly dependent; for every permutation is called a linear (n,p)-normed space.Besides that, we also solve the Alexandrov problem for linear (n,p)-normed spaces.In the third chapter, we first introduce the random normed space and t-norm. Then we prove the Mazur-Ulam theorem under some special conditions in random normed space.