Mazur-Ulam Theorem On Probabilistic 2-normed Spaces

In 1932,Mazur and Ulam proved the famous Mazur-Ulam theorem that every isometry of two real vector spaces is linear up to translation.The Mazur-Ulam theorem has been extensively studied by many authors and it has been obtained a lot of perfect conclusions.In this paper,we study the theorem with similar method on the probabilistic 2-normed spaces,and extend the conclusion to the probabilistic n-normed spaces.We obtain some meaningful results and answers about this theorem partially.We divide this article into five chapters:In the first chapter,we briefly introduce the research background and current situa-tion of the Aleksandrov problem,the Mazur-Ulam theorem and the probabilistic metric spaces.And we give the main conclusion of this thesis.In the second chapter,we introduced some definitions and theorems required in this paper.In the third chapter,we give the conclusions of the Mazur-Ulam theorem on the probabilistic2-normed spaces.We divide this chapter into two parts.In the first part,we give the definition of interior preserving mapping.Due to this result.,we are able to prove the Mazur-Ulam theorem on probabilistic2-normed spaces in a difterent way.Then we show the conclusion of W.Shatanawi and M.Postolache is still valid without the condition "f is collinear preserving mapping".In the second part,we give the definition of probabilistic ca-2-isometry and generalize the conclusion.In the fourth chapter,we show the Mazur-Ulam theorem on probabilistic n-normed spaces.Suppose that X and Y be two probabilistic n-normed spaces.If f:X→ Y is an probabilistic n-isometry,then f is affine.The last chapter,which summarizes this paper and prospects the subject.