The Mazur-Ulam Theorem On (2, P)-normed Spaces

In 1932,Mazur and Ulam states that every isometry of two real vector spaces is linear up to translation.It is called the Mazur-Ulam theorem.Many studiers have tried to study this theorem on other metric spaces later.So far,it has been obtained a lot of perfect conclusions.In this paper,we study the theorem with similar method in the(2,p)-normed linear spaces,and extend the conclusion to the(n,p)-normed linear spaces.We obtain some meaningful results and answers about this theorem partially.We divide this article into four chapters:In the first chapter,we briefly introduce the research background and current situation of the Aleksandrov problem and the Mazur-Ulam theorem,and give the main conclusion of this thesis.In order to discuss the theorem conveniently we give some basic knowledge about(2,p)-normed linear spaces in the second chapter of this paper.Also we give some main conclusions and some property of mapping f.In the third chapter,we give the conclusions of the Aleksandrov problem and the Mazur-Ulam theorem in the(2,p)-normed linear spaces.We divide this chapter into two parts.In the first part,we obtain the results in the(2,p)-normed linear spaces.The main results as follows:(1)Suppose that X and Y be two linear(2,p)-normed linear spaces.If f : X ? Y is an 2-isometry,then f is an affine.(2)Let X and Y be two linear(2,p)-normed linear spaces.If f : X ? Y satisfies(AOPP),and locally 2-Lipschitz.Then f is an affine 2-isometry.In the second part,it is devoted to the conclusion in the(n,p)-normed linear spaces.i.e.suppose that X and Y be two linear(n,p)-normed linear spaces.If f : X ? Y is an n-isometry,then f is an affine.The last chapter,which summarizes this paper and prospects the subject.