On The Aleksandrov Problem In Normed Spaces

In this thesis, we mainly discuss the Aleksandrov Problem in normed spaces. We have divided this thesis into four chapters.In chapter one, we deal with the mappings which preserve unit (some positve) distance, as well as with mppings contracting some distance in p-normed spaces (0<p≤1), and we prove that a mapping preserving two (or three) distances in p-normed spaces is an isometry.In chapter two, we study the Aleksandrov problem in 2-ormed spaces, and we give some conditions under which a mapping is a 2-isometry. Moreover, we discuss the mappings which preserve unit (some positve) distance, as well as with mppings contracting some distance in 2-ormed linear spaces, and obtain some relative conclusions.In chapter three, we study the Aleksandrov problem in n-normed linear spaces, and get a method which proves a mapping is an n-isometry by using mathematical induction.In the last chapter, we make a conclusion of our work, and point out some problems needed to be solved.