Characterizations On Isometries In N-Normed Spaces

The isometry is an important subject in the research of space theory and operator theory.Mazur-Ulam theorem is also important for isometry．After that，many experts proposed a ser-ies of problems including the Aleksandrov problem and the Aleksandrov-Rassias problem，and obtained some corresponding conclusions．In this article，we further expand the scope ofresearch space and field to non-Archimedean field．The first chapter of this paper is about theAleksandrov problem．In section1.1，we sum-marize the background and history about theAleksandrov problem．In section1.2，we consi-der theAleksandrov problem in n-normed linear spaces and generalize the Benz theorem．Weshow some sufficient conditions without the assumption of strictly convexity．The second chapter is about theAleksandrov problem in fuzzy normed spaces．In section2.1，we show the definition and some properties of2-fuzzy n-normed linear spaces．In section2.2，we weaken the conditions of Theorem4.1in [31] by introducing the concept of locally n-Lipschitz mapping and solve the corresponding Aleksandrov problem for such spaces．The third chapter is about Mazur-Ulam theorem．We give the definitions of non-Archim-edean fuzzy2-normed spaces which is a generalization of the concept of non-Archimedeanfuzzy normed spaces and try to give a generalization Mazur-Ulam theorem in fuzzy2-normedspaces and to release surjectivity on isometry．In the end，we generalize the correspondingresults in non-Archimedean fuzzy n-normed spaces．...