On Isometric Extension Problem On The Unit Spheres Of Two-dimensional B Space

The completeness of the linear space has principal significance in the study of the spatial structure.As the well-known equivalence theorem of the completeness of the metric space,the closed-ball set theorem is logically rigorous and easy to understand in determining whether a linear space is complete,so it is widely used by space theory researchers.However,isometric operators has high applicative value in the research of spatial structure.The problem of extension of locally-maintained isometric operators is not only important in theory,but also has high practical applicability.This paper investigates a class of F-norm spaces with a special property in the closed-ball set theorem,two-dimensional b space,and introduces this special property about the closed-sphere.It also studies the isometric mapping and extension problem between such unit spheres.The first chapter introduces the equivalence theorem of metric space completeness in the introduction,namely the closed-ball set theorem,and points out a special property that some metric spaces follow,and then gives the concept of two-dimensional b-space and the spatial property of this class.At the same time,we introduces the famous problem of the extension of isometries on unit spheres is introduced-the Tingley problem.The second chapter discusses a typical two-dimensional b-space--b_E^（2）space and the representation theorem of the onto isometries between unit spheres,and then we proved that this isometry can be linearly extended to the whole space by this representation theorem.In the third chapter,we discuss another general class of two-dimensional b-space--b_p^（2）space,and showed a representation theorem with a special form of expression about the onto isometries between unit spheres ofb_p^（2）space.Then we proved that it is possible to linearly extended to the whole space for this operator by the above representation theorem.