The Isometric Extension Problem On 2-dimensional Normed Linear Spaces

Let E,F be real normed linear spaces,T:E→F be a surjective isometry.Mazur-Ulam theorem proves that T must be an affine.Since then,the relationship between the topological properties and algebra properties of operators sparked great interest to scholars,and expanded similarly Mazur-Ulam type problems.Tingley’s problem is one of the related problems.In 1987,Tingley presented:let E,F be real normed linear spaces,T₀:S_E→S_Fbe a surjective isometry,then is there a linear isometric extension T on E,and T₀=T|_{S E}?In this paper,we obtain some results for Tingley’s problem on 2-dimensional normed space.Details are as follows:In the first chapter,we introduce the research background and current situation of the problem in detail,and the main research results of this paper.In the second chapter,we introduce some of the preparatory knowledge,concepts,and symbols needed later.In the third chapter,we study Tingley’s problem on a 2-dimensional non-strictly convex and non-polygon space E.This chapter consists of three parts.In the section3.1,we study the“monotonicity”of distance between a point and a fixed point on the unit sphere S_E.In the section 3.2,by the property of isometry maintaining maximal convex set,we show that the image of a special subset（contains only four elements）of unit sphere under the isometric map is still the subset.In the section 3.3,we determine the four forms of surjective isometry T₀on the unit sphere S_E.Hence,we can make a extension of T₀,then isometric extension problom is affirmative on E.In the fourth chapter,we give a sufficient condition for Tingley’s problem on a2-dimensional normed linear space with special structure.The last chapter,which summarizes this paper and prospects the subject.