The Research Of Wigner Theorem And Generalized Tingley Problem

The study of space theory occupies an important position in functional analysis.And the study of normed linear space and quasi-normed linear space,which is of great importance.This paper mainly discusses the Wigner type theorem in quasi-normed s space and sn(H)space.The generalized Tingley problem in normed spaces sp(α),sp(α,H)and quasi-normed sn(H)space,obtains the corresponding results.In the first chapter,firstly,the research content of this topic is introduced.Secondly,the development process of Wigner theorem and Tingley problem are introduced in detail.Finally the introduction of phase isometry is introduced,which leads to the following problems to be discussed.In the second chapter,firstly,the definition of quasi-normed s space、sn(H)space and their norm definition form are introduced.Then,on the basis of the existing isometry property,the condition is changed to satisfy the phase isometry.Through discussion and proof of the relevant results.Finally,we can calculate that its phase equivalence to a linear isometry mapping or a surj ective isometry.In the third chapter,firstly,the definition of normed space sp(α)and its norm definition form are introduced.Then,on the basis of the existing linear extension of Tingley problem,the condition is changed to satisfy the phase isometry.Finally,the generalized Tingley problem is deduced and proved for spaces sp(α),sp(α,H),sn(H)respectively.