| In 1932,the famous Mazur-Ulam theorem was proposed.In 1971,Baker proved that the Mazur-Ulam theorem still holds under the premise that the image space is strictly convex after removing the surjective condition.In 1987,D.Tingley proposed famous Tingley’s problem.A positive answer to this question would prove that the metric properties on the unit sphere determine the linear properties on the whole space.In 1931,Wigner first proposed Wigner’s theorem.It has many equivalent forms,one of which is described as follows:Let H and K be compact or real inner product spaces,and let f:H→K be a map.Then f satisfies|<f(x),f(y)>|=|(x,y>|(x,y∈H)if and only if it is phase-equivalent to a linear or an anti-linear isometry,that is,there exists a phase function ε:H→K(K is R or C)with |ε(x)|=1 such that ε·f is a linear or conjugate linear isometry.In this paper,we mainly study Mazur-Ulam theorem and Baker’s result.We combine these and extend Wigner’s theorem to strictly convex normed spaces,and also Tingley’s problem and Wigner’s theorem are naturally combined on CL spaces.In the first chapter,we introduce Tingley’s problem and Wigner’s theorem.In the second chapter,we study Wigner’s theorem of min-phase-isometries in strictly convex normed spaces.We show that if a min-phase-isometry(not necessarily surjective)maps the origin to the origin,it is phase-equivalent to a linear isometry.In the third chapter,we describe the phase-isometries on the unit sphere of CLspaces,and then the following results are proved:Let X be a CL-space,and let Y be a Banach space.Suppose that f:SX→SY be a surjective phase-isometry.Then there is a phase-function ε:SX→{-1,1} such that ε·f:Sx→SY is an isometry which can be extended to be a linear isometry from X onto Y. |
