On The Mazur-Ulam Property Of L~p(Γ)(1≤p<∞,p≠2)

Let X and Y be normed spaces,a mapping f:X→Y is said to be an isometry if‖f(x)-f(y)‖=‖x-y‖((?)x,y∈X).In 1987,D.Tingley raises question:Let X and Y be normed spaces.Suppose that f:S(X)→S(Y)is a surjective isometry.Is there a linear isometry f:X→Y such that f|S(X)=f?This problem is the famous Tingley’s Problem.This paper mainly proves the isometric extension in complex space l~p(r)(1≤p<∞,p≠2),that is,we prove that the surjective isometry f:S(l~p(Γ))→S(X),(1≤p<∞,p≠2)can be extended to a linear isometric operator on the full space.In the first chapter,we mainly introduce the background and development status of Tingley’s Problem and Mazur-Ulam Theorem,give the definition of isometric and Mazur-Ulam Property.In the second chapter,we study the Tingley’s problem between the complex Banach spaces l1(Γ)and complex Banach space X and get the following conclusion:The complex Banach space l1(Γ)satisfies the Mazur-Ulam property,that is,for any complex Banach space X and any surjective isometry f:S(l1(Γ))→S(X).Then,f can be extended to a real linear isometry from l1(Γ)onto X.In the third chapter,we study the Tingley’s problem between the complex Banach spaces l~p(Γ)(1<p<∞,p≠2)and complex Banach space X and get the following conclusion:The complex Banach space l~p(Γ)(1<p<∞,p≠2)satisfies the Mazur-Ulam property,that is,for any complex Banach space X and any surjective isometry f:S(l~pF(Γ))→S(X).Then,f can be extended to a real linear isometry from l~p(Γ)onto X.