ON LINEAR EXTENSION OF 1-LIPSCHITZ MAPPING FROM HILBERT SPACE INTO A NORMED SPACE

In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.

A Note on Linear Extension of Isometries Between the Unit Spheres in β-normed Spaces

In this paper, we give four general results on linear extension of isometries between the unit spheres in β-normed spaces. These results improve the corresponding theorems in β-normed spaces.

Extension of Isometries Between the Unit Spheres of Normed Space E and C(Ω)

In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.

Extension of Isometries on the Unit Sphere of L^p Spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.

A Remark on Extension of Into Isometries

In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, G. G.： The isometric extension of an into mapping from the unit sphere S（l（2）^∞） to S（Lμ^1）. Acta Mathematica Sinica, English Series, 22（6）, 1721-1724 （2006）].

空间s_p(a,L^q)中单位球面上的等距算子延拓问题

讨论了空间Sp(α,L^q)中单位球面上的等距算子延拓问题,得到了空间Sp(α,L^q)中单位球面上的Lamperti等距能延拓到整个Sp(α,L^q)上.

Tingley's Problem on Symmetric Absolute Normalized Norms on R^2

In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make use of the notion of natural extension.Furthermore,using our methods,several sufficient conditions for Tingley＇s problem on symmetric absolute normalized norms on R2 are given.As applications,we present various new examples including the two-dimensional Lorentz sequence space d^（2）（ω,q） and its dual d^（2）（ω,q）＊by simple arguments.

空间L^p(Γ,Σ,μ)(1

文章得到以下结果(它改进了文献[16][18]中的一些结果):设E是一个赋范空间,V0是单位球面S(Lp(Γ,Σ,μ))到单位球面S(E)内的等距映射。如果V0满足下列两个条件:(ⅰ)对于任意的自然数n,实数ξk∈[-1,1]及χAk∈χ(Γ),1≤k≤n,有‖sum from k=1 to n ξkμ(Ai)1/pV0〔(χAi)/(μ(Ai)1/p)〕‖p=sum from k=1 to n|ξk|pμ(Ai),(ⅱ)对于任意的f1,f2∈S(Lp(Γ,Σ,μ))和实数ξ1,ξ2∈[-1,1],有‖ξ1V0(f1)+ξ2V0(f2)‖=1|ξ1V0(f1)+ξ2V0(f2)∈V0[S(Lp(Γ,Σ,μ)],那么V0可延拓为全空间Lp(Γ,Σ,μ)上的等距线性算子。

严格凸、光滑、自反的Banach空间中等距映射的线性延拓

本文主要研究了任意两个严格凸,光滑的自反空间E,F的单位球面S(E)和S(F)之间任意等距映射的线性延拓问题.

空间l^p(Γ)(1

研究了空间l^p(Γ)(1<p<∞)与Banach空间E的单位球面之间的非满等距映射V_O的延拓问题.我们得到,满足一定条件的V_O可线性等距延拓到全空间L^p上.