This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.
Guang Gui DING School of Mathematical Science and LPMC,Nankai University,Tianjin 300071,P.R.China
This article presents a novel method to prove that： let E be an AM-space and if dim E ≥ 3, then there does not exist any odd subtractive.isometric mapping from the unit sphere S（E） into S[L（Ω, μ）]. In particular, there does not exist any real linear isometry from E into L（Ω, μ）.
