本文得到了等距映射的线性延拓的一般结果:设E,F是赋范(或β-严格凸赋β-范)线性空间,若V_0:S_1(E)→S_1(F)是等距,且对任意的x,y∈S_1(E),有‖V_0x-|(?)|V_0y‖≤‖x-|(?)|y‖,(?)∈R,则V_0必可延拓到全空间上等距算子(或线性等距算子)。特别,当E,F是赋范线性空间,V_0是满射或F为严格凸空间时,则V_0必可延拓为全空间的线性等距算子,从而推广了文[3～5]中的相应结果。

In this paper, we obtain the general results on extension of isometries: Let E, F be normed (or β-strictly convex β-normed) spaces. If V_0: S_1(E)→S_1 (F) is an isometry and for all x, y∈S_1(E) we have ‖V_0x-|λ|V_0y‖≤‖x-|λ|y‖, (?)λ∈R, then V_0 can be extended to be an isometry (or linear isometry) on the whole space. In particular, if E, F are a normed spaces, V_0 is surjective or F is strictly convex, then V_0 can be extended to be a linear isometry on the whole space, thus generalizing the corresponding results in [3-5].