The Isometric Extension of the Into Mapping from the Unit Sphere S_1(E) to S_1(l~∞(Г))

This paper considers the isometric extension problem concerning the mapping from the unitsphere S(E)of the normed space E into the unit sphere S(l~∞(Γ)).We find a condition under whichan isometry from S,(E)into S1(l~∞(Γ))can be linearly and isometrically extended to the whole space.Since l~∞(Γ)is universal with respect to isometry for normed spaces,isometric extension problemson a class of normed spaces are solved.More precisely,if E and F are two normed spaces,and ifV:S(E)→S(F)is a surjective isometry,where c(Γ)■(Γ),then Vcan be extended tobe an isometric operator defined on the whole space.