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(L 1 . ). Acta Mathematica Sinica, English Series, 22(6), 1721.1724 (2006)].