Abstract: In this paper, the author gives some properties of non-surjective isometries
between the unit spheres of two l^∞-spaces, and discribes the characters of the range set in this condition. At the end, the author answers the question when the non-surjective isometries between the unit spheres of c₀(\Gamma)$ and l^∞-space can be extended to a linear isometry on the whole space.

Key words: Isometric operator, Extension of isometries