## Isometries on the $C_0 (Î©,E)$ type spaces

J. Math. Sci. Univ. Tokyo
Vol. 2 (1995), No. 1, Page 117--130.

Wang, Risheng
Isometries on the $C_0 (Î©,E)$ type spaces
It is proved that if the unit spheres of $C\sb 0 (X,E)$ and $C\sb 0 (Y,F)$ are isometric, then $X$ and $Y$ are homeomorphic, and the unit spheres of $E$ and $F$ are isometric, generalizing the Banach-Stone's theorem; any isometry between the unit spheres of $C\sb 0 (X,H\sb 1 )$ and $C\sb 0 (Y,H\sb 2 )$ is necessarily the restriction of some linear isometry between $C\sb 0 (X,H\sb 1 )$ and $C\sb 0 (Y,H\sb 2 )$ (Where $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are strictly convex normed spaces and $H\sb 1$ , $H\sb 2$ are real inner spaces).