# ISOMETRIES WITH SMALL BOUND ON $C^1$(X) SPACES

• Jun, Kil-Woung (Department of Mathematics, Chungnam National University, Taejon 305-764) ;
• Lee, Yang-Hi (Department of Mathematics Education, Kong Ju National Teachers College, Kong Ju 314-060)
• Published : 1995.02.01

#### Abstract

For a locally compact Hausdorff space, we denote by $C_0(X)$ the Banach space of all continuous complex valued functions defined on X which vanish at infinity, equipped with the usual sup norm. In case X is compact, we write C(X) instead of $C_0(X)$. A well-known Banach-Stone theorem states that the existence of an isometry between the function spaces $C_0(X)$ and $C_0(Y)$ implies X and Y are homemorphic. D. Amir [1] and M. Cambern [2] independently generalized this theorem by proving that if $C_0(X)$ and $C_0(Y)$ are isomorphic under an isomorphism T satisfying $\left\$\mid$T \right\$\mid$\left\$\mid$T^1 \right\$\mid$< 2$, then X and Y must also be homeomorphic.