Isometries between normed spaces which are surjective on a sphere

Abstract

In this paper, we study the extension of isometric operator between unit spheres of normed spaces, and give an equivalent statement of Tingley problem. We also give another statement of Mazur–Ulam theorem: Let $V : E\rightarrow F$ be an isometric operator, and $V|_{S(E)}$ denotes the operator $V$ restricted to the set $S(E)$. If $V|_{S(E)}$ is an onto isometric operator from $S(E)$ to $S(F)$, then $V$ must be linear.