Isometric Functional Equation

An isometry is a distance-preserving map between metric spaces. For normed spaces E1 and E2, a function $$f:\ E_1 \rightarrow E_2$$ is called an isometry if f satisfies the isometric functional equation $$\| f(x)-f(y)\| = \|x-y\|\ {\rm for\ all}\ x,y \varepsilon E_1$$ . The historical background for Hyers–Ulam stability of isometries will be introduced in Section 13.1. The Hyers–Ulam–Rassias stability of isometries on a restricted domain will be surveyed in Section 13.2. Section 13.3 will be devoted to the fixed point method for studying the stability problem of isometries. In the final section, the Hyers–Ulam–Rassias stability of Wigner equation $$|\langle f(x), f(y)\rangle|= |\langle x, y \rangle|$$ on a restricted domain will be discussed.