A note on the Hyers–Ulam problem

Yunbai Dong¹

¹ School of Mathematics and Computer Science Wuhan Textile University Wuhan 430073, China

Colloquium Mathematicum, Tome 138 (2015) no. 2, pp. 233-239

Résumé

Let $X,Y$ be real Banach spaces and $\varepsilon >0$. Suppose that $f:X\rightarrow Y$ is a surjective map satisfying $|\|f(x)-f(y)\| -\| x-y\| |\leq \varepsilon $ for all $x,y\in X$. Hyers and Ulam asked whether there exists an isometry $U$ and a constant $K$ such that $\| f(x)-Ux\| \leq K\varepsilon $ for all $x\in X$. It is well-known that the answer to the Hyers–Ulam problem is positive and $K=2$ is the best possible solution with assumption $f(0)=U0=0$. In this paper, using the idea of Figiel's theorem on nonsurjective isometries, we give a new proof of this result.

Détail

DOI : 10.4064/cm138-2-7

Keywords: real banach spaces varepsilon suppose rightarrow surjective map satisfying f x y leq varepsilon hyers ulam asked whether there exists isometry constant ux leq varepsilon well known answer hyers ulam problem positive best possible solution assumption paper using idea figiels theorem nonsurjective isometries proof result