A note on the Hyers–Ulam problem
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.