Stability of isometries in $p$-Banach spaces

Abstract

It is known that the isometry equation is stable in Banach spaces. In this paper we investigate stability of isometries in real $p$-Banach spaces, that is Fréchet spaces with $p$-homogenous norms, where $p \in (0,1]$. Let $X,Y$ be $p$-Banach spaces and let $f:X \to Y$ be an {\it $\varepsilon$-isometry}, that is a function such that $|||f(x)-f(y)||-||x-y|| |\leq \varepsilon$ for all $x,y \in X$. We show that if $f$ is a surjective then there exists an affine surjective isometry $U: X \to Y$ and a constant $C_p$ such that $$||f(x)-U(x)||\leq C_p (\varepsilon+\varepsilon^p ||x||^{(1-p)}) for x \in X.$$ We also show that in general the above estimation cannot be improved.