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- W2065795637 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." @default.
- W2065795637 created "2016-06-24" @default.
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- W2065795637 date "2008-01-01" @default.
- W2065795637 modified "2023-09-26" @default.
- W2065795637 title "Stability of isometries in $p$-Banach spaces" @default.
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- W2065795637 doi "https://doi.org/10.7169/facm/1229624655" @default.
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