Functiones et Approximatio Commentarii Mathematici

Stability of isometries in $p$-Banach spaces

Jacek Tabor, Józef Tabor, and Marek Żołdak

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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.

Article information

Funct. Approx. Comment. Math., Volume 38, Number 1 (2008), 109-119.

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]
Secondary: 39B82: Stability, separation, extension, and related topics [See also 46A22]

$p$-homogeneous Fréchet space approximate isometry Hyers-Ulam stability


Tabor, Jacek; Tabor, Józef; Żołdak, Marek. Stability of isometries in $p$-Banach spaces. Funct. Approx. Comment. Math. 38 (2008), no. 1, 109--119. doi:10.7169/facm/1229624655.

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