GENERALIZED STABILITY OF ISOMETRIES ON REAL BANACH SPACES

Title & Authors
GENERALIZED STABILITY OF ISOMETRIES ON REAL BANACH SPACES
Lee, Eun-Hwi; Park, Dal-Won;

Abstract
Let X and Y be real Banach spaces and ${\varepsilon}\;>\;0$, p > 1. Let f : $\small{X\;{\to}\;Y}$ be a bijective mapping with f(0) = 0 satisfying $\small{|\;{\parallel}f(x)-f(y){\parallel}-{\parallel}{x}-y{\parallel}\;|\;{\leq}{\varepsilon}{\parallel}{x}-y{\parallel}^p}$ for all $\small{x\;{\in}\;X}$ and, let $\small{f^{-1}\;:\;Y\;{\to}\;X}$ be uniformly continuous. Then there exist a constant ${\delta}\;>\;0$ and N($\small{{\varepsilon},p}$) such that lim N($\small{{\varepsilon},p}$)=0 and a unique surjective isometry I : X $\small{{\to}}$ Y satisfying $\small{{\parallel}f(x)-I(x){\parallel}{\leq}N({\varepsilon,p}){\parallel}x{\parallel}^p}$ for all $\small{x\;{\in}\;X\;with\;{\parallel}x{\parallel}{\leq}{\delta}}$.
Keywords
(${\varepsilon}$,p)-isometry;isometry;real Banach spaces;
Language
English
Cited by
References
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