Stability of Banach spaces via nonlinear \(\varepsilon\)-isometries

In this paper, we prove that the existence of an \(\varepsilon\)-isometry from a separable Banach space \(X\) into \(Y\) (the James space or a reflexive space) implies the existence of a linear isometry from \(X\) into \(Y\). Then we present a set valued mapping version lemma on non-surjective \(\va...

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Bibliographic Details
Published in:arXiv.org 2014-01
Main Authors: Dai, Duanxu, Dong, Yunbai
Format: Article
Language:English
Subjects:
Banach spaces
Error analysis
Mapping
Perturbation
Smoothness
Online Access:Get full text
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Summary:In this paper, we prove that the existence of an \(\varepsilon\)-isometry from a separable Banach space \(X\) into \(Y\) (the James space or a reflexive space) implies the existence of a linear isometry from \(X\) into \(Y\). Then we present a set valued mapping version lemma on non-surjective \(\varepsilon\)-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by \(\varepsilon\)-isometries.
ISSN:2331-8422
DOI:10.48550/arxiv.1301.3396