Stability of Banach spaces via nonlinear $\varepsilon$-isometries

TL;DR

This paper investigates how approximate isometries between Banach spaces, especially involving the James space and reflexive spaces, imply the existence of exact linear isometries, and explores geometric properties under such perturbations.

Contribution

It establishes that $ ext{ extbackslash epsilon}$-isometries from certain Banach spaces guarantee true isometries and introduces a set-valued mapping lemma for non-surjective cases.

Findings

$ ext{ extbackslash epsilon}$-isometries imply linear isometries in specific Banach spaces

Set valued mapping lemma for non-surjective $ ext{ extbackslash epsilon}$-isometries

Perturbation by $ ext{ extbackslash epsilon}$-isometries affects rotundity and smoothness

Abstract

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.