Stability of Banach spaces via nonlinear \(\varepsilon\)-isometries. (English) Zbl 1325.46010
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.
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
Keywords:
\(\varepsilon\)-isometry; stability; Banach space; rotundity; smoothness; set-valued mappingReferences:
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