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Rolle’s theorem and negligibility of points in infinite dimensional Banach spaces. (English) Zbl 0897.46027
The authors study some aspects of Rolle’s theorem in infinite-dimensional Banach spaces. They first give a kind of approximate Rolle’s theorem which holds in all Banach spaces and is interesting in itself. Further, within the class of all Banach spaces \(X\) with the property \((*)\): \(X\) admits a continuous norm whose dual norm is locally uniformly rotund, they give an affirmative answer to the question whether for every Banach space \(X\) having a \(C^1\) bump function there exists a \(C^1\) diffeomorphism \(\varphi: X\to X\setminus\{0\}\) such that \(\varphi\) is the identity out of a ball. Finally, within the class of Banach spaces with the property \((*)\), they give a characterization of those spaces in which (exact) Rolle’s theorem fails, making use of the preceding result.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
46B20 Geometry and structure of normed linear spaces
46B04 Isometric theory of Banach spaces
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