zbMATH — the first resource for mathematics

On Rolle’s theorem in infinite-dimensional Banach spaces. (English. Russian original) Zbl 0786.46044
Math. Notes 51, No. 3, 311-317 (1992); translation from Mat. Zametki 51, No. 3, 128-136 (1992).
Summary: This note is devoted basically to the construction of a smooth function \(f\) on a real, infinite-dimensional Banach space \(E\) with a smooth norm, such that
1) \(f(x)=0\) for \(\| x\|\geq 1\);
2) \(f'(x)\neq 0\) for \(\| x\|<1\).
We also construct a polynomial \(p\) of degree four on a real, separable, infinite-dimensional Hilbert space, such that
1) \(p(x)=0\) for \(\| x\|=1\);
2) \(p'(x)\neq 0\) for \(\| x\|<1\).
All the results are proved in a constructive manner. These examples are answers to some questions raised by S. B. Stechkin and O. G. Smolyanov.

46G05 Derivatives of functions in infinite-dimensional spaces
Full Text: DOI
[1] O. G. Smolyanov, Analysis on Linear Topological Spaces and Its Applications [in Russian], Moscow State Univ. (1979).
[2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I. Sequence Spaces, Springer, Berlin (1977). · Zbl 0362.46013
[3] J. Diestel, Geometry of Banach Spaces ?Selected Topics, Lecture Notes in Math., No. 485, Springer, Berlin (1975).
[4] E. T. Shavgulidze, ?On a certain diffeomorphism of a locally convex space,? Usp. Mat. Nauk,34, No. 5, 231-232 (1979).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.