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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.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
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References:
[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).
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