# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bessaga, C., Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. acad. polon. sci. Sér. sci. math., 14, 27-31, (1966) · Zbl 0151.17703 [2] Bessaga, C.; Pelczynski, A., Selected topics in infinite-dimensional topology, Monografie matematyczne, (1975), PWN Warsaw · Zbl 0304.57001 [3] Deville, R.; Godefroy, G.; Zizler, V., Smoothness and renormings in Banach spaces, Pitman monographs and surveys in pure and applied mathematics, (1993), Longman Harlow · Zbl 0782.46019 [4] Dobrowolski, T., Smooth andR, Studia math., 65, 115-139, (1979) · Zbl 0421.46012 [5] Ekeland, I., Nonconvex minimization problems, Bull. amer. math. soc. (N.S.), 1, 443-474, (1979) · Zbl 0441.49011 [6] Fabian, M.; Hájek, P.; Vanderwerff, J., On smooth variational principles in Banach spaces, J. math. anal. appl., 197, 153-172, (1996) · Zbl 0868.49012 [7] J. Bés, J. Ferrera, private communication [8] Ferrer, J., Rolle’s theorem fails in l_2, Amer. math. monthly, 103, 161-165, (1996) · Zbl 0888.46017 [9] Phelps, R.R., Convex functions, monotone operators and differentiability, Lecture notes in mathematics, (1993), Springer-Verlag Berlin/New York · Zbl 0921.46039 [10] Shkarin, S.A., On Rolle’s theorem in infinite-dimensional Banach spaces, Mat. zametki, 51, 128-136, (1992) · Zbl 0786.46044
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.