Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 1, 2003

Smooth approximations without critical points

Petr Hájek and Michal Johanis
From the journal Open Mathematics

Abstract

In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.

Keywords: 46B20; 46G05

[1] D. Azagra and M. Cepedello Boiso: “Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds”, preprint, http://arxiv.org/archive/math. Search in Google Scholar

[2] D. Azagra and R. Deville: “James’ theorem fails for starlike bodies”, J. Funct. Anal., Vol. 180, (2001), pp. 328–346. http://dx.doi.org/10.1006/jfan.2000.369610.1006/jfan.2000.3696Search in Google Scholar

[3] D. Azagra, R. Deville, M. Jiménez-Sevilla: “On the range of the derivatives of a smooth mapping between Banach spaces”, to appear in Proc. Cambridge Phil. Soc. Search in Google Scholar

[4] D. Azagra, M. Fabian, M. Jiménez-Sevilla: “Exact filling in figures by the derivatives of smooth mappings between Banach spaces”, preprint. Search in Google Scholar

[5] D. Azagra and M. Jiménez-Sevilla: “The failure of Rolle’s Theorem in infinite dimensional Banach spaces”, J. Funct. Anal., Vol. 182, (2001), pp. 207–226. http://dx.doi.org/10.1006/jfan.2000.370910.1006/jfan.2000.3709Search in Google Scholar

[6] D. Azagra and M. Jiménez-Sevilla: “Geometrical and topological properties of starlike bodies and bumps in Banach spaces”, to appear in Extracta Math. Search in Google Scholar

[7] J.M. Borwein, M. Fabian, I. Kortezov, P.D. Loewen: “The range of the gradient of a continuously differentiable bump”, J. Nonlinear and Convex Anal., Vol. 2, (2001), pp. 1–19. Search in Google Scholar

[8] J.M. Borwein, M. Fabian, P.D. Loewen: “The range of the gradient of a Lipschiz C 1-smooth bump in infinite dimensions”, to appear in Israel Journal of Mathematics. Search in Google Scholar

[9] R. Deville, G. Godefroy, V. Zizler: “Smoothness and renormings in Banach spaces”, Monographs and Surveys in Pure and Applied Mathematics 64, Pitman, 1993. Search in Google Scholar

[10] M. Fabian, O. Kalenda, J. Kolář: “Filling analytic sets by the derivatives of C 1-smooth bumps”, to appear in Proc. Amer. Math. Soc. Search in Google Scholar

[11] M. Fabian, J.H.M. Whitfield, V. Zizler: “Norms with locally Lipschizian derivatives”, Israel Journal of Mathematics, Vol. 44, (1983), pp. 262–276. Search in Google Scholar

[12] T. Gaspari: “On the range of the derivative of a real valued function with bounded support”, preprint. Search in Google Scholar

[13] P. Hájek: “Smooth functions on c 0”, Israel Journal of Mathematics, Vol. 104, (1998), pp. 17–27. Search in Google Scholar

[14] A. Sobczyk: “Projection of the space m on its subspace c 0”, Bull. Amer. Math. Soc., Vol. 47, (1941), pp. 938–947. http://dx.doi.org/10.1090/S0002-9904-1941-07593-210.1090/S0002-9904-1941-07593-2Search in Google Scholar

[15] C. Stegall: “Optimization of functions on certain subsets of Banach spaces”, Math. Ann., Vol. 236, (1978), pp. 171–176. http://dx.doi.org/10.1007/BF0135138910.1007/BF01351389Search in Google Scholar

Published Online: 2003-9-1
Published in Print: 2003-9-1

© 2003 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Scroll Up Arrow