Smooth approximations without critical points

• Autorzy: Petr Hájek , Michal Johanis
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

derivatives; approximation; critical points; 46B20; 46G05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2003
• Tom: 1
• Numer: 3
• Strony: 284-291

Daty

wydano

2003-09-01

online

2003-09-01

Twórcy

autor

Petr Hájek

• Czech Academy of Science

autor

Michal Johanis

• Charles University

Bibliografia

• [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.
• [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.2000.3696
• [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.
• [4] D. Azagra, M. Fabian, M. Jiménez-Sevilla: “Exact filling in figures by the derivatives of smooth mappings between Banach spaces”, preprint.
• [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.2000.3709
• [6] D. Azagra and M. Jiménez-Sevilla: “Geometrical and topological properties of starlike bodies and bumps in Banach spaces”, to appear in Extracta Math.
• [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.
• [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.
• [9] R. Deville, G. Godefroy, V. Zizler: “Smoothness and renormings in Banach spaces”, Monographs and Surveys in Pure and Applied Mathematics 64, Pitman, 1993.
• [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.
• [11] M. Fabian, J.H.M. Whitfield, V. Zizler: “Norms with locally Lipschizian derivatives”, Israel Journal of Mathematics, Vol. 44, (1983), pp. 262–276.
• [12] T. Gaspari: “On the range of the derivative of a real valued function with bounded support”, preprint.
• [13] P. Hájek: “Smooth functions on c 0”, Israel Journal of Mathematics, Vol. 104, (1998), pp. 17–27.
• [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/S0002-9904-1941-07593-2
• [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/BF01351389

Identyfikatory

DOI

10.2478/BF02475210