Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

Duke Mathematical Journal

Volumn 124, Issue 1, 2004, Pages 47-66

  Azagra, Daniel ^a   Boiso, Manuel Cepedello ^b  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4043094897     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-04-12412-1     Document Type: Article

References (26)

1. D. AZAGRA and T. DOBROWOLSKI, Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications, Math. Ann. 312 (1998), 445-463. MR 1654820
2. D. AZAGRA and M. JIMÉNEZ-SEVILLA, The failure of Rolle's theorem in infinite-dimensional Banach spaces, J. Funct. Anal. 182 (2001), 207-226. MR 1829247
3. D. AZAGRA and A. MONTESINOS, Deleting diffeomorphisms with prescribed supports in Banach spaces, preprint, 2003, http://www.mat.ucm.es/~dazagrar/ research.htm
4. S. M. BATES, On the image size of singular maps, I, Proc. Amer. Math. Soc. 114 (1992), 699-705. MR 1074748
5. _, On the image size of singular maps, II, Duke Math. J. 68 (1992), 463-476. MR 1194950
6. _, On smooth rank-1 mappings of Banach spaces onto the plane, J. Differential Geom. 37 (1993), 729-733. MR 1217168
7. _, Toward a precise smoothness hypothesis in Sard's theorem, Proc. Amer. Math. Soc. 117 (1993), 279-283. MR 1112486
8. _, On smooth nonlinear surjections of Banach spaces, Israel J. Math. 100 (1997), 209-220. MR 1469111
9. S. M. BATES and C. G. MOREIRA, De nouvelles perspectives sur le théorème de Morse-Sard, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 13-17. MR 1805620
10. C. BESSAGA, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 27-31. MR 0193646
11. C. BESSAGA and A. PEŁCZYŃSKI, Selected Topics in Infinite-Dimensional Topology, Monogr. Mat. 58, PWN, Warsaw, 1975. MR 0478168
12. H. CARTAN, Calcul différentiel, Hermann, Paris 1967. MR 0223194
13. R. DEVILLE, G. GODEFROY, and V. ZIZLER, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman Sci. Tech., Harlow, England, 1993. MR 1211634
14. T. DOBROWOLSKI, Smooth and R-analytic negligibility of subsets and extension of homeomorphisms in Banach spaces, Studia Math. 65 (1979), 115-139. MR 0557487
15. J. EELLS and K. D. ELWORTHY, Open embeddings of certain Banach manifolds, Ann. of Math. (2) 91 (1970), 465-485. MR 0263120
16. J. EELLS and J. MCALPIN, An approximate Morse-Sard theorem, J. Math. Mech. 17 (1967/1968), 1055-1064. MR 0226680
17. I. KUPKA, Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds, Proc. Amer. Math. Soc. 16 (1965), 954-957. MR 0182024
18. C. G. T. DE ARAVJO MOREIRA, Hausdorff measures and the Morse-Sard theorem, Publ. Mat. 45 (2001), 149-162. MR 1829581
19. A. P. MORSE, The behavior of a function on its critical set, Ann. of Math. (2) 40 (1939), 62-70. MR 1503449
20. N. MOULIS, Sur les variétés Hilbertiennes et les fonctions non dégénérées. Indag. Math. 30 (1968), 497-511. MR 0254876
21. A. SARD, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883-890. MR 0007523
22. _, Images of critical sets, Ann. of Math. (2) 68 (1958), 247-259. MR 0100063
23. _, Hausdorff measure of critical images on Banach manifolds, Amer. J. Math. 87 (1965), 158-174. MR 0173748
24. S. SMALE, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866. MR 0185604
25. J. E. WEST, The diffeomorphic excision of closed local compacta from infinite-dimensional Hilbert manifolds, Compositio Math. 21 (1969), 271-291. MR 0256420
26. H. WHITNEY, A function not constant on a connected set of critical points. Duke Math. J. 1 (1935), 514-517.