×

zbMATH — the first resource for mathematics

On smooth, nonlinear surjections of Banach spaces. (English) Zbl 0898.46044
Summary: It is shown that
(1) every infinite-dimensional Banach space admits a \(C^1\) Lipschitz map onto any separable Banach space, and
(2) if the dual of a separable Banach space \(X\) contains a normalized, weakly null Banach-Saks sequence, then \(X\) admits a \(C^\infty\) map onto any separable Banach space.
Subsequently, we generalize these results to mappings onto larger target spaces.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. M. Bates,A smooth surjective rank-1 endomorphism of a Hilbert space, International Mathematics Research Notices6 (1991), 71–73. · Zbl 0752.58002 · doi:10.1155/S1073792891000107
[2] S. M. Bates,On the image size of singular maps II., Duke Mathematical Journal68 (1992), 463–476. · Zbl 0801.58005 · doi:10.1215/S0012-7094-92-06818-9
[3] S. M. Bates,Lipschitz mappings of Banach-Carnot groups, preprint, 1995.
[4] S. M. Bates and C. C. Pugh,Super singular surjections, Pacific Journal of Mathematics, to appear.
[5] B. Beauzamy and J.-T. Lapresté,Modèles étalés des espaces de Banach, Travaux en Cours, Hermann, Paris, 1984. · Zbl 0553.46012
[6] P. Casazza and T. Shura,Tsirelson’s space, Lecture Notes in Mathematics1363, Springer-Verlag, Berlin, 1989.
[7] R. Deville, G. Godefroy and V. Zizler,Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific, 1993. · Zbl 0782.46019
[8] V. I. Gurarii and N. I. Gurarii,On bases in uniformly convex and uniformly smooth Banach spaces, Izvestiya Akademii Nauk SSSR35 (1971), 210–215.
[9] R. C. James,Superreflexive spaces with bases, Pacific Journal of Mathematics41 (1972), 409–419. · Zbl 0218.46011
[10] M. I. Kadec,On topological equivalence of separable Banach spaces, Doklady Akademii Nauk SSSR162 (1965), 23–25; English translation: Soviet Mathematics Doklady7 (1966) 319–322.
[11] J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Springer-Verlag, Berlin, 1977. · Zbl 0362.46013
[12] H. Rosenthal,Weakly independent sequences and the Banach-Saks property, The Bulletin of the London Mathematical Society8 (1976), 22–24.
[13] S. Sternberg,Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964. · Zbl 0129.13102
[14] H. Torunczyk,Characterizing Hilbert space topology, Fundamenta Mathematicae111 (1981), 247–262. · Zbl 0468.57015
[15] Y. Yomdin,Surjective mappings whose differential is nowhere surjective, Proceedings of the American Mathematical Society111 (1991), 267–270. · Zbl 0727.58005 · doi:10.1090/S0002-9939-1991-1039267-2
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.