Hostname: page-component-cc8bf7c57-n7pht Total loading time: 0 Render date: 2024-12-11T23:02:21.403Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on bump functions that locally depend on finitely many coordinates

Published online by Cambridge University Press:  17 April 2009

M. Fabian  and
V. Zizler
M. Fabian
Affiliation:
Mathematical InstituteCzech Academy of SciencesŽitná 25, 11567 Prague 1Czech Republic, e-mail: [email protected]
V. Zizler
Affiliation:
Department of Mathematical AnalysisFaculty of Mathematics and PhysicsCharles UniversitySokolovská 83, 18600 Prague 8Czech Republic and Department of Mathematics, University of Alberta, Edmonton, T6G 2G1 Alberta, Canada, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 447 - 451
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (Longman, Harlow, 1993).Google Scholar
[2]Habala, P., Hájek, P. and Zizler, V., Introduction to Banach spaces I, II, (Lecture Notes) (Matfyzpress, Charles University, Prague, 1996).Google Scholar
[3]Hájek, P., ‘Smooth norms that depend locally on finitely many coordinates’, Proc. Amer. Math. Soc. 123 (1995), 3817–3821.CrossRefGoogle Scholar
[4]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I, Sequence spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[5]Pechanec, J., Whitfield, J.H.M. and Zizler, V., ‘Norms locally dependent on finitely many coordinates’, An. Acad. Brasil. Ciênc. 53 (1981), 415–417.Google Scholar