Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T21:38:14.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Baire trees, bad norms and the Namioka property

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

Richard Haydon
Richard Haydon
Affiliation:
Brasenose College, Oxford OX1 4AJ, England.
Get access

Extract

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 30 - 42
Copyright
Copyright © University College London 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Davis, W. J., Figiel, T., Johnson, W. B. and Pelczynski, A.. Factoring weakly compact operators. J. Fund. Anal., 17 (1974), 311–327.CrossRefGoogle Scholar
2.Deville, R.. Convergence ponctuelle et uniforme sur un espace compact. Bull. Acad. Polon., 30 (1989), 507–515.Google Scholar
3.Deville, R. and Godefroy, G.. Some applications of projectional resolutions of unity. Proc. London Math. Soc, 67 (1993), 183–199.CrossRefGoogle Scholar
4.Deville, R., Godefroy, G. and Zizler, V.. Smoothness and renormings in Banach spaces (Longman, Harlow, 1993).Google Scholar
5.Haydon, R. G.. A counterexample to several questions about scattered compact spaces. Bull. London Math. Soc, 22 (1990), 261–268.CrossRefGoogle Scholar
6.Haydon, R. G.. Some problems about scattered spaces. Séminaire Initiation à Analyse (1989/1990), No. 9, 10pp.Google Scholar
7.Haydon, R. G.. Trees in renorming theory (in preparation).Google Scholar
8.Namioka, I. and Pol, R.. Mappings of Baire spaces into function spaces and Kadec renormings. Israel J. Math., 78 (1992), 1–20.CrossRefGoogle Scholar
9.Talagrand, M.. Espaces de Baire et espaces de Namioka. Math. Ann., 270 (1985), 159–164.CrossRefGoogle Scholar
10.Todorcevic, S.. Trees and linearly ordered sets. Handbook of set-theoretic topology, ed. Kunen, K. and Vaughan, J. E. (North-Holland, 1984).Google Scholar