Abstract
We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bourgain,On dentability and the Bishop-Phelps property, Isr. J. Math.28 (1977), 265–271.
M. M. Day,Normed Linear Spaces, 3rd ed., Springer, 1972.
J. Diestel,Geometry of Banach Spaces, Springer Lecture Notes485, 1975.
R. Engelking,An Outline of General Topology, North-Holland Publ., Amsterdam, 1968.
J. Lindenstrauss,On operators which attain their norm, Isr. J. Math.1 (1963), 139–148.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Springer Lecture Notes in Math.338, 1977.
J. R. Partington,Norm attaining operators, Isr. J. Math.43 (1982), 273–276.
Z. Semadeni,Banach Spaces of Continuous Functions, Vol. 1, Polish Scientific Publishers, Warsaw, 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schachermayer, W. Norm attaining operators and renormings of Banach spaces. Israel J. Math. 44, 201–212 (1983). https://doi.org/10.1007/BF02760971
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02760971