Abstract

We study the extreme points of the unit ball of a Banach space that remain extreme when considered, under canonical embedding, in the unit ball of the bidual. We give an example of a strictly convex space whose unit vectors are extreme points in the unit ball of the second dual but none are extreme points in the unit ball of the fourth dual. For the space of vector-valued continuous functions on a compact set we show that any function whose values are weak∗^*-extreme points is a weak∗^*-extreme point. We explore the relation between weak∗^*-extreme points and the dual notion of very smooth points. We show that if a Banach space X has a very smooth point in every equivalent norm then X∗X^* has the Radon-Nikodym property.

Author information

Contact details are reproduced from the original publication and may be historical.

S. Dutta

Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

sudipta_r@isical.ac.in

T. S. S. R. K. Rao

Stat-Math Unit, Indian Statistical Institute, R. V. College Post, Bangalore 560059, India

tss@isibang.ac.in

Keywords

  • Higher duals
  • M-ideals
  • extreme points

Mathematics Subject Classification

46B20

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Suggested citation

S. Dutta, T. S. S. R. K. Rao. “On Weak*-Extreme Points in Banach Spaces.” Journal of Convex Analysis 10 (2003), No. 2, 531–539. https://doi.org/10.68381/jca1033

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.