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Journal of Convex Analysis 10 (2003), No. 2, 531--539
Copyright Heldermann Verlag 2003
On Weak*-Extreme Points in Banach Spaces
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
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* has the
Radon-Nikodym property.
Keywords: Higher duals, M-ideals, extreme points.
MSC 2000: 46B20.
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