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Inventiones mathematicae

, Volume 182, Issue 3, pp 585–604 | Cite as

Boundaries for Banach spaces determine weak compactness

  • Hermann PfitznerEmail author
Article

Abstract

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.

The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.

Mathematics Subject Classification (2000)

46B03 46B26 46B50 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Université d’OrléansOrléans Cedex 2France

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