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The compact weak topology on a Banach space

Published online by Cambridge University Press:  14 November 2011

Manuel González  and
Joaquín M. Gutiérrez
Manuel González
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain
Joaquín M. Gutiérrez
Affiliation:
Departamento de Matemática Aplicada, ETS de Ingenieros Industrials, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal, 2, 28006 Madrid, Spain
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Synopsis

The compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 3-4 , 1992 , pp. 367 - 379
Copyright
Copyright © Royal Society of Edinburgh 1992

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