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Angelicity and the boundary problem

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  26 February 2010

B. Cascales  and
G. Godefroy
B. Cascales
Affiliation:
Departmento of Mathemáticas, Universidad de Murcia, Campus de Espinardo, 30.100 Espinardo, Murcia, Spain, [email protected]
G. Godefroy
Affiliation:
Equipe d'Analyse, Université Pierre et Marie Curie. Paris VI, Tour 46-0, Boite 186. 4, place Jussie, 75252 Paris Cedex 05, France, [email protected]
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Abstract

Let K be an arbitrary compact space and C(K) the space of continuous functions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K)* on which every function of C(K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology tp(B) of pointwise convergence on B. It is also shown that this result can be extended to a large class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), tp(B)) is an angelic space in the sense of D. H. Fremlin.

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Mathematika , Volume 45 , Issue 1 , June 1998 , pp. 105 - 112
Copyright
Copyright © University College London 1998

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