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Sur les cones (faiblement complets) contenus dans le dual d'un espace de Banach non-reflexif

Published online by Cambridge University Press:  17 April 2009

Richard Becker
Richard Becker
Affiliation:
Equipe d'Analyse, Universite Paris VI, Tour 46 4ème Etage, 4, Place Jussieu, 75252 - Paris Cedex 05, France
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Abstract

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Let B be a Banach space. consider the convex proper weakly complete cones X contained in B′ with σ(B′, B) such that X ∩ B′, is conic in the sense of Asimow: that is, there exists α ≥ 0 and f ∈ B″ such that ‖ ‖B ≤ f ≤ α·‖ ‖B on X. This class arises in the theory of integral representations.

If B is reflexive, such a cone has a weakly-compact basis. This paper considers the converse problem:- if one requires that X ∩ B′1 be σ(B′, B) metrisable, the existence of X (without a compact σ(B′, B) basis) is equivalent to the statement that B is not a Grothendieck space.

However, in every space C(K) with infinitely compact K, one can find such a cone X. If two such cones in B′ are not too far apart, their sum belongs to this class.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 3 , June 1989 , pp. 321 - 328
Copyright
Copyright © Australian Mathematical Society 1989

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