Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-16T15:31:14.630Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Asimow, L., ‘Directed Banach spaces of affine functions’, Tans. Amer. Math. Soc. 143 (1969), 117–132.CrossRefGoogle Scholar
[2] Becker, R., ‘Sur les cônes situés dans un espace de Banach et qui ne portent que des mesures coniques localisables’: Séminaire d'Initiation à l'Analyse, pp. 16–26. (1984–1985).Google Scholar
[3] Becker, R., ‘Sur lea cônes situés dans un espace de Banach et qui ne portent que des mesures coniques localisables’, C.R.A.S. Paris Math. 301 Sér I (1985), 361–362.Google Scholar
[4] Becker, R., ‘Mesures coniques sur un espace de Banach ou son dual’, J. Austral. Math. Soc. 39 Ser. A (1985), 39–50.CrossRefGoogle Scholar
[5] Becker, R., ‘Sur les mesures coniques localisables’, J. Austral. Math. Soc. 33 Ser. A (1982), 394–400.CrossRefGoogle Scholar
[6] Choquet, G., Lectures on Analysis, 1–3, Mathematics (Lecture Notes series, Benjamin, New York, Amsterdam, 1969).Google Scholar
[7] Diestel, J., Geometry of Banach spaces: Lecture Notes in Mathematics 485 (Springer Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[8] Goullet De Rugy, A., ‘Sur les cônes engendrés par une famille de convexes compacts’, Bull. Sc. Math. (2)97 (1973), 242–251.Google Scholar
[9] Hagler, J. and Johnson, W.B., ‘On Banach spaces whose dual Balls are not weak* sequentially compact’, Israel J. Math. 28 (1977), 325–330.CrossRefGoogle Scholar
[10] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, vol 1: Ergebnisse der Mathematik und ihrer Grenzgebeite 92 (Springer - Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[11] Rogalski, M., Espaces de Banach réticulés et problème de Dirichlet (Publication mathématiques d'Orsay, 1972).Google Scholar
[12] Schaffer, H.H., Topological vector spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1966).Google Scholar
[13] Thomas, R., ‘Integral representations in conuclear spaces’: Vector spaces measures and applications II Proceedings, Dublin 1977 (Lecture Notes in Mathematics 645, Springer Verlag, Berlin, Heidelberg, New York). pp. 172–179.Google Scholar