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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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References

1.Bourgain, J., Fremlin, D. and Talagrand, M.. Pointwise compact sets of Baire measurable functions. Amer. J. Math., 100 (1978), 845–886.CrossRefGoogle Scholar
2.Bourgain, J. and Talagrand, M.. Compacité extremale. Proc. Amer. Math. Soc, 80 (1980), 68–70.CrossRefGoogle Scholar
3.Cascales, B., Manjabacas, G. and Vera, G.. A Krein-Smulian type result in Banach spaces. Oxford Quarterly Journal Math. (2), 48 (1997), 161–167.Google Scholar
4.Cascales, B. and Vera, G.. Topologies weaker than the weak topology of a Banach space. J. Math. Anal. Appl., 182 (1994), 41–68.CrossRefGoogle Scholar
5.Deville, R., Godefroy, G. and Zizler, V.. Smoothness and renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64 (1993).Google Scholar
6.Floret, K.. Weakly compact sets. LNM Vol. 801 (Springer-Verlag, 1980).CrossRefGoogle Scholar
7.Gamelin, T.. Uniform algebras (Prentice-Hall, 1969).Google Scholar
8.Godefroy, G.. Some applications of Simons' inequality. Murcia. Functional Analysis Seminar II.Google Scholar
9.Godefroy, G.. Boundaries of a convex set and interpolation sets. Math. Ann., 277 (1987), 173–184.CrossRefGoogle Scholar
10.Godefroy, G.. Five lectures in geometry of Banach spaces. In Seminar on Functional Analysis, 1987, chapter 1, pages 9–67. Universidad de Murcia, 1988.Google Scholar
11.Grothendieck, A.. Criteres de compacite dans les espaces fontionneles generaux. Amer. Math., 74 (1952), 168–186.Google Scholar
12.Haydon, R.. Compactness in Cs(T) and applications. Pub. Dep. Math. (Lyon), 9 (1972), 105–113.Google Scholar
13.James, R. C.. Weakly compact sets. Trans. Amer. Math. Soc, 113 (1964), 120–140.CrossRefGoogle Scholar
14.Kelley, J. L. and Namioka, I.. Linear Topological Spaces. (Springer, Berlin, 1963).Google Scholar
15.Larsen, R.. Banach algebras, an introduction. Pure and Applied Math. Vol. 24 (Marcel Decker Inc. New York, 1973).Google Scholar
16.Rodé, G.. Superkonvexitat and schwache Kompaktheit. Arch. Math., 36 (1981), 62–72.Google Scholar
17.Simons, S.. A convergence theorem with boundary. Pacific J. Math., 40 (1972), 703–708.Google Scholar
18.Stegall, C.. Applications of descriptive topology in functional analysis (University of Linz, 1985).Google Scholar
19.Watson, S.. A compact Hausdorff space without -points in which Gδ-sets have interior. Proc. Amer. Math. Soc, 123 (1995), 2575–2577.Google Scholar