Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-02T20:38:48.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Some more characterizations of Banach spaces containing l1

Published online by Cambridge University Press:  24 October 2008

Richard Haydon
Richard Haydon
Affiliation:
Brasenose College, Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:

a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 2 , September 1976 , pp. 269 - 276
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alfsen, E. M.Compact convex sets and boundary integral (Berlin, Springer, 1971).CrossRefGoogle Scholar
(2)Bishop, E. and Phelps, R. R.The support functionals of a convex set. Proc. Symp. Pure Maths. (Convexity) A.M.S. 7 (1963), 27–35.CrossRefGoogle Scholar
(3)Choquet, G.Lectures in Analysis, vol II (New York, Benjamin, 1969).Google Scholar
(4)Day, M. M.Normed linear spaces (Berlin, Springer, 1958).CrossRefGoogle Scholar
(5)Ellis, A. J. (ed.). Facial structure of compact convex sets (Report of NATO Advanced Study Institute, University College of Swansea, 1972).Google Scholar
(6)Huff, R. E. and Morris, P.Dual spaces with the Krein–Mil'man property have the Radon–Nikodym property. Proc. Amer. Math. Soc. (to appear).Google Scholar
(7)James, R. C. A conjective about l 1 subspace (to appear).Google Scholar
(8)Lindenstrauss, J. and Stegall, C.Examples of separable spaces which do not contain l 1 and whose duals are non-separable. Studia Math. (to appear).Google Scholar
(9)Odell, E. and Rosenthal, H. P. A double-dual characterization of separable Bach spaces containing l 1 (to appear).Google Scholar
(10)Rosenthal, H. P.A characterization of Banach spaces containing l 1. Proc. Nat. Acad. Sci. 71 (1974), 2411–2413.CrossRefGoogle Scholar
(11)Rosenthal, H. P. Point-wise compact subset of the first Baire class (to appear).Google Scholar
(12)Schwartz, L.Radon measures on arbitrary topological spaces (Oxford University Press, for Tata Institute, Bombay).Google Scholar
(13)Semadeni, Z. Bach spaces of continuous functions Monografie Matematyczne 55 (PWN, Warsaw, 1971).Google Scholar
(14)Stegall, C.The Radon–Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc. 206 (1975), 213–223.CrossRefGoogle Scholar