Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T21:56:18.104Z Has data issue: false hasContentIssue false
  • English
  • Français

Convex series

Published online by Cambridge University Press:  24 October 2008

G. J. O. Jameson
G. J. O. Jameson
Affiliation:
University of Warwick, Coventry, England
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:

(i) CS-closed if it contains the sum of every convergent convex series of its elements;

(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 37 - 47
Copyright
Copyright © Cambridge Philosophical Society 1972

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)Banach, S.Théorie des Opérations Linéaires (Warsaw, 1932).Google Scholar
(2)Ellis, A. J.The duality of partially ordered normed linear spaces. J. London Math. Soc. 39 (1964), 730–744.CrossRefGoogle Scholar
(3)Ganijng, D. J. H.The β– and γ–duality of sequence spaces. Proc. Cambridge Philos. Soc. 63 (1967), 963–981.Google Scholar
(4)Jameson, G. J. O.Ordered linear 8paces. Springer Lecture Notes no. 141 (Berlin, 1970).CrossRefGoogle Scholar
(5)Klee, V. L.Convex sets in linear spaces II. Duke Math. J. 18 (1951), 875–883.Google Scholar
(6)Köthe, G.Topologische lineare Räume I (Berlin, 1960).CrossRefGoogle Scholar
(7)Tukey, J. W.Some notes on the separation of convex sets. Portugal. Math. 3 (1942), 95–102.Google Scholar