Hostname: page-component-7dd5485656-k8lnt Total loading time: 0 Render date: 2025-10-30T12:08:13.311Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological convexity spaces

Published online by Cambridge University Press:  20 January 2009

Victor Bryant
Victor Bryant
Affiliation:
Department of Pure Mathematics, The University of Sheffield
Rights & Permissions [Opens in a new window]

Extract

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.

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 2 , September 1974 , pp. 125 - 132
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

(1) Bryant, V. W. and Webster, R. J., Generalizations of the theorems of Radon, Helly and Caratheodory, Monatsh. Math. 73 (1969), 309–315.CrossRefGoogle Scholar
(2) Bryant, V. W. and Webster, R. J., Convexity Spaces I: The basic properties, J. Math. Anal. Appl. 37 (1972), 206–215.CrossRefGoogle Scholar
(3) Bryant, V. W. and Webster, R. J., Convexity Spaces II: Separation, J. Math. Anal. Appl. 43 (1973), 321–327.CrossRefGoogle Scholar
(4) Klee, V., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443–466.Google Scholar
(5) Prenowitz, W., A contemporary approach to classical geometry, Amer. Math. Monthly 68 (1961), 1–67.CrossRefGoogle Scholar
(6) Schaefer, H. H., Topological Vector Spaces (Macmillan, New York, 1966).Google Scholar
(7) Stone, M. A.,, Convexity (Mimeographed Lecture notes, Chicago, 1946).Google Scholar
(8) Valentine, F., Convex Sets (McGraw-Hill, New York, 1964).Google Scholar