Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-31T20:19:04.283Z Has data issue: false hasContentIssue false
  • English
  • Français

A factor theorem for locally convex differentiability spaces

Published online by Cambridge University Press:  17 April 2009

Roger Eyland  and
Bernice Sharp
Roger Eyland
Affiliation:
Department of Pure Mathematics, University of Sydney, New South Wales 2006, Australia
Bernice Sharp
Affiliation:
Australian Catholic University, 40 Edward Street North Sydney NSW 2060, Australia
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.

The main result of this paper is that a continuous convex function with domain in a locally convex space factors through a normed space. In a recent paper by Sharp, topological linear spaces are categorised according to the differentiability properties of their continuous convex functions; we show that under suitable conditions the classification is preserved by linear maps. A technique for deducing results for locally convex spaces from Banach space theory is an immediate consequence. Examples are given and Asplund C(S) spaces are characterised.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 101 - 113
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Asplund, E., ‘Fréchet Differentiability of Convex Functions’, Acta Math 121 (1968), 31–47.CrossRefGoogle Scholar
[2]Borwein, J.M., ‘Continuity and Differentiability Properties of Convex Operators’, Proc. London Math. Soc. 3 (1982), 420–444.CrossRefGoogle Scholar
[3]Čoban, M. and Kenderov, P., ‘Generic Gateaux Differentiability of Convex Functional in C(T) and the Topological Properties of T’, in Mathematics and Education in Mathematics, 1986: Proceedings of the Fifteenth Spring Conference of the Union of Bulgarian Mathematicians, pp. 141–149, 1986.Google Scholar
[4]Dugundji, J., Topology (Allyn and Bacon, Boston, 1970).Google Scholar
[5]Giles, J.R., Convex Analysis with Application in Differentiation of Convex Functions: Research Notes in Mathematics 58 (Pitman, 1982).Google Scholar
[6]Gillman, L. and Jerison, M., Rings of Continuous Functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[7]Kelley, J.L., General Topology: Graduate Texts in Mathematics 27 (Springer-Verlag, Berlin, Heidelberg, New York, 1955).Google Scholar
[8]Kelley, J.L. and Namioka, I., Linear Topological Spaces (Van Nostrand, Princeton, N.J., 1963).CrossRefGoogle Scholar
[9]Larman, D.G. and Phelps, R., ‘Gateaux Differentiabilty of Convex Functions on Banach Spaces’, London Math. Soc (2) 20 (1979), 115–127.CrossRefGoogle Scholar
[10]Namioka, and Phelps, R., ‘Banach Spaces which are Asplund Spaces’, Duke Math. J. 42 (1975), 735–750.CrossRefGoogle Scholar
[11]Phelps, R., Convex Functions, Monotone Operators and Differentiability: Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1989).CrossRefGoogle Scholar
[12]Robertson, A.P. and Robertson, W., Topological Vector Spaces (Cambridge University Press, 1966).Google Scholar
[13]Sharp, B., Ph.D. Thesis (University of Newcastle, Australia, 1989).Google Scholar
[14]Sharp, B., ‘The Differentiability of Convex Functions on Topological Linear Spaces’, Bull. Austral. Math. Soc. 42 (1990), 201–213.CrossRefGoogle Scholar
[15]Wilansky, A., Functional Analysis (Blaisdell, 1964).Google Scholar