Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T06:33:31.221Z Has data issue: false hasContentIssue false
  • English
  • Français

The differentiability of convex functions on topological linear spaces

Published online by Cambridge University Press:  17 April 2009

Bernice Sharp
Bernice Sharp
Affiliation:
Australian Catholic University, 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.

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 2 , October 1990 , pp. 201 - 213
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Asplund, E., ‘Fréchet differentiability of convex functions’, Acta Math 121 (1968), 31–47.CrossRefGoogle Scholar
[2]Avrebukh, V.I. and Smulyanov, O.G., ‘The theory of differentiation in linear topological spaces’, Russian Math. Surveys 22 6 (1967), 201–258.CrossRefGoogle Scholar
[3]Averbukh, V.I. and Smulyanov, O.G., ‘The various definitions of the derivative in linear topological spaces’, Russian Math. Surveys 23 4 (1968), 67–113.CrossRefGoogle Scholar
[4]Borwein, J.M., ‘Continuity and differentiability properties of convex operators’, Proc. London Math. Soc. 3 (1982), 420–444.CrossRefGoogle Scholar
[5]Borwein, J.M., ‘Generic differentiability of order-bounded convex operators’, J. Austral. Math. Soc. Ser. B 28 (1986), 22–29.CrossRefGoogle Scholar
[6]Čoban, M. and Kenderov, P., ‘Dense Gateaux differentiability of the sup-norm in C(T) and the topological properties of T’, C.R.Acad. Bulgare Sci. 38 (1985), 1603–1604.Google Scholar
[7]Giles, J.R., Convex analysis with application in differentiation of convex functions: Research Notes in Mathematics 58 (Pitman, 1982).Google Scholar
[8]Kelley, J.L. and Namioka, I., Linear topological spaces (Van Nostrand, Princeton, N.J., 1963).CrossRefGoogle Scholar
[9]Köthe, G., Topological vector spaces 1 (Springer-Verlag, Berlin, Heidelberg, New York, 1969).Google Scholar
[10]Larman, D.G. and Phelps, R., ‘Gateaux differentiabilty of convex functions on Banach spaces’, J. London Math. Soc (2) 20 (1979), 115–127.CrossRefGoogle Scholar
[11]Mazur, S., ‘Über konvexe Mengen in linearen normierten Räumen’, Studia Math 4 (1933), 70–84.CrossRefGoogle Scholar
[12]Namioka, and Phelps, R., ‘Banach spaces which are Asplund spaces’, Duke Math. J. 42 (1975), 735–750.CrossRefGoogle Scholar
[13]Phelps, R., Convex Functions, Monotone Operators and Differentiability: Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1989).CrossRefGoogle Scholar
[14]Robertson, A.P. and Robertson, W., Topological Vector Spaces (Cambridge University Press, Cambridge, 1966).Google Scholar
[15]Sharp, B., Ph.D. Thesis (University of Newcastle, NSW Australia, 1989).Google Scholar
[16]Yamamuro, S., Differential calculus in topological linear spaces: Lecture Notes in Mathematics 374 (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar