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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
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Abstract

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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

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