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A new minimax theorem and a perturbed James's theorem

Published online by Cambridge University Press:  17 April 2009

M. Ruiz Galán  and
S. Simons
M. Ruiz Galán
Affiliation:
Departamento de Matemática Aplicada, E. U. Arquitectura Técnica, Universidad de Granada, 18071 Granada, Spain
S. Simons
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106–3080, United States of America
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Abstract

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The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y, where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of James's (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstine's theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 1 , August 2002 , pp. 43 - 56
Copyright
Copyright © Australian Mathematical Society 2002

References

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