Quadrivariate existence theorems and strong representability

Abstract

In this article, we give conditions under which we can compute the conjugate of a convex function on the product of two Fréchet spaces defined in terms of another convex function on the product of two (possibly different) Fréchet spaces. We use this result to give simple proofs of some (both old and new) results for Banach spaces, and deduce some (both old and new) stability results for strongly representable multifunctions. We take as our starting point a result on closed convex cones in the product of two Fréchet spaces.

Keywords:

• Fenchel conjugate
• Fréchet space
• Banach space
• monotone multifunction
• Baire's theorem

AMS Subject Classifications:

• 47H05
• 47N10
• 46A04

Acknowledgements

The author would like to thank Benar Svaiter for sending him a preprint of 5, Maicon Marques Alves for sending him a preprint of 7, and Constantin Zălinescu for making him aware of 3, and sending him a preprint of 16. Part of the research in this article was done while the author was visiting Centre de Recerca Matemàtica in Barcelona, Spain. The author would like to thank the Centre for its hospitality.