Phelps’ lemma, Danes̆’ drop theorem and Ekeland’s principle in locally convex spaces

Hamel, Andreas

doi:10.1090/S0002-9939-03-07066-7

Phelps’ lemma, Danes̆’ drop theorem and Ekeland’s principle in locally convex spaces
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Proc. Amer. Math. Soc. 131 (2003), 3025-3038 Request permission

Abstract:

A generalization of Phelps’ lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland’s principle and Danes̆’ drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland’s principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

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