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

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## 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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## Additional Information

**Andreas H. Hamel**- Affiliation: Department of Mathematics and Computer Sciences, Martin-Luther-University Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06099 Halle, Germany
- Email: hamel@mathematik.uni-halle.de
- Received by editor(s): May 17, 2001
- Published electronically: April 30, 2003
- Communicated by: Jonathan M. Borwein
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 3025-3038 - MSC (2000): Primary 49J40, 46A03
- DOI: https://doi.org/10.1090/S0002-9939-03-07066-7
- MathSciNet review: 1993209