Cybernetics and Systems Analysis, Vol. 38, No. 3, 2002 ON THE DIEUDONNÉ THEOREM IN REFLEXIVE BANACH SPACES 1 S. Adly, a E. Ernst, b and M. Théra c UDC 519.6 The converse part of the Dieudonné theorem on the closedness of the difference of two closed convex sets is establish. It is proved that, in a reflexive Banach setting, the sufficient conditions given by Dieudonné when applied for the weak topology, are also necessary. Keywords: Dieudonné theorem, reflexive Banach space, weak topology. 1. INTRODUCTION AND NOTATION Dieudonné [5] proved the following famous result: the algebraic difference C C 1 2 - of two closed convex subsets C 1 and C 2 of a Hausdorff topological vector space is closed provided the recession condition C C 1 2 0 ∞ ∞ ∩ = { } is satisfied and at least one of the sets C 1 and C 2 is locally compact. These conditions are obviously not necessary. Just put C C K 1 2 =- = , where K is a closed but not a locally compact convex set, and note that C C K 1 2 2 - = is closed, although the second condition of the Dieudonné Theorem fails to be satisfied. This note concerns with the broadest condition which should be added to the recession condition C C 1 2 0 ∞ ∞ ∩ = { } to ensure the closedness of the algebraic difference of closed convex sets in reflexive Banach spaces. Namely, we want to determine the class of those closed convex sets C 1 containing no lines for which the algebraic difference C C 1 2 - is closed for every closed convex set C 2 fulfilling the recession condition C C 1 2 0 ∞ ∞ ∩ = { }. This class is characterized (Theorem 1) by a geometrical condition: the set C 1 belongs to the above class if and only if it is well-positioned in the sense defined by the authors in [2]. As in a reflexive Banach space, a closed convex set is well-positioned if and only if it is weakly locally compact (Proposition 1), the strongest Dieudonné-type Theorem in a reflexive Banach setting (in the sense developed before) reads as follows: The algebraic difference of two closed convex sets containing no lines is closed when the two following conditions are fulfilled: a) C C 1 2 0 ∞ ∞ ∩ = { }, and b) at least one of the sets C C 1 2 , is weakly locally compact. Since their proofs are rather technical (the reader is refered to a forthcoming paper of the authors [3]), we will admit Lemma 2 and Theorem 3. They are unpublished results establishing some geometrical characterizations of well-positioned sets. 339 1060-0396/02/3803-0339$27.00 © 2002 Plenum Publishing Corporation a LACO, Université de Limoges, 123, avenue A. Thomas, 87060 Limoges Cedex, France, adly@unilim.fr. b Laboratoire de modélisation en mécanique et thermodynamique (LMMT), Casse 322, faculté de sciences et téchniques de Saint Jérôme, avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20, emil.ernst@univ.u-3mrs.fr. c LACO, Université de Limoges, 123, avenue A. Thomas, 87060 Limoges Cedex, France, michel.thera@unilim.fr. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 34-39, May-June, 2002. Original article submitted May 16, 2002. 1 Research of Michel Théra has been partially supported by the French Chilean Scientific Cooperation Program ECOS under grant C00E05 and by NATO-CLG 978488.