Local completeness of locally pseudoconvex spaces and Borwein–Preiss variational principle

J. H. Qiu¹ ; S. Rolewicz²

¹Department of Mathematics Suzhou University Suzhou, Jiangsu 215006 People's Republic of China
²Institute of Mathematics Polish Academy of Sciences P.O. Box 21, /Sniadeckich 8 00-956 Warszawa, Poland

Studia Mathematica, Tome 183 (2007) no. 2, pp. 99-115

Résumé

The notion of local completeness is extended to locally pseudoconvex spaces. Then a general version of the Borwein–Preiss variational principle in locally complete locally pseudoconvex spaces is given, where the perturbation is an infinite sum involving differentiable real-valued functions and subadditive functionals. From this, some particular versions of the Borwein–Preiss variational principle are derived. In particular, a version with respect to the Minkowski gauge of a bounded closed convex set in a locally convex space is presented. In locally convex spaces it can be shown that the relevant perturbation only consists of a single summand if and only if the bounded closed convex set has the quasi-weak drop property if and only if it is weakly compact. From this, a new description of reflexive locally convex spaces is obtained.

Détail

DOI : 10.4064/sm183-2-1

Keywords: notion local completeness extended locally pseudoconvex spaces general version borwein preiss variational principle locally complete locally pseudoconvex spaces given where perturbation infinite sum involving differentiable real valued functions subadditive functionals particular versions borwein preiss variational principle derived particular version respect minkowski gauge bounded closed convex set locally convex space presented locally convex spaces shown relevant perturbation only consists single summand only bounded closed convex set has quasi weak drop property only weakly compact description reflexive locally convex spaces obtained