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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-2-1,
author = {J. H. Qiu and S. Rolewicz},
title = {Local completeness of locally pseudoconvex spaces and Borwein-Preiss variational principle},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {99-115},
zbl = {1139.46003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-2-1}
}
J. H. Qiu; S. Rolewicz. Local completeness of locally pseudoconvex spaces and Borwein-Preiss variational principle. Studia Mathematica, Tome 178 (2007) pp. 99-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-2-1/