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

Tom 183 / 2007

J. H. Qiu, S. Rolewicz Studia Mathematica 183 (2007), 99-115 MSC: 46A55, 49J45. DOI: 10.4064/sm183-2-1


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.


  • J. H. QiuDepartment of Mathematics
    Suzhou University
    Suzhou, Jiangsu 215006
    People's Republic of China
  • S. RolewiczInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21, /Sniadeckich 8
    00-956 Warszawa, Poland

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