Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces

Volume 55 / 2007

P. Holický, O. F. K. Kalenda, L. Veselý, L. Zajíček Bulletin Polish Acad. Sci. Math. 55 (2007), 211-217 MSC: 46B10, 46B03. DOI: 10.4064/ba55-3-3

Abstract

On each nonreflexive Banach space $X$ there exists a positive continuous convex function $f$ such that $1/f$ is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that $X$ is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.

Authors

  • P. HolickýFaculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75 Praha 8,
    Czech Republic
    e-mail
  • O. F. K. KalendaFaculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75, Praha 8
    Czech Republic
    e-mail
  • L. VeselýDipartimento di Matematica “F. Enriques”
    Università degli Studi di Milano
    Via C. Saldini 50
    20133 Milano, Italy
    e-mail
  • L. ZajíčekFaculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75, Praha 8
    Czech Republic
    e-mail

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