, Volume 16, Issue 3, pp 543–563

New results on q-positivity

  • Y. García Ramos
  • J. E. Martínez-Legaz
  • S. Simons

DOI: 10.1007/s11117-012-0191-7

Cite this article as:
García Ramos, Y., Martínez-Legaz, J.E. & Simons, S. Positivity (2012) 16: 543. doi:10.1007/s11117-012-0191-7


In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.


q-Positive setsSymmetrically self-dual spacesMonotonicitySymmetrically self-dual Banach spacesLipschitz mappings

Mathematics Subject Classification


Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Y. García Ramos
    • 1
  • J. E. Martínez-Legaz
    • 2
  • S. Simons
    • 3
  1. 1.Centro de Investigación de la Universidad del PacíficoLimaPeru
  2. 2.Universitat Autònoma de BarcelonaBarcelonaSpain
  3. 3.University of CaliforniaSanta BarbaraUSA