Abstract

We show how our previous paper "Weak compactness of sublevel sets in complete locally convex spaces" [J. Convex Analysis 26/3 (2019) 739–751] allows to derive with quite direct arguments a large number of recent results established in Banach spaces for connexions between subdifferential and weak compactness of sublevel sets. In fact, the arguments extend the results to locally convex spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Pedro Pérez-Aros

Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Rancagua, Chile

pedro.perez@uoh.cl

Lionel Thibault

Institut Montpelliérain A. Grothendieck, Université de Montpellier, France
and: Centro de Modelamiento Matematico, Universidad de Chile, Santiago, Chile

lionel.thibault@umontpellier.fr

Keywords

  • Convex function
  • conjugate function
  • inf-convolution
  • infcompact function
  • weak compactness
  • subdifferential
  • sublevel

Mathematics Subject Classification

46A25, 46A04, 46A50

References

  1. F. Delbaen, J. Orihuela: Mackey constraints for James’ compactness theorem and risk measures, J. Math. Analysis Appl. 485 (2020), art. no. 123764.https://doi.org/10.1016/j.jmaa.2019.123764Google Scholar
  2. F. Delbaen, J. Orihuela: On the range of the subdifferential in non reflexive Banach spaces, J. Funct. Analysis 281 (2021), art. no. 108915.https://doi.org/10.1016/j.jfa.2020.108915Google Scholar
  3. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-7515-7Google Scholar
  4. D. J. Farrell, W. B. Moors: An application of the generalised James’ weak compactness theorem, J. Convex Analysis 28 (2021) 795–802.https://doi.org/10.68381/jca28045Google Scholar
  5. W. B. Moors: An application of the generalised James’ weak compactness theorem part II, J. Convex Analysis 29/4 (2022) 1161–1166.https://doi.org/10.68381/jca29064Google Scholar
  6. W. B. Moors, N. O. Tan: An abstract variational theorem, J. Convex Analysis 26/4 (2019) 1125–1144.https://doi.org/10.68381/jca26062Google Scholar
  7. J. J. Moreau: Fonctionnelles Convexes, Lecture Notes, Collège de France (1966–1967); 2nd ed. by Facoltà di Ingegneria, Università di Roma ”Tor Vergata” (2003).Google Scholar
  8. P. Pérez-Aros, L. Thibault: Weak compactness of sublevel sets in complete locally convex spaces, J. Convex Analysis 26/3 (2019) 739–751.https://doi.org/10.68381/jca26039Google Scholar
  9. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, River Edge (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

P. Pérez-Aros, L. Thibault. “More on Connexions Between Subdifferential and Weak Compactness of Sublevels.” Journal of Convex Analysis 30 (2023), No. 1, 317–328. https://doi.org/10.68381/jca30017

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press.