Abstract

We provide a short proof of following theorem, due to Delbaen and Orihuela and independently, Pérez-Aros and Thibault. Let AA be a nonempty closed and bounded convex subset of a Banach space (X,∥⋅∥)(X,\|\cdot\|) and let WW be a nonempty weakly compact subset of (X,∥⋅∥)(X, \|\cdot\|). If we have x0∗∈{x∗∈X∗:sup⁡a∈Ax∗(a)<0}   and   argmax(y∗∣A)≠∅x_0^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0\}\ \ \ \text{and}\ \ \ \mathrm{argmax}(y^*|_A) \not= \varnothing for each y∗∈{x∗∈X∗:sup⁡a∈Ax∗(a)<0y^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0 and sup⁡w∈W∣(x∗−x0∗)(w)∣<1}\sup_{w \in W} |(x^*-x^*_0)(w)|<1\}, then AA is weakly compact.

Author information

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

David J. Farrell

Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand

Warren B. Moors

Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand

w.moors@auckland.ac.nz

Keywords

  • Weak compactness
  • James' weak compactness theorem

Mathematics Subject Classification

46B20; 46B26, 49A50, 49A51

References

  1. A. Brøndsted, R. T. Rockafellar: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8Google Scholar
  2. B. Cascales, J. Orihuela, A. Pérez: One-sided James’ compactness theorem, J. Math. Analysis Appl. 445(2) (2017) 1267–1283.https://doi.org/10.1016/j.jmaa.2016.03.080Google Scholar
  3. F. Delbaen, J. Orihuela: Mackey constraints for James’s compactness theorem and risk measures, J. Math. Analysis Appl. 485(1) (2020) 123764, 15 pp.https://doi.org/10.1016/j.jmaa.2019.123764Google Scholar
  4. N. Dunford, J. T. Schwartz: Linear Operators. I: General Theory, Interscience Publishers, New York (1958).Google Scholar
  5. R. C. James: Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964) 129–140.https://doi.org/10.1090/S0002-9947-1964-0165344-2Google Scholar
  6. W. B. Moors: On a one-sided James’ theorem, J. Math. Analysis Appl. 449 (2017) 528–530.https://doi.org/10.1016/j.jmaa.2016.12.019Google Scholar
  7. W. B. Moors: Weak compactness of sublevel sets, Proc. Amer. Math. Soc. 145 (2017) 3377–3379.https://doi.org/10.1090/proc/13466Google Scholar
  8. W. B. Moors: A gentle introduction to James’ weak compactness theorem and beyond, Methods Functional Analysis Topology 25(1) (2019) 35–83.Google Scholar
  9. 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
  10. J. Orihuela: Conic James’ compactness theorem, J. Convex Analysis 25(4) (2018) 1335–1344.https://doi.org/10.68381/jca25081Google Scholar
  11. J. Orihuela, M. J. Ruiz Galán: A coercive James’s weak compactness theorem and nonlinear variational problems, Nonlinear Analysis 75(2) (2012) 598–611.https://doi.org/10.1016/j.na.2011.08.062Google Scholar
  12. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd ed., Lecture Notes in Mathematics 1364, Springer, Berlin (1993).Google Scholar
  13. 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
  14. J. Saint-Raymond: Weak compactness and variational characterization of the convexity, Mediterranean J. Math. 10(2) (2013) 927–940.https://doi.org/10.1007/s00009-012-0226-0Google Scholar

Suggested citation

D. J. Farrell, W. B. Moors. “An Application of the Generalised James' Weak Compactness Theorem.” Journal of Convex Analysis 28 (2021), No. 3, 795–802. https://doi.org/10.68381/jca28045

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.