Abstract

In a recent paper [Separable reduction in the theory of Fréchet subdifferentials, Set-Valued Var. Anal. 21(4) (2013) 661–-671] we presented the separable reduction for a general statement covering practically all important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; all statements being considered in the Fréchet sense. As in earlier studies of various separable reduction techniques, this was done with the help of suitable cofinal families of separable subspaces. In this paper we show that such reductions can be done with the help of a subclass of cofinal families known as rich families, recently articulated (and used) by J. M. Borwein and W. B. Moors [Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000) 215–221] and by J. Lindenstrauss, D. Preiss, and J. Tiser [Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces, Ann. Math. Studies 179, Princeton University Press 2012]. The most advantageous feature of rich families is that the intersection of even countably many of them is again a rich family. This means that in case we need separable reduction of a combination of properties and know that each of them is reducible by elements of a certain rich family, then all we need to do is to take the intersection of rich families associated with the properties instead of devising a new (and typically fairly complicated) proof.

Author information

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

Marián Fabian

Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

fabian@math.cas.cz

Keywords

  • Separable reduction
  • cofinal family
  • rich family
  • Fréchet subdifferential
  • Fréchet normal cone
  • fuzzy calculus
  • extremal principle

References

  1. J. M. Borwein, W. B. Moors: Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000) 215–221.https://doi.org/10.1090/S0002-9939-99-05001-7Google Scholar
  2. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Longman, Harlow (1993).Google Scholar
  3. R. Engelking: General Topology, PWN, Warszawa (1977).Google Scholar
  4. M. Fabian: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carol., Math. Phys. 30 (1989) 51–56.Google Scholar
  5. M. Fabian, A. Ioffe: Separable reduction in the theory of Fréchet subdifferentials, Set-Valued Var. Anal. 21(4) (2013) 661–671.https://doi.org/10.1007/s11228-013-0256-1Google Scholar
  6. M. Fabian, N. V. Zhivkov: A characterization of Asplund spaces with help of local ε-supports of Ekeland and Lebourg, C. R. Acad. Bulg. Sci. 38 (1985) 671–674.Google Scholar
  7. A. D. Ioffe: On subdifferentiability spaces, Ann. New York Acad. Sci. 410 (1983) 107–121.https://doi.org/10.1111/j.1749-6632.1983.tb23308.xGoogle Scholar
  8. A. D. Ioffe: Metric regularity and subdifferential calculus, Usp. Mat. Nauk 55(3) (2000) 103–162 (in Russian); Russ. Math. Surv. 55(3) (2000) 501–558 (in English).https://doi.org/10.4213/rm292https://doi.org/10.1070/RM2000v055n03ABEH000292Google Scholar
  9. A. D. Ioffe: Separable reduction revisited, Optimization 60(1–2) (2011) 211–221.https://doi.org/10.1080/02331934.2010.526714Google Scholar
  10. A. D. Ioffe: On the general theory of subdifferentials, Adv. Nonlinear Anal. 1 (2012) 47–120.https://doi.org/10.1515/ana-2011-0003Google Scholar
  11. A. D. Ioffe: Convexity and variational analysis, in: Computational and Analytical Mathematics, D. Bailey et al. (ed.), Springer, New York (2013) 411–444.https://doi.org/10.1007/978-1-4614-7621-4_19Google Scholar
  12. J. Lindenstrauss, D. Preiss, J. Tišer: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Ann. Math. Stud. 179, Princeton University Press, Princeton (2012).https://doi.org/10.23943/princeton/9780691153551.001.0001Google Scholar
  13. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation, Vol. 1, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  14. J. P. Penot: A short proof of the separable reduction theorem, Demonstr. Math. 4 (2010) 653–663.https://doi.org/10.1515/dema-2010-0313Google Scholar
  15. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd Ed., Lect. Notes Math. 1364, Springer, Berlin (1993).Google Scholar
  16. D. Preiss: Gâteaux differentiable functions are somewhere Fréchet differentiable, Rend. Circ. Mat. Palermo, II. Ser. 33 (1984) 122–133.https://doi.org/10.1007/BF02844417Google Scholar
  17. L. Zajíček: Generic Fréchet differentiability on Asplund spaces via a.e. strict differentiability on many lines, J. Convex Analysis 19 (2012) 23–48.https://doi.org/10.68381/jca19002Google Scholar

Suggested citation

M. Fabian, A. Ioffe. “Separable Reductions and Rich Families in the Theory of Fréchet Subdifferentials.” Journal of Convex Analysis 23 (2016), No. 3, 631–648. https://doi.org/10.68381/jca23024

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.