A General Scheme for Separably Reducible Properties

Marián Fabian; Alexander D. Ioffe; Julian P. Revalski

doi:10.68381/jca33048

Abstract

We propose a general scheme for studying separably reducible properties in metric spaces and then apply it to obtain separable determinacy of Lipschitz property and the separable determinacy of slopes.

Author information

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

Marián Fabian

Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

fabian@math.cas.cz

Alexander D. Ioffe

Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

alexander.ioffe38@gmail.com

Julian P. Revalski

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

revalski@math.bas.bg

Keywords

Separable reduction
cofinal family
rich family
Lipschitz property
slopes

Mathematics Subject Classification

46B26, 49J52; 26E15

References

J. Borwein, W. Moors: Separable determinantion of integrability and minimality of the Clark subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000) 215–221.https://doi.org/10.1090/S0002-9939-99-05001-7 Google Scholar
M. Cúth, M. Fabian: Asplund spaces characterized by rich families and separable reduction of Fréchet subdifferentiability, J. Funct. Analysis 270/4 (2016) 1361–1378.https://doi.org/10.1016/j.jfa.2015.12.005 Google Scholar
M. Fabian: Separable reductions and rich families in the theory of Fréchet subdifferentials, Lecture Notes, Spring School on Variational Analysis, Paseky nad Jizerou, Czech Republic, April 19–25 (2015).Google Scholar
M. Fabian, A. Ioffe: Separable reduction in the theory of Fréchet subdifferentials, Set-Valued Var. Analysis 21 (2013) 661–671.https://doi.org/10.1007/s11228-013-0256-1 Google Scholar
M. Fabian, A. Ioffe: Separable reductions and rich families in theory of Fréchet subdifferentials, J. Convex Analysis 23 (2016) 631–648.https://doi.org/10.68381/jca23024 Google Scholar
M. Fabian, A. Ioffe, J. P. Revalski: Separable reduction of local metric regularity, Proc. Amer. Math. Soc. 146 (2018) 5157–5167.https://doi.org/10.1090/proc/14071 Google Scholar
J. Lindenstrauss, D. Preiss, J. Tišer: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Ann. Math. Studies No. 179, Princeton University Press, Princeton (2012).https://doi.org/10.23943/princeton/9780691153551.001.0001 Google Scholar
D. Preiss: Gâteaux differentiable functions are somewhere Fréchet differentiable, Rend. Circ. Math. Palermo 33 (1984) 122–133.https://doi.org/10.1007/BF02844417 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.

[1] J. Borwein, W. Moors: Separable determinantion of integrability and minimality of the Clark subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000) 215–221.https://doi.org/10.1090/S0002-9939-99-05001-7 Google Scholar

[2] M. Cúth, M. Fabian: Asplund spaces characterized by rich families and separable reduction of Fréchet subdifferentiability, J. Funct. Analysis 270/4 (2016) 1361–1378.https://doi.org/10.1016/j.jfa.2015.12.005 Google Scholar

[3] M. Fabian: Separable reductions and rich families in the theory of Fréchet subdifferentials, Lecture Notes, Spring School on Variational Analysis, Paseky nad Jizerou, Czech Republic, April 19–25 (2015).Google Scholar

[4] M. Fabian, A. Ioffe: Separable reduction in the theory of Fréchet subdifferentials, Set-Valued Var. Analysis 21 (2013) 661–671.https://doi.org/10.1007/s11228-013-0256-1 Google Scholar

[5] M. Fabian, A. Ioffe: Separable reductions and rich families in theory of Fréchet subdifferentials, J. Convex Analysis 23 (2016) 631–648.https://doi.org/10.68381/jca23024 Google Scholar

[6] M. Fabian, A. Ioffe, J. P. Revalski: Separable reduction of local metric regularity, Proc. Amer. Math. Soc. 146 (2018) 5157–5167.https://doi.org/10.1090/proc/14071 Google Scholar

[7] J. Lindenstrauss, D. Preiss, J. Tišer: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Ann. Math. Studies No. 179, Princeton University Press, Princeton (2012).https://doi.org/10.23943/princeton/9780691153551.001.0001 Google Scholar

[8] D. Preiss: Gâteaux differentiable functions are somewhere Fréchet differentiable, Rend. Circ. Math. Palermo 33 (1984) 122–133.https://doi.org/10.1007/BF02844417 Google Scholar