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Journal of Convex Analysis 23 (2016), No. 3, 631--648
Copyright Heldermann Verlag 2016

Separable Reductions and Rich Families in the Theory of Fréchet Subdifferentials

Marián Fabian
Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic

Alexander Ioffe
Department of Mathematics, Technion, Haifa 32000, Israel

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.

Keywords: Separable reduction, cofinal family, rich family, Fréchet subdifferential, Fréchet normal cone, fuzzy calculus, extremal principle.

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