Separable reductions and rich families in the theory of Fréchet subdifferentials (Q2821967)

For a Banach space \(X\) denote by \(\mathcal{S}(X)\) the family of all closed separable subspaces of \(X\). A subset \(\mathcal{R}(X)\) of \(\mathcal{S}(X)\) is called a rich family of separable subspaces ifNEWLINENEWLINE(i)\; \(\mathcal{R}(X)\) is cofinal in \(\mathcal{S}(X)\), that is, for every \(Z\in \mathcal{S}(X)\) there exists \(Y\in \mathcal{R}(X)\) such that \(Z\subset Y\); (ii)\; if \(Y_1\subset Y_2\subset \dots\) are in \(\mathcal{R}(X)\), then cl\(\left(\cup_{k=1}^\infty Y_k\right)\in \mathcal{R}(X)\).NEWLINENEWLINEThe notion can be defined in a more abstract framework, namely as a cofinal subset \(R\) of a partially ordered set \((T,\preccurlyeq)\), such that that every increasing sequence in \(R\) has a supremum in \(T\) belonging to \(R\).NEWLINENEWLINEThis notion, first considered by \textit{J. M. Borwein} and \textit{W. B. Moors} in [Proc. Am. Math. Soc. 128, No. 1, 215--221 (2000; Zbl 0937.49009)], turned out to be a useful tool in proving separable reduction properties. Roughly speaking, separable reduction means that a property concerning a Banach space \(X\) holds whenever it holds for (sufficiently large) separable subspaces of \(X\), allowing to take advantage of the good topological and geometric properties of separable Banach spaces. The strength of this notion comes from the fact that the intersection of a countable family of rich families in a Banach space is again a rich family. This means that in case one needs separable reduction for a combination of properties, one can prove that each of them is reducible by elements of a certain rich family and then take the intersection of these families, instead of devising a different (usually fairly complicated) proof for each case.NEWLINENEWLINEIn a previous paper, [Set-Valued Var. Anal. 21, No. 4, 661--671 (2013; Zbl 1284.49016)], the authors proved a general result having as consequences a lot of previously known results on separable reduction for Fréchet subdifferentials. The aim of the present paper is to show that these results can be proved via rich families of separable subspaces in a simpler and unitary way.NEWLINENEWLINE The topic is further pursued in a paper by \textit{M. Cúth} and \textit{M. Fabian} [J. Funct. Anal. 270, No. 4, 1361--1378 (2016; Zbl 1352.46019)].