Radon–Nikodým compact spaces of low weight and Banach spaces

Abstract

Abstract. We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.The concept of Radon-Nikodým compact, due to Reynov [13], has its origin in Banach space theory, and it is defined as a topological space which is homeomorphic to a weak * compact subset of the dual of an Asplund space, that is, a dual Banach space with the Radon-Nikodým property (topological spaces will be here assumed to be Hausdorff). In [10], the following characterization of this class is given: compact space K is Radon-Nikodým compact if and only if there is a lower semicontinuous metric d on K which fragments K.Recall that a map f : X × X → R on a topological space X is said to fragment X if for each (closed) subset L of X and each ε > 0 there is a nonempty relatively open subset U of L of f -diameter less than ε, i.e. sup{f (x, y) : x, y ∈ U } < ε. Also, a map g : Y → R from a topological space to the real line is lower semicontinuous if {y : g(y) ≤ r} is closed in Y for every real number r.It is an open problem whether a continuous image of a Radon-Nikodým compact is Radon-Nikodým. Arvanitakis [2] has taken the following approach to this problem: if K is a Radon-Nikodým compact and π : K → L is a continuous surjection, then we have a lower semicontinuous fragmenting 2000 Mathematics Subject Classification: Primary 46B26; Secondary 46B22, 46B50, 54G99.

Locally uniformly convex norms in Banach spaces and their duals

“…Every continuous image of a Radon-Nikodým compact is strongly fragmented. There are other definitions [1,7] which have recently been shown to be equivalent to strong fragmentability [2,14] and Arvanitakis's terminology in which strongly fragmented compacta are called quasi-Radon-Nikodým seems to have become standard. It is not known whether every quasi-Radon-Nikodým compact is Radon-Nikodým; a positive answer would of course settle the problem of continuous images.…”

“…(Notice that our hypotheses do ensure that the LUR hypothesis holds for Ω(·, K, l).) If (L, s) is a node of our tree then there are three possibilities: (1) if the oscillation of f on L is smaller than then (L, s) has no immediate successors in the tree (that is to say, (L, s) is a maximal element); (2) if the oscillation of f on L is at least and there is a good choice (M, N ) of type τ (s) on L then we introduce into Υ two immediate successors, (L ∩ X(M, N ), s + 1) and (L ∩ X(M, N ), s + 1), of (L, s) (notice that, by Proposition 6.3, the LUR hypothesis holds for the Ω functions associated with these two new nodes); (3) if the oscillation of f on L is at least but no good choice of type τ (s) exists, then we introduce just one immediate successor (L, s + 1) of (L, s) into the tree.…”

Weak compactness and σ-Asplund generated Banach spaces

“…Here, (i) was proved by Avilés [3,Theorem 20] (see also [8,Theorem 7]), while (ii) follows from [3,Theorem 20], [13,Lemma 4], [8,Theorem 7], and Remark 8 below.…”

The number of weakly compact sets which generate a Banach space

“…It has been observed in [5] that such spaces must have large enough density character, namely greater than or equal to cardinal b. In Section 7, we address the natural question now: What is the relation between CG(X) and CG(Y ) for Y a subspace of X?…”