A weak Asplund space whose dual is not weak$^*$ fragmentable

Abstract

Abstract. Under the assumption that there exists in the unit interval [0,1] an uncountable set A with the property that every continuous mapping from a Baire metric space B into A is constant on some non-empty open subset of B, we construct a Banach space X such that (X * , weak * ) belongs to Stegall's class but (X * , weak * ) is not fragmentable.

Separated sets and Auerbach systems in Banach spaces

“…On the other hand, let us stress the fact that the two notions are actually distinct, as it was first proved in [57]. Let us also refer to [55] for a simplified proof of the example and to [36,39,41,56] for related results.…”

Schmidt-Like Spherical Aberration Corrector for Large Spectral Region Intended for the Space Optics

“…only regarding the behaviour of ω, the exponent of ω and ω 1 and in addition, if we assume 2 ℵ 0 = 2 ℵ 1 , however this is strongly opposite to the solution of the Dual Baer test R-module problem,see e.g. [JT20], since the principle of Namioka [PW01], established below implies CH and this implies the negation of the usual uniformization of ℵ 1 that we assume, which is the famous weak diamond at ℵ 1 , discovered by Shelah and Devlin [KDSSh78] and we know very well, that in this case a module with size ℵ 1 with proj.dim = 1 cannot be R-projective module over any ring, not the case implicitly said and as it is well -known by the algebraic persons, this is the situation, equivalent to the existence of a Q-set with size ℵ 1 over the real line and, for instance, assuming that Principle B! holds, i.e we have a model with existence of a Stegall Banach space, whose dual is not weak * -fragmentable, but with positive solution of the Dual Baer test R-module problem, see Trlifaj [JT20].…”

“…Kenderov [PW01], firstly establishes a construction in ZFC about conditions about weak Asplundness of a space, not weak * -fragmentable -we will make a note on the result below in the examples -of course any constructions about non-existence of a weak asplund space, not weak * fragmentable should rely on additional set-theoretic assumption. Now, after this humble clarification and modification of A!…”

Some more recent results concerning weak Asplund spaces

“…Although the exact relationship between fragmentable spaces, Stegall spaces, weak Asplund spaces, and Gâteaux differentiability spaces remains unclear, several partial results are known. For instance, the authors in [5] have provided an example (with the aid of some additional set-theoretic assumptions) of a Banach space X such that (X * ,weak * ) is in Stegall's class but is not fragmentable while in [4] the author has given an example (also with the aid of some additional set-theoretic assumptions) of a Banach space X such that X is weak Asplund but (X * ,weak * ) is not in Stegall's class. Moreover, in this paper we give an example (in ZFC) of a Gâteaux differentiability space that is not weak Asplund.…”

“…(i) From Theorem 2.1, it follows that K A is not fragmentable. On the other hand, it is shown in[5, Theorem 3] that for a Banach space X, (X * ,weak * ) is in Stegall's class if and only if it is in Stegall's class with respect to the class of all Baire metric spaces. The result then follows from Theorem 3.3.…”