Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of ${\ell }^{\infty }\left(\mathbb{N}\right)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musiał....

σ-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as σ-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness...

2000 Mathematics Subject Classification: 46B30, 46B03.It is shown that most of the well known classes of nonseparable Banach spaces related to the weakly compact generating can be characterized by elementary properties of the closure of the coefficient space of Markusevic bases for such spaces. In some cases, such property is then shared by all Markusevic bases in the space.

We prove that there exist weakly countably determined spaces of complexity higher than coanalytic. On the other hand, we also show that coanalytic sets can be characterized by the existence of a cofinal adequate family of closed sets. Therefore the Banach spaces constructed by means of these families have at most coanalytic complexity.