Compacta in product of lines from a functional-analytic point of view

M. Fabian, G. Godefroy, V. Montesinos and V. Zizler

December 1, 2003

Along several papers we developed a technique of splitting certain subsets of a Banach space X in such a way that it allows us to detect whether or not X belongs to one of several classes of non-separable Banach spaces. Those subsets sometimes are Markushevich bases, sometime just total and countably supported subsets. Classes of Banach spaces considered included the weakly compactly generated, the subspaces of weakly compactly generated, the Vašák or weakly countably determined and the weakly Lindelöf determined. The technique is reminiscent of the way several authors (S. Argyros, S. Mercourakis, V. Farmaki, S. P. Gu’lko, G. A. Sokolov,...) used to split the set of coordinates in representing compact sets as subsets of products of lines. It is not surprising, then, that we can recover those representations by using our approach. This is done here for the Eberlein and the Gul’ko compacta.