Abstract: A Markushevich basis {xa,fa}a � G for a Banach space X is a biorthogonal system in X such that [` span ]{xa}a � G = X and {fa}a � G is separating points in X. We will call such a basis a s- shrinking Markushevich basis if for every e > 0 the set G can be split into G = �n = 1�Gne such that (Gne)� � BX**e, for each n � IN, where BX**e is the e- ball in X** and (Gen)� is the set of all accummulation points of {xa}a � Gen in X**. We will give a sketch of a proof to the fact that a Banach space X admits a s- shrinking Markushevich basis if and only if X is a subspace of a weakly compactly generated space Z i.e. such a space Z that there is a weakly compact set K in Z whose span is dense in Z. If this happens for X, we will sketch a proof that then any normalized Markushevich basis for X is s- shrinking. We will show how this approach yields a short proof to the known result of Y. Benyamini, M. E. Rudin and M. Wage, namely that a continuous image of an Eberlein compact is an Eberlein compact. The compact space K is an Eberlein compact if K is homeomorphic to a weakly compact set in a Banach space (in its weak topology). We will discuss several open problems in this area.

The talk will be based on a recent joint work of M. Fabian, V. Montesinos and V. Zizler.