Bases and Structure of the Space

Abstract

From the hypothesis of the existence of a basis of a certain type for a Banach space X, conclusions can be drawn on the structure of X, e. g. properties of X like weak sequential completeness, separability, reflexivity, weak conditional compactness and the dimension (finite or infinite). In the first section some results are established on the first two properties of X or of X*. The following paragraph has to do with reflexivity. In both sections the hypothesis of an unconditional basis plays an important role, while in the last paragraph criteria for finite dimension of X are given in terms of absolutely convergent and uniform bases for X.