Weak-star point of continuity property and Schauder bases

Ginés López-Pérez¹ ; José A. Soler-Arias¹

¹ Universidad de Granada Facultad de Ciencias Departamento de Análisis Matemático 18071 Granada, Spain

Studia Mathematica, Tome 219 (2013) no. 3, pp. 225-236

Résumé

We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we show that a dual space has the Radon–Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results of Dutta and Fonf (2008) obtained for the separable case. Also, as a consequence of the above characterization, the following result of Rosenthal (2007) is deduced: every seminormalized basic sequence in a Banach space with the point of continuity property has a boundedly complete subsequence.

Détail

DOI : 10.4064/sm219-3-3

Keywords: characterize weak star point continuity property subspaces dual spaces separable predual deduce weak star point continuity property determined subspaces schauder basis natural setting dual spaces separable banach spaces consequence above characterization dual space has radon nikodym property only every seminormalized topologically weak star null tree has boundedly complete branch which improves results dutta fonf obtained separable consequence above characterization following result rosenthal deduced every seminormalized basic sequence banach space point continuity property has boundedly complete subsequence