Some types of convergence related to the reconstruction property in Banach spaces

Abstract

Casazza and Christensen [Canad. Math. Bull., 51 (2008), 348--358] introduced and studied the reconstruction property in Banach spaces. In this paper, we discuss different types of convergence of series related to the reconstruction property in Banach space. First we discuss the uniform convergence of series associated with the reconstruction property in Banach spaces. Necessary and sufficient conditions for the uniform convergence of certain series related to the reconstruction property in Banach spaces are given. A sufficient condition for a Banach space to be finite dimensional in terms of the uniform convergence of a series related to the reconstruction property in Banach spaces is obtained. Motivated by a series of papers by Casazza, we discuss unconditional convergence of series associated with the reconstruction property in Banach spaces. A necessary condition in this direction is given. An absolute type reconstruction property in Banach spaces is also discussed which depends on the absolute convergence of series related to the reconstruction property in Banach spaces.