UNIFORM CONVERGENCE IN THE DUAL OF A VECTOR-VALUED SEQUENCE SPACE

Abstract

In this note the authors establish several results concerning the uniform convergence of series in vector-valued sequence spaces. Corollaries include sufficient conditions for the weak sequential completeness of β-duals of sequence spaces, versions of the Uniform Boundedness Theorem and the Banach-Steinhaus Theorem for elements of operator-valued β-duals, and a characterization of weakly convergent sequences in β-duals. A further appli- cation establishes a vector-valued version of the Hahn-Schur Lemma.