A projection property and weak sequential completeness of $\alpha$-duals

In [7] the Hahn-Schur Theorem from summability was used to give a simple proof of a result of Bennett on the weak sequential completeness of the $\alpha$-dual of a monotone sequence space ([2]). In this note we point out that the same method of proof employed in [7] can be used to give a generalization of Bennett’s result to a wider class of sequence spaces than the monotone spaces. As was the case in [7], the methods are also applicable to vector-valued sequence spaces.