Abstract
Let E be a Banach space, lambda a perfect sequence space, and M an Orlicz function. Denote by lambda(E, M)(r) the space of all weakly (M, lambda) -summable sequences from E that are the limit of their finite sections. In this paper, we describe the continuous linear functionals on lambda(E, M)(r) in terms of strongly (N, lambda*)-summable sequences in the dual E* of E, and then we give a characterization of the reflexivity of lambda(E, M) in terms of that of lambda and of E and the AK-property.