Some Results on Interval-Valued Riesz-Summable Series

In this study, the concept of Riesz summability for interval-valued series is introduced, and the corresponding interval-valued Riesz-summable series space RIcs is constructed. In contrast to the sequence setting, Riesz summability of an interval-valued series is defined by applying the Riesz transformation to its sequence of partial sums. A metric based on the Hausdorff metric is defined on RIcs, and the resulting metric space is proved to be complete. It is also shown that every convergent interval-valued series is Riesz-summable to the same interval and that the inclusion of the convergent series space in RIcs is proper. Furthermore, RIcs is proved to be a quasilinear space and, when equipped with an appropriate norm, a normed quasilinear space. The metric associated with this norm is shown to coincide with the previously defined metric. In addition, the solid-floored property of RIcs is established. These results provide a systematic theoretical framework for the metric, algebraic, topological, and order-related analysis of interval-valued Riesz-summable series and contribute to the further development of summability theory in interval-valued settings.