be a sequence of positive numbers and
. For the weighted mean transformation
the authors introduce the Riesz sequence spaces
is a bounded sequence with
and study some of their properties, such as completeness with respect to a paranorm and characterization of the continuous and the Köthe-Toeplitz duals. It is proved that these spaces are linearly isomorphic to the complete paranormed sequence spaces of Maddox,
, respectively. Some matrix mappings related to the Riesz spaces are characterized.