Rigidity of topological duals of spaces of formal series with respect to product topologies

Even in spaces of formal power series is required a topology in order to legitimate some operations, in particular to compute infinite summations. In general the topologies considered are just a product of the topology of the base field, an inverse limit topology or a topology induced by a pseudo-valuation. As our main result we prove the following phenomenon: the (left and right) topological dual spaces of formal power series equipped with the product topology with respect to any Hausdorff division ring topology on the base division ring, are all the same, namely just the space of polynomials. As a consequence, this kind of rigidity forces linear maps, continuous with respect to any (and then to all) those topologies, to be defined by very particular infinite matrices similar to row-finite matrices.