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Even in spaces of formal power series is required a topology in order tolegitimate some operations, in particular to compute infinite summations. Manytopologies can be exploited for different purposes. Combinatorists andalgebraists may think to usual order topologies, or the product topologyinduced by a discrete coefficient field, or some inverse limit topologies.Analysists will take into account the valued field structure of real or complexnumbers. As the main result of this paper we prove that the topological dualspaces of formal power series, relative to the class of product topologies withrespect to Hausdorff field topologies on the coefficient field, are all thesame, namely the space of polynomials. As a consequence, this kind of rigidityforces linear maps, continuous for any (and then for all) of those topologies,to be defined by very particular infinite matrices similar to row-finitematrices.

Rigidity of the topological dual of spaces of formal series with respect to product topologies