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

TL;DR

This paper proves that the topological duals of spaces of formal power series, under various product topologies, are all identical to the space of polynomials, revealing a rigidity that constrains continuous linear maps.

Contribution

It establishes that the dual spaces are all the same (polynomials) under different product topologies, showing a rigidity in their structure.

Findings

Dual spaces are all the space of polynomials.

Continuous linear maps are represented by row-finite matrices.

Rigidity constrains the form of continuous maps across topologies.

Abstract

Even in spaces of formal power series is required a topology in order to legitimate some operations, in particular to compute infinite summations. Many topologies can be exploited for different purposes. Combinatorists and algebraists may think to usual order topologies, or the product topology induced by a discrete coefficient field, or some inverse limit topologies. Analysists will take into account the valued field structure of real or complex numbers. As the main result of this paper we prove that the topological dual spaces of formal power series, relative to the class of product topologies with respect to Hausdorff field topologies on the coefficient field, are all the same, namely the space of polynomials. As a consequence, this kind of rigidity forces linear maps, continuous for any (and then for all) of those topologies, to be defined by very particular infinite matrices similar to row-finite matrices.