Strongly linear algebra is an enrichment of linear algebra that allows infinite sums of a formal flavor. I will give some applications of this approach, which can be seen as extensions of topological methods, to the algebra of generalised formal series (such as transseries or multivariate formal series in commuting or non-commuting variables), and operators on these structures.

The first application is a formal version of the Lie correspondence that applies to objects that are “formally nilpotent” without being nilpotent or topologically nilpotent.

The second (related) application is a general result for treating the problem of normalisation of formal vector fields using asymptotic differential algebra.

This will be based on joint work with Lothar Sebastian Krapp, Salma Kuhlmann, Daniel Panazzolo, Michele Serra, and Vincenzo Mantova.