Asymptotic extension of topological modules and algebras

Abstract

Given a topological R–module or algebra E and an asymptotic scale M⊂R Λ, we define a natural M–extended topology on the sequence space E Λ, and the M–extension of E as the Hausdorff space associated with the subspace of sequences for which multiplication is continuous with respect to this topology. Colombeau's generalized functions and similar constructions are obtained as special cases, but this new approach also allows the iteration of the construction, which was not possible with previously existing theories. We use only the topology, i.e. neigbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly useful in non-metrizable spaces. Many ideas commonly used in generalized function spaces (functoriality, association, sheaf structure, algebraic analysis, …) can be applied to a large extent; but reasoning on a category-theoretic level allows the establishment of several results so far only known for particular cases, to the whole class of such spaces.