Topological interpretation of function spaces stable under a general operation (tentative version)

Abstract

Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson, Sternfeld and Weit. They showed that, with suitable assumptions, a real or complex function space on which a non-affine continuous function operates is like an algebra. In this note, however, the point of view is somewhat different: the scalars are any field K, and from the outset no topology or continuity are assumed. Rather, a function space operated by a non-additively-affine (multivariate) function induces a topology on the set on which it is defined and the family of functions which operate on it induces topologies on K^n. These are used to derive a density property and to investigate "homomorphisms" of such spaces w.r.t. the "operations".