Fréchet algebras of finite type (Q1884693)

A Fréchet algebra is a locally multiplicatively convex, complete, topological algebra the topology of which arises from an increasing countable family \(\{p_j\}\) of submultiplicative seminorms such that the intersection of their kernels is just zero. Let \(\chi\) be an inverse (projective) limit sequence of Banach algebras and let \(\lim_{\leftarrow}\chi\) be equipped with a Fréchet algebra topology. An inverse limit sequence \(\chi\) is an Arens-Michael representation for a Fréchet algebra \(F\) if \(F\) and \(\lim_{\leftarrow}\chi\) are isomorphic as Fréchet algebras. A Fréchet algebra is of finite type if each continuous seminorm has a finite-dimensional cokernel. The author studies Fréchet algebras having an Arens-Michael representation and classifies them using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type into any Fréchet space is, in fact, closed.