Polynomially significant properties and equivalence of topologies on fully nuclear spaces

Abstract

We relate the equivalence of the topologies τ o and τ ω on a fully nuclear space E having the bounded approximation property with the polynomially significant properties on E′ b , using the localization property of Defant and Govaerts (A. Defant and W. Govaerts (1986). Tensor products and spaces of vector-valued continuous functions. Manuscripta Math., 55, 433–449). This allows us to give examples of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on . We also give an example of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on for every open polydisc U in E× F′ ] . The conditions for equivalence of topologies are expressed in terms of the linear invariants (DN), and given in Vogt (D. Vogt (1983). Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist. J. reine u. angew. Math., 345, 182–200.) and Meise and Vogt (R. Meise and D. Vogt (1986). Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces. Studia Math., 83, 147–175.).