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Every Separable Complex Fréchet Space with a Continuous Norm is Isomorphic to a Space of Holomorphic Functions

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  13 March 2020

José Bonet
José Bonet*
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain
*
e-mail: jbonet@mat.upv.es
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Abstract

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

Keywords

Spaces of holomorphic functionsFréchet spacescontinuous normbounded approximation property

MSC classification

Primary: 46A04: Locally convex Fréchet spaces and (DF)-spaces
Secondary: 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A32: Spaces of linear operators; topological tensor products; approximation properties
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 8 - 12
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Copyright
© Canadian Mathematical Society 2020

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