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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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References

Bayart, F. and Matheron, É., Dynamics of linear operators. Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511581113CrossRefGoogle Scholar
Berenstein, C. A. and Gay, R., Complex analysis and special topics in harmonic analysis. Springer, New York, 1995. https://doi.org/10.1007/978-1-4613-8445-8CrossRefGoogle Scholar
Berenstein, C. A., Li, B. Q., and Vidras, A., Geometric characterization of interpolating varieties for the (FN)-space ${A}_p^{\it 0} $ of entire functions. Canad. J. Math. 47(1995), 28–43. https://doi.org/10.4153/CJM-1995-002-9CrossRefGoogle Scholar
Bierstedt, K. D. and Bonet, J., Some aspects of the modern theory of Fréchet spaces. RACSAM Rev. R. Acad. Cienc. Exactas Fas. Ser. A. Mat. 97(2003), 159–188.Google Scholar
Bonet, J., Lusky, W., and Taskinen, J., Monomial basis in Korenblum type spaces of analytic functions. Proc. Amer. Math. Soc. 146(2018), no. 12, 5269–5278. https://doi.org/10.1090/proc/14195CrossRefGoogle Scholar
Bonet, J. and Peris, A., Hypercyclic operators on non-normable Fréchet spaces. J. Funct. Anal. 159(1998), 587–595. https://doi.org/10.1006/jfan.1998.3315CrossRefGoogle Scholar
Braun, R. W., Meise, R., and Taylor, B. A., Ultradifferentiable functions and Fourier analysis. Results Math. 17(1990), 206–237. https://doi.org/10.1007/BF03322459CrossRefGoogle Scholar
Hedenmalm, H., Korenblum, B., and Zhu, K., Theory of Bergman spaces. Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-0497-8Google Scholar
Jarchow, H., Locally convex spaces. B. G. Teubner, Stuttgart, 1981.CrossRefGoogle Scholar
Köthe, G., Topological vector spaces. II. Grundlehren der Mathematischen Wissenschaften, 237, Springer-Verlag, New York and Berlin, 1979.Google Scholar
Mashreghi, J. and Ransford, T., Linear polynomial approximation schemes in Banach holomorphic functions spaces. Ann. Math. Phys. 9(2019), 899–905. https://doi.org/10.1007/s13324-019-00312-yCrossRefGoogle Scholar
Meise, R. and Taylor, B. A., Sequence space representations for (FN)-algebras of entire functions modulo closed ideals. Studia Math. 85(1987), 203–227. https://doi.org/10.4064/sm-85-3-203-227Google Scholar
Meise, R. and Vogt, D.. Introduction to functional analysis. Oxford Graduate Texts in Mathematics, 2, The Clarendon Press, Oxford University Press, New York, 1997.Google Scholar
Carreras, P. Pérez and Bonet, J., Barrelled locally convex spaces. North-Holland Mathematics Studies, 131, North-Holland Publishing Co., Amsterdam, 1987.Google Scholar
Rudin, W., Real and complex analysis. McGrawn-Hill, New York, 1974.Google Scholar
Vogt, D., A nuclear Fréchet space of ${C}^{\infty }$-functions which has no basis. Note Mat. 25(2005/06), 187–190.Google Scholar
Vogt, D., A nuclear Fréchet space consisting of ${C}^{\infty }$-functions and failing the bounded approximation property. Proc. Amer. Math. Soc. 138(2010), 1421–1423. https://doi.org/10.1090/S0002-9939-09-10166-1CrossRefGoogle Scholar
Vogt, D., Non-natural topologies on spaces of holomorphic functions. Ann. Polon. Math. 108(2013), no. 3, 215–217. https://doi.org/10.4064/ap108-3-1CrossRefGoogle Scholar
Zhu, K., Operator theory on function spaces. Mathematical Surveys and Monographs, 138, American Mathematical Society, 2007. https://doi.org/10.1090/surv/138Google Scholar
Zhu, K., Analysis on Fock spaces. Graduate Texts in Mathematics, 263, Springer, New York, 2012. https://doi.org/10.1007/978-1-4419-8801-0Google Scholar