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Linearization of weighted (LB)-spaces of entire functions on Banach spaces

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Given a decreasing sequence of weights V on a Banach space X, we consider the weighted inductive limits of spaces of entire functions V H(X) and V H 0(X). We prove that V H(X) is the strong dual of a Fréchet space F for a particular class of sequence of weights, and we study some conditions to ensure that the equality V H 0(X)′′ = V H(X) holds. The existence of a predual of V H(X) leads to a linearization of this space of holomorphic functions. In fact, we show that there exists a holomorphic function \({\Delta: X \rightarrow F}\) with the following universal property: for each Banach space E and each function \({f \in V H(X, E),}\) there is a unique continuous linear operator \({T_f: F\rightarrow E}\) such that \({T_f \circ \Delta = f.}\) The correspondence \({f\rightarrow T_f}\) is an isomorphism between the space V H(X, E) and the space \({\mathcal{L}_i(F, E)}\) of all continuous linear operators from F into E endowed with the inductive limit topology.

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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, 46071, Valencia, Spain

    María J. Beltrán

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  1. María J. Beltrán
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Correspondence to María J. Beltrán.

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The author was partially supported by MEC and FEDER Project MTM2010-15200, grant F.P.U. AP2008-00604 and Conselleria d’Educació de la GVA, Project GV/2010/040.

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Beltrán, M.J. Linearization of weighted (LB)-spaces of entire functions on Banach spaces. RACSAM 106, 275–286 (2012). https://doi.org/10.1007/s13398-011-0049-z

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