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1 results

  • 15 - Infinite Dimensional Holomorphy

  • from Part 2 - Advanced Toolbox
  • Andreas Defant, Carl V. Ossietzky Universität Oldenburg, Germany, Domingo García, Universitat de València, Spain, Manuel Maestre, Universitat de València, Spain, Pablo Sevilla-Peris
  • Book:
    Dirichlet Series and Holomorphic Functions in High Dimensions
    Published online:
    19 July 2019
    Print publication:
    08 August 2019, pp 351-409

  • Summary

    We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Fréchet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous, if and only if it has representation as a series of homogeneous polynomials (known as Taylor expansion). A function is weak holomorphic if the composition with every functional is holomorphic. A function is holomorphic if and only if it is weak holomorphic. Analytic functions are holomorphic.