15 - Infinite Dimensional Holomorphy

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.