Vector-valued holomorphic functions in several variables

TL;DR

This paper provides explicit proofs of classical theorems for vector-valued holomorphic functions in several variables, extending integral formulas to locally complete spaces using Pettis-integration.

Contribution

It generalizes integral formulas for vector-valued holomorphic functions to locally complete spaces, filling gaps in the literature for finitely many variables.

Findings

Proves identity theorem for vector-valued holomorphic functions.

Establishes Liouville's theorem in the vector-valued setting.

Shows density of polynomials in the $E$-valued polydisc algebra.

Abstract

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.