Normed domains of holomorphy

TL;DR

This paper explores the concept of domains of holomorphy in several complex variables when functions are restricted to Banach spaces, offering new insights and results for the classical theory.

Contribution

It introduces a novel perspective on domains of holomorphy in $ ext{C}^n$ with Banach space restrictions, including new results and a fresh view for the case $n=1$.

Findings

Positive results on domains of holomorphy with Banach space restrictions

Negative results highlighting limitations in the theory

A new perspective on the one-dimensional case

Abstract

We treat the classical concept of domain of holomorphy in $\CC^n$ when the holomorphic functions considered are restricted to lie in some Banach space. Positive and negative results are presented. A new view of the case $n = 1$ is considered.