Envelopes of open sets and extending holomorphic functions on dual Banach spaces

TL;DR

This paper explores the properties of open set envelopes in dual Banach spaces and their impact on extending holomorphic functions, revealing limitations and connections to convex set closures.

Contribution

It introduces new insights into the extension of holomorphic functions in dual Banach spaces and examines the role of convex set closures in this context.

Findings

Extension of holomorphic functions often not possible in certain dual Banach spaces

Examples of absolutely convex sets illustrating extension limitations

Connections established between envelopes and iterated weak* closures

Abstract

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.