Cartan-Thullen theorem and Levi problem in context of generalised convexity

Abstract

We demonstrate that the Cartan-Thullen theorem and its generalisation to the context of generalised convexity, which we establish herein, can be regarded as consequences of the classical theorems of functional analysis: the Banach-Steinhaus theorem and the Banach-Alaoglu theorem. Furthermore, we characterise the domains of holomorphy, and their generalisations, as the spaces that are complete, or as the spaces exhaustible by suitably defined polytopes. We also provide an abstract analogue of the Levi problem and its elementary resolution. Our results allow for a novel characterisation of Stein spaces as the holomorphically complete spaces, as well as a proof that the Bremermann-Lelong lemma is equivalent to the positive answer to the Levi problem. Another contribution of ours is the introduction of the analogues of the notions of the complex analysis to the setting of generalised convexity.