  We demonstrate that the Cartan-Thullen theorem and its generalisation to the
context of generalised convexity, which we establish herein, can be regarded as
a consequence of the classical theorems of functional analysis. 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 analogue of the Levi problem and its elementary
solution. Our result allows for a novel characterisation of Stein spaces.
