On Holomorphy Versus Linearity in Classifying Locally Convex Spaces

Abstract

This chapter provides an overview of the fact that in the holomorphic approach the corresponding concepts have been introduced as holomorphically bornological, holomorphically barreled, holomorphically infrabaralled, and holomorphically Mackey spaces that are more restricted classes than the corresponding linear ones. In the linear theory of locally convex spaces, it is classical to study bornological, barreled, infrabarreled, and Mackey spaces. An interesting highlight is the holomorphic Banach-Steinhaus on a Fréchet space that contains, as a particular case, the classical linear Banach-Steinhaus theorem on such a space. A holomorphically bornological space is also a bornological space. A semimetrizable space is a holomorphically bornological space. A Silva space is known to be essentially the same thing as the dual of a Fréchet-Schwartz space, or FS-space; thus, it is also known as a DFS-space. A Silva space is a holomorphically bornological space. Any inductive limit of bornological spaces is a bornological space.