Some classes of topological vector spaces associated with the Closed graph theorem

Abstract

Introduction: Given two topological vector spaces E and F and a linear mapping t : E → F with a closed graph, t may or may not be continuous. When such a linear mapping t is necessarily continuous, the closed graph theorem is said to hold for E and F . For example, if E and F are Banach spaces, then every linear mapping with a closed graph of E into F is necessarily continuous. The main aim of this thesis is to give precise descriptions of certain topological vector spaces that can serve as domain spaces, and also those that can serve as range spaces for a closed graph theorem. This is motivated by the works of M. Mahowald [35], N. Adasch [1], V. Eberhardt [12, 14, 11] and N.J. Kalton [25], Chapter 2 of the thesis is concerned with the concept of essential separability which turns out to be a useful variation of separability. We look at various characterizations of essential separability and link it up with the well-known concepts of weak compactness and weak relative compactness (Section 2.2). In Chapter 3, we introduce the class of 6-barrelled spaces which serve as domain spaces for some closed graph theorems (Theorems 3.1.2 and 4,1.3). We show that in the separated case, 6-barrelled spaces can be characterized in terms of essential separability (Theorem 3.1.1). We establish also some of the basic permanence properties of 6-barrelled spaces including the countable codimensional subspace property (Theorem 3.1.6). It is seen that the class of separated δ-barrelled spaces is a proper subclass of Kalton's domain spaces and strictly contains the class of separated barrelled spaces (Example 3.1.1(a), (d)). Also in this Chapter, conditions under which a δ-barrelled space is barrelled are considered. In Chapter 4, those locally convex spaces which can serve as range spaces in our closed graph theorem in which the domain space is an arbitrary δ-barrelled space with its Mackey topology are considered. These are the infra- δ-spaces. We also look at the domain spaces (δ-spaces) for the corresponding open mapping theorem (Theorem 4.1.4). Finally, Chapter 5 deals with some topics that are closely related to the concept of δ-barrelledness. In particular, we look at the closed graph theorem when the range space is not assumed to be complete. Then we generalize δ-barrelledness to δ-ultrabarrelledness in general topological vector spaces. In a way similar to the characterization of δ-barrelledness, we obtain a characterization of δ-ultrabarrelledness by means of a closed graph theorem (Theorem 5.2.2). We end the Chapter with a generalization of some of our concepts to arbitrary infinite cardinals.