Some aspects of B-completeness and the closed graph theorem

Abstract

One of the deepest results of Functional Analysis is the closed graph theorem of Banach which was originally proved for Banach spaces and has by now been generalised in many ways. Our aim in this thesis is to give some new forms of the closed graph theorem, describe some of its applications and find out more about B-complete spaces and algebras. Chapter I of the thesis deals with preliminary materials. In Chapter II we give a new and simpler proof of a result of Savgulidze and Smoljanov which asserts that E » $ is B-complete whenever E is a B-complete locally convex space and $ is the topological direct sum of countably many copies of the scalar field. In our proof we have avoided explicit use of dual spaces, which allows us to extend our results to certain non-locally convex situations and topological algebras. He discuss non-locally convex cases in Section 3. In Chapter III we introduce classes 01 and of topological vector spaces which serve as range and domain spaces for a closed graph theorem. We assume that (R contains Fxji for every F e (K and prove that JD contains each countable codimensional subspace of each of its elements. Using this fact along with the results of Chapter II we are able to extract some of the known results and some new results on the inheritance of topological vector space properties by subspaces of countable codimension. Also in this chapter, we give a closed graph theorem for webbed spaces. In Chapter IV we have defined a new class of locally convex spaces called M(a)-barrelled spaces. These spaces serve as domain space for a closed graph theorem where the range space is an a-weakly compactly generated Banach space. The connection between this class of locally convex spaces and the class of domain spaces for a closed graph theorem with range space a Banach space of density character at most a is considered. In Chapter V we are concerned with B- and B^-completeness in locally convex algebras and obtain various extensions of our E x $ results in this setting. In particular in Section 2 we looked at the possibility of B- or B^-completeness of E x ij> when it has the usual pointwise multiplication. Similar work has been carried out in Sections 3 and 4 for the unitization of a B-complete or a B - complete algebra E and also for E x $ when it has a new multiplication which is a natural generalization of the unitization.