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Title: Contributions to the theory of linear topological spaces
Author: Iyahen, Sunday Osarumwense
Awarding Body: Keele University
Current Institution: Keele University
Date of Award: 1967
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This thesis is mainly concerned with linear topoJogical spaces in which local convexity is not assumed. In particular it contains a study of the closed graph and open mapping theorems in this context, together with results analogous to the Banach-Steinhaus theorem. Many of the techniques and notions used to study these important theorems in locally convex spaces are no longer effective for general linear topological spaces and much of this thesis is taken up with the development of alternative methods and definitions. The first of these is the notion of a 1:-inductive limit of linear topological spaces. This plays much the same part in the theory of general linear topological spaces as an inductive limit does for locally convex spaces, and natural analogues are proved for most of the known results on inductive limits. After this has been introduced, it is shown that the *-inductive limit topology of a sequence of locally convex spaces is locally convex. Then a study is made of ultrabarrelled spaces, which replace barrelled spaces in certain theorems when local convexity is not assumed. Also ultrabornological and quasi-ultrabarrelled spaces are defined and studied. Any *-inductive limit of members of one of these classes has the same property. In particular, any *-inductive limit of complete metric linear spaces has the three properties. However, an uncountable direct sum of Banach spaces has none of these properties and none of these properties passes on to closed linear subspaces. Ultrabarrelled spaces are characterised in terms of closed linear maps into complete metric linear spaces and similar characterisations are given for ultrabornological and quasi-ultrabarrelled spaces in terms of bounded and closed bounded linear maps, respectively. These notions find application in the study of two-norm spaces. The next section of the thesis looks at semiconvex spaces, spaces in which there is a neighbourhood base of the origin consisting of semiconvex sets. For these, there can be defined a type of inductive limit topology which is in some respects intermediate between that of the ordinary inductive limit of locally convex spaces and *-inductive limit of general linear topological spaces. Such is called a *-inductive limit topology. Similarly there are spaces (called hyperbarrelled spaces) fitting naturally between barrelled spaces and ultrabarrelled spaces, with analogues for bornological and quasi-barrelled spaces. A thorough study is made of these, in which results rather similar to those already found for ultrabarrelled spaces are obtained. For example, hyperbarrelled spaces are characterised in terms of closed linear maps into complete separated locally bounded spaces. It is also shown that any product of separated hyperbarrelled spaces is hyperbarrelled. Finally, the problem of characterising the sorts of spaces that can be range spaces in various forms of the closed graph theorem is considered. Various general classes Dr (A₁;A, T) and D(A₁;A, T) of linear topological spaces are defined, generalising in a natural way the Br-complete and B-complete spaces. These are used to find extensions of the known closed graph and open mapping theorems. The notions are also meaningful for commutative topological groups and, for these, analogues of the known theorems are proved.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics