Keele Research Repository

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