ABSTRACT
Chapter 2 is dedicated to the study of linear topological spaces. As far as more advanced purely algebraic concepts are concerned, thorough treatments are presented of convex and affine sets, balloons and cones, and quotient spaces. The studies of purely topological concepts mainly concentrate on supremum, weak and product topologies. The concept of linear topological spaces or topological vector spaces is very general and of great importance in functional analysis. It combines the purely topological concepts of topological spaces with the purely algebraic concepts of linear or vector spaces in a natural way such that the vector space operations of addition and multiplication by scalars are continuous. Various important properties of linear topological spaces are established and a characterization of linear topologies is given in terms of their open sets. Furthermore, the chapter deals with properties of closed maps, and presents the closed graph lemma in linear topological spaces, and Baire’s category theorem in complete metric spaces. Finally, the last two sections study locally convex spaces and quotient topologies.
