Topological Vector Spaces

In this chapter we start our investigations on general topological vector spaces by introducing the basic concepts and giving the standard descriptions of linear topologies by means of particular neighbourhood bases of the zero vector. This is followed by a brief discussion of boundedness and of continuity of linear forms in 2.3. In the sections 2.4–2.6 we consider projective topologies as a first general method to generate new linear topologies from given ones. The usual universal characterization of cartesian products is supplemented in 2.5 by a rather exceptional one which, however, will turn out to be a convenient and powerful tool in subsequent discussions.