ABSTRACT
Chapter 3 deals with linear metric and linear semimetric spaces, the concepts of paranormed spaces and Schauder bases, and their most important properties. Among other things, it is shown that the quotient of a paranormed space is a paranormed space, and that quotient paranorms preserve completeness. The highlights of the chapter are the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach–Steinhaus theorem, which are generally considered as being the main results in functional analysis, apart from the Hahn–Banach extension theorem which is presented in Chapter 4. Furthermore, the chapter contains studies of useful properties of seminorms, further results related to local convexity, and the Minkowski functional and its role in defining a seminorm or norm on a linear space. Finally, a sufficient condition is established for the metrizability of a linear topology, and also a criterion is given for a linear topology to be generated by a seminorm.
