Introduction

This chapter begins by stating some basic conventions, definitions and notation that will be used throughout the work. Additional standard notations will be introduced from time to time, as needed. The reader should consult the index of notation for reference. Many of the ideas presented in the first section will be familiar to some readers. They are mentioned for the sake of review and to fix our notation. Also, of course, some standard concepts are defined in slightly different ways by different authors, and we wish to make clear our own conventions. The chapter concludes with a number of examples discussed in some depth. We urge readers to acquaint themselves with these since an abstract theory, such as that presented in this work, lacks substance without knowledge of examples.

The first section deals primarily with basic elementary results on normed, semi-normed, or topological linear spaces and algebras. Such topics as ideals, homomorphisms, quotient norms, etc. are discussed, and the role of semi-norms in locally convex topological linear spaces is quickly surveyed. The unitization of an algebra and an important convention about it are also introduced.