Functional analysis in asymmetric normed spaces / Stefan Cobzas

The main aim of this book is to present the basic results on asymmetric normed spaces. The author points out that Duffin and Karlovitz, in 1968, proposed the term "asymmetric norm'', and a systematic study of the properties of asymmetric normed spaces started with the papers of Romaguera and his collaborators. For classical functional analysis in normed and locally convex spaces, basic topological tools come from metric and uniform spaces. For the study of asymmetric normed and locally convex spaces the basic topological tools come from quasi-metric and quasi-uniform spaces. Hence the first chapter of this book contains a thorough presentation of related results from quasi-metric and quasi-uniform spaces. As a quasi-metric, in a natural way, induces two topologies, the first chapter contains an introduction to bitopological spaces. Also topological properties of asymmetric seminormed and asymmetric locally convex spaces are included in the first chapter.
This book consists of two chapters. It is to be noted that the set of all lower semi-continuous linear functionals on an asymmetric normed space X need not be a linear space. In spite of such existing differences many important results like Hahn-Banach and Kreĭn-Milman type theorems, Open mapping and Closed Graph theorems and the Banach-Steinhauss principle from the symmetric case have their counterparts in asymmetric normed spaces. These types of results are included in Chapter 2. Chapter 2 also contains some interesting results involving weak topologies and applications to best approximation theory. This book can be used as a reference book by research scholars and can also be used as an introductory textbook at the master's level. (MSN)