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Functional analysis in asymmetric normed spaces. (English) Zbl 1266.46001
Frontiers in Mathematics. Basel: Birkhäuser (ISBN 978-3-0348-0477-6/pbk; 978-3-0348-0478-3/ebook). x, 219 p. (2013).
Asymmetric topology has become an interesting area in the last decades, fundamentally because of its applications in several areas like Economics or Theoretical Computer Science. But probably one of the most interesting facts has been to demonstrate one more time how general topology results in combination with classical results in functional analysis can become very useful and beautiful tools for attacking problems in different contexts. As the author mentions in the introduction of this book, the apparently innocent modification of the symmetry axiom of a metric (a norm in the case of asymmetric normed spaces) drastically changes the whole theory. This would be of interest of mathematicians in itself, but it also results in a powerful tool to analyze several contexts where the breaking of symmetry is a natural framework, as is the case, for example, of the complexity distance between programs, algorithms or data bases in Computer Science.
The book is structured in two chapters. The first one is devoted to the basic topological properties of quasi-metric and quasi-uniform spaces: bitopological spaces, asymmetric locally convex spaces, compactness and completeness in quasi-metric and quasi-uniform spaces, and completions of quasi-metric and quasi-uniform spaces. The second chapter is devoted to what the author calls Asymmetric Functional Analysis that corresponds to the theory of asymmetric normed spaces (also asymmetric locally convex spaces and asymmetric norms on lattices) and the theory of operators and functions that operate between them. Here, the asymmetric topologies on lattices are treated and an important collection of classical results are derived: Hahn-Banach type theorems and separation of convex sets, Extreme points and Krein-Milman theorem, The Open Mapping and the Closed Graph Theorems, and The Banach-Steinhaus Principle. The book also deals with weak topologies and compactness in asymmetric normed spaces and asymmetric locally convex spaces, Schauder type theorem, the bidual, reflexivity and a Goldstine type theorem, asymmetric moduli of rotundity and smoothness and applications to best approximation. The last part of the chapter is devoted to spaces of semi-Lipschitz functions.
The book takes advantage of survey papers of H. P. A. Künzi in its first chapter and of the author himself in the case of the second one (as he acknowledges in the introduction) and, of course, an extensive bibliography. In fact, this book contains a very important bibliographic review of the state of art and it is also a tremendous effort to unify the notation (including an index or glossary of terms at the end of the book) that must be recognized. The book is addressed to researchers and postgraduate students.
Reviewer’s remarks. It would be very interesting to include in future editions chapters devoted to the different applications of asymmetric topology and asymmetric normed spaces and the functional analysis developed on them. As I have mentioned, these ideas have found application in different contexts like Computer Science, Economics or Optimization. I think it would be an important step in the development of these ideas that have been really fruitful in these areas.

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
54-02 Research exposition (monographs, survey articles) pertaining to general topology
46B99 Normed linear spaces and Banach spaces; Banach lattices
54E35 Metric spaces, metrizability
54E55 Bitopologies
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