Defects of Properties in Mathematics: Quantitative Characterizations

This book is devoted to a research method, called by the authors the quantitative study of the defects of properties. The authors recall some basic definitions from various fields of mathematics. Instead of defining ‘object x has property P’, the authors define a quantity E(x), called the defect of property P, which measures ‘how far’ x is from having the property P.
Chapter 1 is a detailed overview of the topics discussed in the rest of the book: this is handy for the readers new to the theory, who can get the whole picture, skipping the more technical parts on a first reading. In the next relatively independent chapters the authors study properties in set theory, topology, measure theory, real function theory, functional analysis and algebra. In the final chapter, the authors study properties in complex analysis, geometry, number theory and fuzzy logic. The book deals with a great variety of problems in a very readable way. New ideas are motivated by many examples and remarks, and most of the chapters conclude with applications. The only small objection I have is that applications should have deserved more space. In the bibliographical remarks in every chapter, the authors give full references and explicitly state which results are new.
This book is a good overview of the theory of quantitative characterisations, the introduced concepts are elegant and the methods of proof are (the authors believe) ‘rather elementary’. This makes the material accessible to undergraduate and graduate students, while researchers may find the new concepts very motivating.

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