This book deals with a branch of modern functional analysis which has arisen only in the last 10 years, although it has its origin in a 1932 paper by Pontrjagin and Schnirelman.

In general there is quite a big difference between the level of recent research and the level of lectures as they are given to students. The question arises if this is in the nature of the subject, or if it is mainly a problem of producing an appropriate representation of the subject. Concerning ‘Entropy, compactness and the approximation of operators’, we came to the opinion that it should be possible to represent the subject at a level which makes reference only to the results of an introductory course on functional analysis. We have tried to write the book in the corresponding style and have listed in the introduction the concepts necessary for an understanding of the book. A few facts beyond the standard elementary knowledge of functional analysis are used without proof. However, a reader who is only interested in the fundamental relations between entropy quantities, approximation quantities, and eigenvalues can leave out the more difficult passages. By reading only sections 1.1, 1.2, 1.3, 1.4 of chapter 1, section 2.1 of chapter 2, section 3.1 of chapter 3, and section 4.2 of chapter 4, he or she will get an impression of the main ideas of the book and will be able to follow the applications of the general results in chapter 5.