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Inequalities. A journey into linear analysis. (English) Zbl 1135.26014
Cambridge: Cambridge University Press (ISBN 978-0-521-87624-7/hbk; 978-0-521-69973-0/pbk). ix, 335 p. (2007).
This excellent book provides an introduction to a selection of inequalities that relates to problems in linear analysis. The book begins with Cauchy’s inequality and ends with Grothendieck’s inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner’s inequality, and many, many more. The author put these inequalities in proper context and also show their usefulness to obtain properties of functions spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. The book contains brief notes and remarks at the end of each chapter, which include suggestions for further reading and also a collection of exercises, of varied nature: some are five-finger exercises, but some establish results that are even deeper. The book is very interesting, well written, readable and should be of interest to more than just analysts.

26D15 Inequalities for sums, series and integrals
42B25 Maximal functions, Littlewood-Paley theory