General introduction This book provides an elementary exposition intended for students who have completed four years' study of university level mathematics. A knowledge of the elements of functional analysis, Fourier analysis and distribution theory (including, in particular, Fourier analysis in S and S ) is assumed. Chapter 0 contains a reminder of the notation, concepts and main results used in the remainder of the book (with references). On the other hand, no knowledge of partial differential equations is needed, although it will be beneficial to have received an initiation to the topic. The book stems from a course on Tseudo-differential operators and the Nash-Moser theorem', presented at the Ecole Normale Superieure (ENS) from October 1986 onwards, to second-year students studying for the degree of Master of Fundamental and Applied Mathematics and Computer Science. Although the topics covered largely form the subject of research litera- ture, we have striven to avoid any scholarly discussions, Veiled references' and sibylline allusions, which might open chasms beneath the reader's foot- steps. A particular presentation of the subject is selected and developed in each chapter: the commentary at the end of each chapter indicates the sources, differing approaches, certain current extensions, and the connec- tions between the topics handled. Finally, we have assembled numerous exercises, divided into two classes. Elementary exercises are intended to help readers assimilate the course and monitor their progress. Other more complex exercises, marked with an asterisk (*), present recent developments which have sometimes only been published in journal articles: we crave their authors' forgiveness for this simplification! These exercises, unlike those in certain famous treatises, can 1
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