Treatise on Analysis: 001

FOUNDATIONS OFMODERN ANALYSISEnlarged and Corrected PrintingJ. DIEUDONNEThis book is the first volume of a treatise which will eventually consist offour volumes. It is also an enlarged and corrected printing, essentiallywithout changes, of my Foundations of Modern Analysis, published in1960. Many readers, colleagues, and friends have urged me to write a sequelto that book, and in the end I became convinced that there was a place fora survey of modern analysis, somewhere between the minimum tool kitof an elementary nature which I had intended to write, and specialistmonographs leading to the frontiers of research. My experience of teachinghas also persuaded me that the mathematical apprentice, after taking the firststep of Foundations, needs further guidance and a kind of general birdseyeview of his subject before he is launched onto the ocean of mathematicalliterature or set on the narrow path of his own topic of research.Thus I have finally been led to attempt to write an equivalent, for themathematicians of 1970, of what the Cours dAnalyse of Jordan, Picard,and Goursat were for mathematical students between 1880 and 1920.It is manifestly out of the question to attempt encyclopedic coverage, andcertainly superfluous to rewrite the works of N. Bourbaki. I have thereforebeen obliged to cut ruthlessly in order to keep within limits comparable tothose of the classical treatises. I have opted for breadth rather than depth, inthe opinion that it is better to show the reader rudiments of many branchesof modern analysis rather than to provide him with a complete and detailedexposition of a small number of topics.Experience seems to show that the student usually finds a new theorydifficult to grasp at a first reading. He needs to return to it several times beforehe becomes really familiar with it and can distinguish for himself whichare the essential ideas and which results are of minor importance, and onlythen will he be able to apply it intelligently. Front Cover; Foundations of Modern Analysis; Copyright Page; Preface to the Enlarged and Corrected Printing; Preface; Contents; Notations; Chapter I. Elements of the Theory of Sets; Chapter II. Real Numbers; Chapter III. Metric Spaces; Chapter IV. Additional Properties of the Real Line; Chapter V. Normed Spaces; Chapter VI. Hilbert Spaces; Chapter VII. Spaces of Continuous Functions; Chapter VIII. Differential Calculus; Chapter IX. Analytic Functions; Appendix to Chapter IX. Application of Analytic Functions to Plane Topology; Chapter X. Existence Theorems. Appendix. Elements of Linear AlgebraReferences; Index; Pure and Applied Mathematics. This volume is intended to instruct first year graduate student or exceptionally advanced undergraduates in the elementary background for all branches of modern mathematics involving analysis, and to train the student in the axiomatic method. The emphasis is on the conceptual aspects of the topics treated, with problems at the ends of sections. The fundamental concepts, as presented, should afford a student a thorough grounding in the abstract and axiomatic way of thinking that undergirds all current mathematics research This volume, the eighth out of nine, continues the translation of "Treatise on Analysis" by the French author and mathematician, Jean Dieudonne. The author shows how, for a voluntary restricted class of linear partial differential equations, the use of Lax/Maslov operators and pseudodifferential operators, combined with the spectral theory of operators in Hilbert spaces, leads to solutions that are much more explicit than solutions arrived at through "a priori" inequalities, which are useless applications.