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J. Dieudonne, FOUNDATIONS OF MODERN ANALYSIS Enlarged and Corrected Printing (four volumes)

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Product Description(of the first volume, announcing the four volumes)

FOUNDATIONS OF MODERN ANALYSIS Enlarged and Corrected Printing, J. DIEUDONNE This book is the first volume of a treatise which will eventually consist of four volumes. It is also an enlarged and corrected printing, essentially without changes, of my Foundations of Modern Analysis, published in 1960. 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 specialist monographs 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 mathematical literature 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 the mathematicians 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, and certainly superfluous to rewrite the works of N. Bourbaki. I have therefore been 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 detailed exposition 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. The chapters of this treatise arevi PREFACE TO THE ENLARGED AND CORRECTED PRINTING therefore samples rather than complete theories: indeed, I have systematically tried not to be exhaustive. The works quoted in the bibliography will always enable the reader to go deeper into any particular theory.However, I have refused to distort the main ideas of analysis by presenting them in too specialized a form, and thereby obscuring their power andgenerality. It gives a false impression, for example, if differential geometryis restricted to two or three dimensions, or if integration is restricted to Lebesgue measure, on the pretext of making these subjects more accessible or intuitive. On the other hand I do not believe that the essential content of the ideas involved is lost, in a first study, by restricting attention to separable metrizable topological spaces. The mathematicians of my own generation were certainly left to banish, hypotheses of countability wherever they were not needed: this was the only way to get a clear understanding.

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