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Elementary Real and Complex Analysis (Dover Books on Mathematics)

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Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, series, the derivative, higher derivatives, the integral and more. Each chapter contains a problem set (hints and answers at the end), while a wealth of examples and applications are found throughout the text. Over 340 theorems fully proved. 1973 edition.

It was with great delight that I learned of the imminent publication of an English-language edition of my introductory course on mathematical analysis under the editorship of Dr. R. A. Silverman. Since the literature already includes many fine books devoted to the same general subject matter, I would like to take this opportunity to point out the special features of my approach.

Mathematical analysis is a large "continent" concerned with the concepts of function, derivative, and integral. At present this continent consists of many "countries" such as differential equations (ordinary and partial), integral equations, functions of a complex variable, differential geometry, calculus of variations, etc. But even though the subject matter of mathematical analysis can be regarded as well-established, notable changes in its structure are still under way. In Goursat's classical "cours d'analyse" of the twenties all of analysis is portrayed on a kind of "great plain," on a single level of abstraction. In the books of our day, however, much attention is paid to the appearance in analysis of various "stages" of abstraction, i.e., to various "structures" (Bourbaki's term) characterizing the mathematicological foundations of the original constructions. This emphasis on foundations clarifies the gist of the ideas involved, thereby freeing mathematics from concern with the idiosyncracies of each object under consideration. At the same time, an understanding of the nub of the matter allows one to take account immediately of new objects of a different individual nature but of exactly the same "structural depth."
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