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Vladimir A. ZorichMathematicalAnalysis [email protected] Springer

Vladimir A. ZorichMoscow State UniversityDepartment of Mathematics (Mech-Math)Vorobievy Gory119992 MoscowRussiaTranslator:Roger CookeBurlington, VermontUSAe-mail: [email protected] of Russian edition:Matematicheskij Analiz (Part 1,4th corrected edition, Moscow, 2002)MCCME (Moscow Center for Continuous Mathematical Education Publ.)Cataloging-in-Publication Data applied forA catalog record for this book is available from the Library of Congress.Bibliographic information published by Die Deutsche BibliothekDie Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data is available in the Internet at http://dnb.ddb.deMathematics Subject Classification (2000): Primary 00A05Secondary: 26-01,40-01,42-01,54-01,58-01ISBN 3-540-40386-8 Springer-Verlag Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part of thematerial is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilm or in any other way, and storage in databanks. Duplication of this publication or parts thereof is permitted only under the provisionsof the German Copyright Law of September 9,1965, in its current version, and permissionfor use must always be obtained from Springer-Verlag. Violations are liable for prosecutionunder the German Copyright Law.Springer-Verlag is a part of Springer Science* Business Springer-Verlag Berlin Heidelberg 2004Printed in GermanyThe use of general descriptive names, registered names, trademarks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.Cover design: design & production GmbH, HeidelbergTypeset by the translator using a Springer ETgX macro packagePrinted on acid-free paper46/3indb- 5 4 3 2

PrefacesPreface to the English EditionAn entire generation of mathematicians has grown up during the time between the appearance of the first edition of this textbook and the publicationof the fourth edition, a translation of which is before you. The book is familiar to many people, who either attended the lectures on which it is based orstudied out of it, and who now teach others in universities all over the world.I am glad that it has become accessible to English-speaking readers.This textbook consists of two parts. It is aimed primarily at universitystudents and teachers specializing in mathematics and natural sciences, andat all those who wish to see both the rigorous mathematical theory andexamples of its effective use in the solution of real problems of natural science.Note that Archimedes, Newton, Leibniz, Euler, Gauss, Poincare, who areheld in particularly high esteem by us, mathematicians, were more than meremathematicians. They were scientists, natural philosophers. In mathematicsresolving of important specific questions and development of an abstract general theory are processes as inseparable as inhaling and exhaling. Upsettingthis balance leads to problems that sometimes become significant both inmathematical education and in science in general.The textbook exposes classical analysis as it is today, as an integral partof the unified Mathematics, in its interrelations with other modern mathematical courses such as algebra, differential geometry, differential equations,complex and functional analysis.Rigor of discussion is combined with the development of the habit ofworking with real problems from natural sciences. The course exhibits thepower of concepts and methods of modern mathematics in exploring specific problems. Various examples and numerous carefully chosen problems,including applied ones, form a considerable part of the textbook. Most of thefundamental mathematical notions and results are introduced and discussedalong with information, concerning their history, modern state and creators.In accordance with the orientation toward natural sciences, special attentionis paid to informal exploration of the essence and roots of the basic conceptsand theorems of calculus, and to the demonstration of numerous, sometimesfundamental, applications of the theory.

VIPrefacesFor instance, the reader will encounter here the Galilean and Lorentztransforms, the formula for rocket motion and the work of nuclear reactor, Euler's theorem on homogeneous functions and the dimensional analysisof physical quantities, the Legendre transform and Hamiltonian equationsof classical mechanics, elements of hydrodynamics and the Carnot's theorem from thermodynamics, Maxwell's equations, the Dirac delta-function,distributions and the fundamental solutions, convolution and mathematicalmodels of linear devices, Fourier series and the formula for discrete codingof a continuous signal, the Fourier transform and the Heisenberg uncertaintyprinciple, differential forms, de Rham cohomology and potential fields, thetheory of extrema and the optimization of a specific technological process,numerical methods and processing the data of a biological experiment, theasymptotics of the important special functions, and many other subjects.Within each major topic the exposition is, as a rule, inductive, sometimesproceeding from the statement of a problem and suggestive heuristic considerations concerning its solution, toward fundamental concepts and formalisms.Detailed at first, the exposition becomes more and more compressed as thecourse progresses. Beginning ab ovo the book leads to the most up-to-datestate of the subject.Note also that, at the end of each of the volumes, one can find the listof the main theoretical topics together with the corresponding simple, butnonstandard problems (taken from the midterm exams), which are intendedto enable the reader both determine his or her degree of mastery of thematerial and to apply it creatively in concrete situations.More complete information on the book and some recommendations forits use in teaching can be found below in the prefaces to the first and secondRussian editions.Moscow, 2003V. Zorich

PrefacesVIIPreface to the Fourth Russian EditionThe time elapsed since the publication of the third edition has been too shortfor me to receive very many new comments from readers. Nevertheless, someerrors have been corrected and some local alterations of the text have beenmade in the fourth edition.Moscow, 2002V. ZorichPreface to the Third Russian editionThis first part of the book is being published after the more advanced Part2 of the course, which was issued earlier by the same publishing house. Forthe sake of consistency and continuity, the format of the text follows thatadopted in Part 2. The figures have been redrawn. All the misprints thatwere noticed have been corrected, several exercises have been added, and thelist of further readings has been enlarged. More complete information on thesubject matter of the book and certain characteristics of the course as a wholeare given below in the preface to the first edition.Moscow, 2001V. ZorichPreface to the Second Russian EditionIn this second edition of the book, along with an attempt to remove the misprints that occurred in the first edition, 1 certain alterations in the expositionhave been made (mainly in connection with the proofs of individual theorems), and some new problems have been added, of an informal nature as arule.The preface to the first edition of this course of analysis (see below) contains a general description of the course. The basic principles and the aimof the exposition are also indicated there. Here I would like to make a fewremarks of a practical nature connected with the use of this book in theclassroom.Usually both the student and the teacher make use of a text, each for hisown purposes.At the beginning, both of them want most of all a book that contains,along with the necessary theory, as wide a variety of substantial examples1No need to worry: in place of the misprints that were corrected in the platesof the first edition (which were not preserved), one may be sure that a host ofnew misprints will appear, which so enliven, as Euler believed, the reading of amathematical text.

VIIIPrefacesof its applications as possible, and, in addition, explanations, historical andscientific commentary, and descriptions of interconnections and perspectivesfor further development. But when preparing for an examination, the studentmainly hopes to see the material that will be on the examination. The teacherlikewise, when preparing a course, selects only the material that can and mustbe covered in the time alloted for the course.In this connection, it should be kept in mind that the text of the presentbook is noticeably more extensive than the lectures on which it is based. Whatcaused this difference? First of all, the lectures have been supplemented byessentially an entire problem book, made up not so much of exercises as substantive problems of science or mathematics proper having a connection withthe corresponding parts of the theory and in some cases significantly extending them. Second, the book naturally contains a much larger set of examplesillustrating the theory in action than one can incorporate in lectures. Thirdand finally, a number of chapters, sections, or subsections were consciouslywritten as a supplement to the traditional material. This is explained in thesections "On the introduction" and "On the supplementary material" in thepreface to the first edition.I would also like to recall that in the preface to the first edition I tried towarn both the student and the beginning teacher against an excessively longstudy of the introductory formal chapters. Such a study would noticeablydelay the analysis proper and cause a great shift in emphasis.To show what in fact can be retained of these formal introductory chapters in a realistic lecture course, and to explain in condensed form the syllabusfor such a course as a whole while pointing out possible variants dependingon the student audience, at the end of the book I give a list of problemsfrom the midterm exam, along with some recent examination topics for thefirst two semesters, to which this first part of the book relates. From this listthe professional will of course discern the order of exposition, the degree ofdevelopment of the basic concepts and methods, and the occasional invocation of material from the second part of the textbook when the topic underconsideration is already accessible for the audience in a more general form.2In conclusion I would like to thank colleagues and students, both knownand unknown to me, for reviews and constructive remarks on the first editionof the course. It was particularly interesting for me to read the reviews ofA. N. Kolmogorov and V. I. Arnol'd. Very different in size, form, and style,these two have, on the professional level, so many inspiring things in common.Moscow, 19972V. ZorichSome of the transcripts of the corresponding lectures have been published and Igive formal reference to the booklets published using them, although I understandthat they are now available only with difficulty. (The lectures were given andpublished for limited circulation in the Mathematical College of the IndependentUniversity of Moscow and in the Department of Mechanics and Mathematics ofMoscow State University.)

PrefacesIXProm the Preface to the First Russian EditionThe creation of the foundations of the differential and integral calculus byNewton and Leibniz three centuries ago appears even by modern standardsto be one of the greatest events in the history of science in general andmathematics in particular.Mathematical analysis (in the broad sense of the word) and algebra haveintertwined to form the root system on which the ramified tree of modernmathematics is supported and through which it makes its vital contact withthe nonmathematical sphere. It is for this reason that the foundations ofanalysis are included as a necessary element of even modest descriptions ofso-called higher mathematics; and it is probably for that reason that so manybooks aimed at different groups of readers are devoted to the exposition ofthe fundamentals of analysis.This book has been aimed primarily at mathematicians desiring (as isproper) to obtain thorough proofs of the fundamental theorems, but who areat the same time interested in the life of these theorems outside of mathematics itself.The characteristics of the present course connected with these circumstances reduce basically to the following:In the exposition. Within each major topic the exposition is as a rule inductive, sometimes proceeding from the statement of a problem and suggestiveheuristic considerations toward its solution to fundamental concepts and formalisms.Detailed at first, the exposition becomes more and more compressed asthe course progresses.An emphasis is placed on the efficient machinery of smooth analysis. Inthe exposition of the theory I have tried (to the extent of my knowledge) topoint out the most essential methods and facts and avoid the temptation ofa minor strengthening of a theorem at the price of a major complication ofits proof.The exposition is geometric throughout wherever this seemed worthwhilein order to reveal the essence of the matter.The main text is supplemented with a rather large collection of examples,and nearly every section ends with a set of problems that I hope will significantly complement even the theoretical part of the main text. Followingthe wonderful precedent of Polya and Szego, I have often tried to presenta beautiful mathematical result or an important application as a series ofproblems accessible to the reader.The arrangement of the material was dictated not only by the architect