Georgi E. Shilov - Elementary Real and Complex Analysis

Elementary Real and
Complex AnalysisGeorgi E. Shilov TRANSLATED AND EDITED BY Richard A. SilvermanREVISED ENGLISH EDITIONDOVER PUBLICATIONS, INC.New YorkCopyright Copyright 1973 by the Massachusetts Institute of Technology. All rights reserved. Bibliographical Note This Dover edition, first published in 1996, is an unabridged, corrected republication of the work first published in English by The MIT Press, Cambridge, Massachusetts, 1973, as Volume 1 of the two-volume course Mathematical Analysis. E. (Georgi Evgenevich) [Matematicheski analiz. Chasti 12. Chasti 12.

English] Elementary real and complex analysis/Georgi E. Shilov; revised English edition translated and edited by Richard A. Silverman, p. cm. Originally published in English: Cambridge, Mass.: MIT Press, 1973. Includes index.

E-ISBN-13: 978-0-486-68922-7 (pbk.) ISBN-10: 0-486-68922-0 (pbk.) 1. Mathematical analysis. I. Silverman, Richard A. II. Title.

QA300.S4552 1996

515dc20	95-37030
CIP

Manufactured in the United States by Courier Corporation
68922006
www.doverpublications.com ContentsPreface 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 Goursats classical cours danalyse 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 (Bourbakis 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.

Consider, for example, Picards proof of the existence and uniqueness of the solution of a differential equation in which the desired function is successively approximated on a given interval by other functions in accordance with certain rules. This proof had been known for some time when Banach and others formulated the fixed point method. The latter plainly reveals the nub of Picards proof, namely the presence of a contraction operator in a certain metric space. In this regard, the specific context of Picards problem, i.e., numerical functions on an interval, a differential equation, etc., turns out to be quite irrelevant. As a result, the fixed point method not only makes the geometrical proof of Picards theorem more transparent, but, by further developing the key idea of Picards proof, even leads to the proof of existence theorems involving neither functions on an interval nor a differential equation. Considerations of the same kind apply equally well to the geometry of Hilbert space, the study of differentiable functionals, and many other topics.

Analysis presented from this point of view can be found, for example, in the superb books by J. Dieudonn. However it seems to me that Dieudonns books, for all their formal perfection, require that the readers mathematical I. Q. be too high. Thus, for my part, I have tried to accomodate the interests of a larger population of those concerned with mathematics.

Therefore in many cases where Dieudonn instantly and almost miraculously produces deep classical results from general considerations, so that the reader can only take off his hat in silent admiration, the reader of my course is invited to climb with me from the foothills of elementary topics to successive levels of abstraction and then look down from above on the various valleys which now come into his field of view. Perhaps this approach is thornier, but in any event the mathematical traveler will thereby acquire the training needed for further exploration on his own. The present course begins with a systematic study of the real numbers, understood to be a set of objects satisfying certain definite axioms. There are other approaches to the theory of real numbers where things I take as axioms are proved, starting from set theory and the axioms for the natural numbers (for example, a rigorous treatment in this vein can be found in Landaus famous course). Both treatments have a key deficiency, namely the absence of a proof of the compatibility of the axioms. Evidently modern mathematics lacks a construction of the real numbers which is free of this shortcoming.

The whole question, far from being a mere technicality, involves the very foundations of mathematical thought. In any event, this being the case, it is really not very important where one starts a general treatment of analysis, and my choice is governed by the consideration that the starting point bear as close a resemblance as possible to analytic constructions proper. The concepts of a mathematical structure and an isomorphism are introduced in is devoted to improper integrals, and makes full use of the technique of analytic functions now at our disposal. Each chapter is equipped with a set of problems; hints and answers to most of these problems appear at the end of the book. To a certain extent, the problems help to develop necessary technical skill, but they are primarily intended to illustrate and amplify the material in the text. G.E.S.

Real Numbers 1.1. Set-Theoretic Preliminaries Such sets are said to be finite. In mathematics one must often deal with sets consisting of a number of objects which is not finite. The simplest examples of such sets are the set 1, 2, 3, ... of all natural numbers (positive integers) and the set of all the points on a line segment (precise definitions of these objects will be given later). Such sets are said to be infinite.

To the category of sets we also assign the empty set, namely, the set containing no elements of all. As a rule, sets will be denoted by large letters A, B, C, ... and elements of sets will be denoted by small letters. By a A (or A a) we mean that a is an element of the set A, while by a A (or a A) we mean that a is not an element of the set A. By AB (or BA) we mean that every element of the set A is an element of the set

Next page