Analysis in Euclidean Space

by Kenneth Hoffman

Overview

Developed for an introductory course in mathematical analysis at MIT, this text focuses on concepts, principles, and methods. Its introductions to real and complex analysis are closely formulated, and they constitute a natural introduction to complex function theory.
Starting with an overview of the real number system, the text presents results for subsets and functions related to Euclidean space of n dimensions. It offers a rigorous review of the fundamentals of calculus, emphasizing power series expansions and introducing the theory of complex-analytic functions. Subsequent chapters cover sequences of functions, normed linear spaces, and the Lebesgue interval. They discuss most of the basic properties of integral and measure, including a brief look at orthogonal expansions. A chapter on differentiable mappings addresses implicit and inverse function theorems and the change of variable theorem. Exercises appear throughout the book, and extensive supplementary material includes a Bibliography, List of Symbols, Index, and an Appendix with background in elementary set theory.

Product Details

ISBN-13:	9780486841410
Publisher:	Dover Publications
Publication date:	07/17/2019
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	448
File size:	23 MB
Note:	This product may take a few minutes to download.

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PREFACE

This textbook has been developed for use in the two-semester introductory course in mathematical analysis at the Massachusetts Institute of Technology. The aim of the course is to introduce the student to basic concepts, principles, and methods of mathematical analysis.

The presumed mathematical background of the students is a solid calculus course covering one and (some of ) several variables, plus (perhaps) elementary differential equations and linear algebra. The linear algebra background is not necessary until the second semester, since it enters the early chapters only through certain examples and exercises which utilize matrices. At M.I.T. the introductory calculus course is condensed into one year, after which the student has available a one-semester course in differential equations and linear algebra. Thus, over half the students in the course are sophomores. Since many students enter M.I.T. having had a serious calculus course in high school, there are quite a few freshmen in the course. The remainder of the students tend to be juniors, seniors, or graduate students in fields such as physics or electrical engineering . Since very little prior experience with rigorous mathematical thought is assumed, it has been our custom to augment the lectures by structured tutorial sessions designed to help the students in learning to deal with precise mathematical definitions and proofs. It is to be expected that at many institutions the text would be suitable for a junior, senior, or graduate course in analysis, since it does assume a considerable technical facility with elementary mathematics as well as an affinity for mathematical thought.

The presentation differs from that found in existing texts in two ways. First, a concerted effort is made to keep the introductions to real and complex analysis close together. These subjects have been separated in the curriculum for a number of years, thus tending to delay the introduction to complex function theory. Second, the generalizations beyond Rn are presented for subsets of normed linear spaces rather than for metric spaces. The pedagogical advantage of this is that the original material can be developed on the familiar terrain of Euclidean space and then simply observed to be largely valid for normed linear spaces, where the symbolism is just like that of Rn. The students are prepared for the generalization in much the same way that high-school algebra prepares one for manipulation in a commutative ring with identity.

The first semester covers the bulk of the first five chapters. It emphasizes the Four C's: completeness, convergence, compactness, and continuity. The basic results are presented for subsets of and functions on Euclidean space of n dimensions. This presentation includes (of course) a rigorous review of the intellectual skeleton of calculus, placing greater emphasis on power series expansions than one normally can in a calculus course. The discussion proceeds (in Chapter 5) into complex power series and an introduction to the theory of complex-analytic functions. The review of linear geometry in Section 1.6 is usually omitted from the formal structure of the first semester. The instructor who is pressed for time or who is predisposed to separate real and complex analysis may also omit all or part of Sections 5.5-5.10 on analytic functions and Fourier series without interrupting the flow of the remainder of the text.

The second semester begins with Chapter 6. It reviews the main results of the first semester, the review being carried out in the context of (subsets of and functions on) normed linear spaces. The author has found that the student is readily able to absorb the fact that many of the arguments he or she has been exposed to are formal and are therefore valid in the more general context. It is then emphasized that two of the most crucial results from the first semester — the completeness of Rn and the Heine-Borel theorem — depend on finite-dimensionality. This leads naturally to a discussion of (i) complete (Banach) spaces, the Baire category theorem and fixed points of contractions, and (ii) compact subsets of various normed linear spaces, in particular, equicontinuity and Ascoli's theorem . From there the course moves to the Lebesgue integral on Rn, which is developed by completing the space of continuous functions of compact support. Most of the basic properties of integral and measure are discussed, and a short presentation of orthogonal expansions (especially Fourier series) is included. The final chapter of the notes deals with differentiable maps on Rn, the implicit and inverse function theorems, and the change of variable theorem. This chapter may be presented earlier if the instructor finds it desirable, since the only dependence on Lebesgue integration is the proof of the change of variable theorem.

A few final remarks. Some mathematicians will look at these notes and say, "How can you teach an introductory course in analysis which never mentions partial differential equations or calculus of variations?" Others will ask, "How can you teach a basic course in analysis which devotes so little attention to applications, either to mathematics or to other fields of science?" The answer is that there is no such thing as the introductory course in analysis. The subject is too large and too important to allow for that. The three most viable foci for organization of an introductory course seem to be (i) emphasis on general concepts and principles, (ii) emphasis on hard mathematical analysis (the source of the general ideas), and (iii) emphasis on applications to science and engineering. This text was developed for the first type of course. It can be very valuable for a certain category of students, principally the students going on to graduate school in mathematics, physics, or (abstract) electrical engineering, etc. It is not, and was not intended to be, right for all students who may need some advanced calculus cir analysis beyond the elementary level.

Thanks are due to many people who have contributed to the development of this text over the last eight years. Colleagues too numerous to mention used the classroom notes and pointed out errors or suggested improvements. Three must be singled out: Steven Minsker, David Ragozin, and Donald Wilken. Each of them assisted the author in improving the notes and managing the pedagogical affairs of the M.I.T. course. I am especially grateful to David Ragozin, who wrote an intermediate version of the chapter on Lebesgue integration. I am indebted to Mrs. Sophia Koulouras, who typed the original notes, and to Miss Viola Wiley, who typed the revision and the final manuscript. Finally, my thanks to Art Wester and the staff of Prentice-Hall, Inc.

Kenneth Hoffman

Preface to the Student

This textbook will introduce you to many of the general principles of mathematical analysis. It assumes that you have a mathematical background which includes a solid course (at least one year) in the calculus of functions of one and several variables, as well as a short course in differential equations. It would be helpful if you have been exposed to introductory linear algebra since many of the exercises and examples involve matrices. The material necessary for following these exercises and examples is summarized in Section 1.6, but a linear algebra background is not essential for reading the book since it does not enter into the logical development in the text until Chapter 6.

You will meet a large number of concepts which are new to you, and you will be challenged to understand their precise definitions, some of their uses, and their general significance. In order to understand the meaning of this in quantitative terms, thumb through the Index and see how many of the terms listed there you can describe precisely. But it is the qualitative impact of the definitions which will loom largest in your experience with this book. You may find that you are having difficulty following the "proofs" presented in the book or even in understanding what a "proof" is. When this happens, look to the definitions because the chances are that your real difficulty lies in the fact that you have only a hazy understanding of the definitions of basic concepts or are suffering from a lack of familiarity with definitions which mean exactly what they say, nothing less and nothing more.

You will also learn a lot of rich and beautiful mathematics. To make the learning task more manageable, the notes have been provided with supplementary material and mechanisms which you should utilize:

1. Appendix: Note that the text proper is followed by an Appendix which discusses sets, functions, and a bit about cardinality (finite, infinite, countable, and uncountable sets). Read the first part on sets and functions and then refer to the remainder when it comes up in the notes.

2. Bibliography: There is a short bibliography to which you might turn if you're having trouble or want to go beyond the notes.

3. List of Symbols: If a symbol occurs in the notes which you don't recognize, try this list.

4. Index: The Index is fairly extensive and can lead you to various places where a given concept or result is discussed.

One last thing. Use the Exercises to test your understanding. Most of them come with specific instructions or questions, "Find this", "Prove that", "True or false?". Occasionally an exercise will come without instructions and will be a simple declarative sentence, "Every differentiable function is continuous". Such statements are to be proved. Their occurrence reflects nothing more than the author's attempt to break the monotony of saying, "Prove that ..." over and over again. The exercises marked with an asterisk are (usually) extremely difficult. Don't be discouraged if some of the ones without asterisks stump you. A few of them were significant mathematical discoveries not so long ago.

KENNETH HOFFMAN

(Continues…)

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Excerpted from "Analysis In Euclidean Space"
by .
Copyright © 2003 Kenneth Hoffman.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Preface
Preface to the Student
1. Numbers and Geometry
2. Convergence and Compactness
3. Continuity
4. Calculus Revisited
5. Sequences and Functions
6. Normed Linear Spaces
7. The Lebesgue Integral
8. Differentiable Mappings
Appendix: Elementary Set Theory
List of Symbols
Bibliography
Index