174 Graduate Texts in Mathematics S. Axler F.W. Gehring K.A. Ribet Springer Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to 33 HIRSCH. Differential Topology. Axiomatic Set Theory. 2nd ed. 34 SprrzER. Principles of Random Walk. 2 OXTOBY. Measure and Category. 2nd ed. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 35 WERMER. Banach Algebras and Several 4 HILTON/STAMMBACH. A Course in Complex Variables. 2nd ed. Homological Algebra. 2nd ed. 36 KELLEY/NAMIOKA et al. Linear 5 MAC LANE. Categories for the Working Topological Spaces. Mathematician. 37 MONK. Mathematical Logic. 6 HUGHES/PIPER. Projective Planes. 38 GRAUERT/FRITZSCHE. Several Complex 7 SERRE. A Course in Arithmetic. Variables. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C*-Algebras. 9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENY/SNELL/KNAPP. Denumerable and Representation Theory. Markov Chains. 2nd ed. 10 COHEN. A Course in Simple Homotopy 41 APOSTOL. Modular Functions and Theory. Dirichlet Series in Number Theory. 11 CONWAY. Functions of One Complex 2nded. Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite 12 BEALS. Advanced Mathematical Analysis. Groups. 13 ANDERSON/FULLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous of Modules. 2nd ed. Functions. 14 GoLUBiTSKY/GunjJEMlN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry. and Their Singularities. 45 LOBVE. Probability Theory I. 4th ed. 15 BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed. Analysis and Operator Theory. 47 MOKE. Geometric Topology in 16 WINTER. The Structure of Fields. Dimensions 2 and 3. 17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHS/WU. General Relativity for 18 HALMOS. Measure Theory. Mathematicians. 19 HALMOS. A Hilbert Space Problem Book. 49 GRUENBERG/WEIR. Linear Geometry. 2nd ed. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem. 21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential 22 BARNES/MACK. An Algebraic Introduction Geometry. to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry. 23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic. 24 HOLMES. Geometric Functional Analysis 54 GRAVER/WATKINS. Combinatorics with and Its Applications. Emphasis on the Theory of Graphs. 25 HEWITT/STROMBERG. Real and Abstract 55 BROWN/PEARCY. Introduction to Operator Analysis. Theory I: Elements of Functional 26 MANES. Algebraic Theories. Analysis. 27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An 28 ZARISKI/SAMUEL. Commutative Algebra. Introduction. Vol.1. 57 CROWELL/FOX. Introduction to Knot 29 ZARISKI/SAMUEL. Commutative Algebra. Theory. Vol.H. 58 KOBLTTZ. p-adic Numbers, p-adic 30 JACOBSON. Lectures in Abstract Algebra I. Analysis, and Zeta-Functions. 2nd ed. Basic Concepts. 59 LANG. Cyclotomic Fields. 31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in II. Linear Algebra. Classical Mechanics. 2nd ed. 32 JACOBSON. Lectures in Abstract Algebra m Theory of Fields and Galois Theory. Douglas S. Bridges Foundations of Real and Abstract Analysis Springer Douglas S. Bridges Department of Mathematics University of Waikato Hamilton New Zealand S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Department of Mathematics San Francisco State East Hall University of California University University of Michigan at Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720 USA USA USA Mathematics Subject Classification (1991): 26-01, 28-01, 46Axx, 46-01 Library of Congress Cataloging-in-Publication Data Bridges, D.S. (Douglas S.), Foundations of real and abstract analysis / Douglas S. Bridges. p. cm. — (Graduate texts in mathematics ; 174) Includes index. ISBN 0-387-98239-6 (hardcover : alk. paper) 1. Mathematical analysis. I. Title. II. Series. QA300.B69 1997 515-dc21 97-10649 © 1998 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf- ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-387-98239-6 Springer-Verlag New York Berlin Heidelberg SPIN 10524519 Dedicated to the memory of my parents: Douglas McDonald Bridges and Allison Hogg Sweet Analytics, ’tis thou hast ravished me. Faustus (Marlowe) The stone which the builders refused is become the head stone of the corner. Psalm cxviii, 22. ...from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. The origin of species (Darwin) Preface The core of this book, Chapters 3 through 5, presents a course on metric, normed,andHilbertspacesatthesenior/graduatelevel.Themotivationfor each of these chapters is the generalisation of a particular attribute of the EuclideanspaceRn:inChapter3,thatattributeisdistance;inChapter4, length; and in Chapter 5,inner product. In addition to thestandard topics that, arguably, should form part of the armoury of any graduate student in mathematics, physics, mathematical economics, theoretical statistics,..., this part of the book contains many results and exercises that are seldom found in texts on analysis at this level. Examples of the latter are Wong’s Theorem(3.3.12)showingthattheLebesguecoveringpropertyisequivalent to the uniform continuity property, and Motzkin’s result (5.2.2) that a nonempty closed subset of Euclidean space has the unique closest point property if and only if it is convex. Thesadrealitytodayisthat,perceivingthemasoneoftheharderparts oftheirmathematicalstudies,studentscontrivetoavoidanalysiscoursesat almost any cost, in particular that of their own educational and technical deprivation. Many universities have at times capitulated to the negative demand of students for analysis courses and have seriously watered down their expectations of students in that area. As a result, mathematics ma- jors are graduating, sometimes with high honours, with little exposure to anything but a rudimentary course or two on real and complex analysis, often without even an introduction to the Lebesgue integral. For that reason, and also in order to provide a reference for material that is used in later chapters, I chose to begin this book with a long chapter providing a fast–paced course of real analysis, covering conver- x Preface gence of sequences and series, continuity, differentiability, and (Riemann and Riemann–Stieltjes) integration. The inclusion of that chapter means that the prerequisite for the book is reduced to the usual undergraduate sequence of courses on calculus. (One–variable calculus would suffice, in theory, but a lack of exposureto moreadvanced calculuscourseswould in- dicate a lack of the mathematical maturity that is the hidden prerequisite for most senior/graduate courses.) Chapter 2 is designed to show that the subject of differentiation does not end with the material taught in calculus courses, and to introduce the Lebesgue integral. Starting with the Vitali Covering Theorem, the chap- terdevelopsatheoryofdifferentiation almost everywhere thatunderpinsa beautifulapproachtotheLebesgueintegralduetoF.Riesz[39].Oneminor disadvantage of Riesz’s approach is that, in order to handle multivariate integrals,itrequiresthetheoryofset–valuedderivatives,atopicsufficiently involved and far from my intended route through elementary analysis that I chose to omit it altogether. The only place where this might be regarded as a serious omission is at the end of the chapter on Hilbert space, where I require classical vector integration to investigate the existence of weak solutions to the Dirichlet Problem in three–dimensional Euclidean space; since that investigation is only outlined, it seemed justifiable to rely only on the reader’s presumed acquaintance with elementary vector calculus. Certainly, one–dimensional integration is all that is needed for a sound in- troductiontotheL spacesoffunctionalanalysis,whichappearinChapter p 4. Chapters 1 and 2 form Part I (Real Analysis) of the book; Part II (Ab- stract Analysis) comprises the remaining chapters and the appendices. I have already summarised the material covered in Chapters 3 through 5. Chapter 6, the final one, introduces functional analysis, starting with the Hahn–Banach Theorem and the consequent separation theorems. As well as the common elementary applications of the Hahn–Banach Theorem, I have included some deeper ones in duality theory. The chapter ends with the Baire Category Theorem, the Open Mapping Theorem, and their con- sequences.Heremostoftheapplicationsarestandard,althoughoneortwo unusual ones are included as exercises. The book has a preliminary section dealing with background material needed in the main text, and three appendices. The first appendix de- scribes Bishop’s construction of the real number line and the subsequent development of its basic algebraic and order properties; the second deals briefly with axioms of choice and Zorn’s Lemma; and the third shows how someofthematerialinthechapters—inparticular,Minkowski’sSeparation Theorem—can be used in the theory of Pareto optimality and competitive equilibria in mathematical economics. Part of my motivation in writing AppendixCwastoindicatethat“mathematicaleconomics”isafardeeper subjectthanissuggestedbytheundergraduatetextsoncalculusandlinear algebra that are published under that title. Preface xi I have tried, wherever possible, to present proofs so that they translate mutatis mutandis into their counterparts in a more abstract setting, such as that of a metric space (for results in Chapter 1) or a topological space (for results in Chapter 3). On the other hand, some results first appear as exercises in one context before reappearing as theorems in another: one exampleofthisistheUniformContinuityTheorem,whichfirstappearsas1 Exercise (1.4.8:8) in the context of a compact interval of R, and which is provedlater,asCorollary(3.3.13),inthemoregeneralsettingofacompact metric space. I hope that this procedure of double exposure will enable students to grasp the material more firmly. The text covers just over 300 pages, but the book is, in a sense, much larger,sinceitcontainsnearly750exercises,whichcanbeclassifiedintoat least the following, not necessarily exclusive, types: • applications and extensions of the main propositions and theorems; • results that fill in gaps in proofs or that prepare for proofs later in the book; • pointers towards new branches of the subject; • deep and difficult challenges for the very best students. The instructor will have a wide choice of exercises to set the students as assignmentsortestquestions.Whicheveronesareset,aswiththelearning of any branch of mathematics it is essential that the student attempt as many exercises as the constraints of time, energy, and ability permit. It is important for the instructor/student to realise that many of the exercises—especially in Chapters 1 and 2—deal with results, sometimes major ones, that are needed later in the book. Such an exercise may not clearly identify itself when it first appears; if it is not attempted then, it will provide revision and reinforcement of that material when the student needs to tackle it later. It would have been unreasonable of me to have includedmajorresultsasexerciseswithoutsomeguidelinesforthesolution of the nonroutine ones; in fact, a significant proportion of the exercises of all types come with some such guideline, even if only a hint. AlthoughChapters3through6makenumerousreferencestoChapters1 and2,Ihavetriedtomakeiteasyforthereadertotacklethelaterchapters without ploughing through the first two. In this way the book can be used as a text for a semester course on metric, normed, and Hilbert spaces. (If 1AreferenceoftheformProposition(a.b.c)istoPropositionc inSectionb of Chaptera;onetoExercise(a.b.c:d)istothedthexerciseinthesetofexercises with reference number (a.b.c); and one to (B3) is to the 3rd result in Appendix B. Within each section, displays that require reference indicators are numbered in sequence: (1),(2),.... The counter for this numbering is reset at the start of a new section.