[PDF] A Guide to Advanced Real Analysis

View - Download A Guide to Advanced Real Analysis

Previous PDF

Next PDF

“bevbook" — 2010/12/8 — 16:35 — page i — #1

A Guide

to

Advanced Real Analysis

"bevbook" - 2011/2/15 - 16:16 - page ii - #2 c?2009 by The Mathematical Association of America (Incorporated)

Library of CongressCatalog Card Number 2009927192

Print Edition ISBN 978-0-88385-343-6

Electronic Edition ISBN 978-0-88385-915-5

Printed in the United States of America

Current Printing (last digit):

10 9 8 7 6 5 4 3 2 1

“bevbook" — 2010/12/8 — 16:35 — page iii — #3

The Dolciani Mathematical Expositions

NUMBER THIRTY-SEVEN

MAA Guides # 2

A Guide

to

Advanced Real Analysis

Gerald B. Folland

University of Washington

Published and Distributed by

The Mathematical Association of America

“bevbook" — 2010/12/8 — 16:35 — page iv — #4

DOLCIANI MATHEMATICAL EXPOSITIONS

Committee on Books

Paul Zorn,Chair

Dolciani Mathematical Expositions Editorial Board

Underwood Dudley,Editor

Jeremy S. Case

Rosalie A. Dance

Tevian Dray

Patricia B. Humphrey

Virginia E. Knight

Mark A. Peterson

Jonathan Rogness

Thomas Q. Sibley

Joe Alyn Stickles

“bevbook" — 2010/12/8 — 16:35 — page v — #5 The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematicsat Hunter College of the City Uni- versity of New York.In makingthe gift, ProfessorDolciani,herselfan exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. TheAssociation,for its part, wasdelightedto acceptthegraciousgestureinitiat- ing the revolving fund for this series from one who has servedthe Association with distinction, both as a member of the Committee on Publicationsand as a member of the Board of Governors. It was with genuine pleasure that theBoard chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimu- lating mathematical content. Typically, they contain an ample supply of exercises, many with accompanyingsolutions. They are intended to besufficiently elementary for the undergraduate and even the mathematically inclinedhigh-school student to understand and enjoy, but also to be interesting and sometimes challenging to the more advancedmathematician.

1.Mathematical Gems,Ross Honsberger

2.Mathematical Gems II,Ross Honsberger

3.Mathematical Morsels,Ross Honsberger

4.Mathematical Plums,Ross Honsberger (ed.)

5.Great Moments in Mathematics(Before 1650), Howard Eves

6.Maxima and Minima without Calculus,Ivan Niven

7.Great Moments in Mathematics(After 1650), Howard Eves

8.Map Coloring, Polyhedra, and the Four-Color Problem,David Barnette

9.Mathematical Gems III,Ross Honsberger

10.More Mathematical Morsels,Ross Honsberger

11.Old and New Unsolved Problems in Plane Geometry and Number Theory,

Victor Klee and Stan Wagon

12.Problems for Mathematicians, Young and Old,Paul R. Halmos

13.Excursionsin Calculus:AnInterplayoftheContinuousandtheDiscrete,Robert

M. Young

14.The Wohascum County Problem Book,George T. Gilbert, Mark Krusemeyer,

and Loren C. Larson

15.Lion Hunting and Other Mathematical Pursuits: A Collectionof Mathematics,

Verse, and Stories by Ralph P. Boas, Jr.,edited by Gerald L. Alexandersonand

Dale H. Mugler

16.Linear Algebra Problem Book,Paul R. Halmos

17.From Erdosto Kiev: Problems of Olympiad Caliber,Ross Honsberger

“bevbook" — 2010/12/8 — 16:35 — page vi — #6

18.Which Way Did the BicycleGo? ...and Other Intriguing Mathematical Myster-

ies,JosephD. E. Konhauser,Dan Velleman, and Stan Wagon

19.In P´olya"s Footsteps: MiscellaneousProblems and Essays,Ross Honsberger

20.Diophantusand Diophantine Equations,I. G. Bashmakova(Updated by Joseph

Silverman and translated by Abe Shenitzer)

21.Logic as Algebra,Paul Halmos and Steven Givant

22.Euler: The Master of Us All,William Dunham

23.TheBeginningsandEvolutionofAlgebra,I. G. BashmakovaandG.S. Smirnova

(Translated by Abe Shenitzer)

24.Mathematical Chestnuts from Around the World,Ross Honsberger

25.Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures,

Jack E. Graver

26.Mathematical Diamonds,Ross Honsberger

27.Proofs that Really Count: The Art of Combinatorial Proof,Arthur T. Benjamin

and Jennifer J. Quinn

28.Mathematical Delights,Ross Honsberger

29.Conics,Keith Kendig

30.Hesiod"s Anvil: falling and spinning through heaven and earth,Andrew J.

Simoson

31.A Gardenof Integrals,Frank E. Burk

32.A Guide to Complex Variables(MAA Guides #1), Steven G. Krantz

33.Sink or Float? Thought Problemsin Math and Physics, Keith Kendig

34.Biscuits of NumberTheory, Arthur T. Benjamin and Ezra Brown

35.Uncommon Mathematical Excursions: Polynomia and Related Realms, Dan

Kalman

36.When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B.

Nelsen

37.A Guide to AdvancedReal Analysis(MAA Guides #2), Gerald B. Folland

MAA Service Center

P.O. Box 91112

Washington, DC 20090-1112

1-800-331-1MAA FAX: 1-301-206-9789

“bevbook" — 2010/12/8 — 16:35 — page vii — #7

Preface

The term “real analysis" refers, in the first place, to the classical theory of functions of one and several real variables: limits and continuity, differen- tiation, the Riemann integral, infinite series, and relatedtopics. However, it has come to encompass some theories of a more abstract naturethat have extended the ideas of real-variable theory to much more general settings, a development whichinturnhas shednew lightonconcrete, “classical" prob- lems. This more advanced part of real analysis is the subjectof the present book. This book is addressed, therefore, to people who are alreadyfamiliar with classical real-variable theory. (Many books are available on that sub- ject; the old classic is Rudin [16], and the most engaging of the recent ones is K¨orner [10]. In addition, an MAA Guide to it by Steven Krantz [11] is appearing along with this one.) In accordance with the philosophy of the MAA Guides, my aim is to give an account of the subject withinabrief text that will provide an overview for the novice and a refresher for those who have already studiedit. Essential definitions,major theorems, and key ideas of proofs are included; technical details are not. Thus, most of the formally stated results in the book are followed by sketches of proofswhose de- gree of completeness varies widely. The results for which little or no proof is provided fall into two categories, which are distinguished by the labels “Proposition" and “Theorem." If the result is called a proposition,its proof is easy, and the reader is encouraged to try it as an exercise.If it is called a theorem, its proof is long and not susceptible to condensation into a short sketch. Of course, this presentation works only if the reader has a resource for filling in the gaps. I take my own book [6] as a standard reference for a more complete account of the material in this book, simply because I am most familiar with it. All the results stated here are provedin [6] except those for which an explicit reference is given to some other source. Lang vii “bevbook" — 2010/12/8 — 16:35 — page viii — #8 viiiPreface [12], Royden [15], and Rudin [17] are other books that cover most of the same material. This book is not, however, merely a condensed version of [6].One of the main problems for a textbook writer, as for a novelist or historian, is to figure out a way of turning a body of material whose parts have many in- terconnections into a linear narrative, and the solutionisgenerally far from unique. I have taken the opportunityafforded bythe nature of MAA Guides to arrange the topics in a quite different way than I did in [6]. Perhaps read- ers who examine both texts will gain something from comparing the two perspectives.

Gerald B. Folland

Seattle, April 2009

“bevbook" — 2010/12/8 — 16:35 — page ix — #9

Contents

Preface. .... ... .... .... .... .... ... .... .... .... ... .... .... .... ... vii Prologue: Notation, Terminology, and Set Theory ... .... .... ... .. 1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . 2 Zorn"s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Topology... .... ... .... .... .... .... ... .... .... .... ... .... .... 5

1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Topological spaces and continuousmaps . . . . . . . . . . 9

1.3 Neighborhoodbases and convergence . . . . . . . . . . . . 13

1.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Measure and Integration: General Theory. .... .... ... .... .... 21

2.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Convergence of functions and convergence of integrals .. . 31

2.4 Product measures and the Fubini-Tonellitheorem . . . . . .34

2.5 Relations between (signed and complex) measures . . . . . 36

3 Measure and Integration: Constructions and Special Examples 41

3.1 Constructionof measures . . . . . . . . . . . . . . . . . . 41

3.2 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Regular Borel measures and functions on the real line . . .52

3.4 Hausdorff measure . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Regular Borel measures on LCH spaces . . . . . . . . . . . 59

4 Rudiments of Functional Analysis..... .... ... .... .... .... .... 63

4.1 Normed vector spaces and bounded linear maps . . . . . . . 63

4.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Other topological vector spaces . . . . . . . . . . . . . . . 71

ix “bevbook" — 2010/12/8 — 16:35 — page x — #10 xContents

5 Function Spaces. ... .... .... ... .... .... .... ... .... .... .... ... 75

5.1Lpspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Spaces of continuousfunctions . . . . . . . . . . . . . . . 80

6 Topics in Analysis on Euclidean Space. .... ... .... .... .... .... 85

6.1 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2 Fourier series and transforms . . . . . . . . . . . . . . . . . 89

6.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 94

Bibliography .. .... .... ... .... .... .... ... .... .... .... ... .... .... 101 Index. ... .... ... .... .... .... ... .... .... .... .... ... .... .... .... ..103 About the Author. .... ... .... .... .... .... ... .... .... .... ... .... . 107 “bevbook" — 2010/12/8 — 16:35 — page 1 — #11

Prologue

Notation, Terminology, and Set Theory

In this prologue we set the stage by briefly discussing some points of no- tation and terminology and a few facts from set theory that will be used throughout the book.

Numbers

We set

NDthe set of positive integers,

ZDthe set of integers,

RDthe set of real numbers,

CDthe set of complex numbers.

We often enlarge the real number system by adjoining two “elements at infinity,"1(also calledC1for emphasis) and?1. In the extended systemR[ f1g DOE?1;1?, every setEhas a least upper bound or supremumand a greatest lower bound orinfimum, denoted respectively by supEand infE. Moreover, every infiniteseries with nonnegativeterms has a well-defined sum inOE0;1?, namely, the supremum of its partial sums. IfzDxCiyis a complex number, its complex conjugatex?iy is denoted by z, and its absolute value or moduluspzzDpx2Cy2is denoted byjzj. The spaces of orderedn-tuples of real or complex numbers are denoted byRnandCn. IfuD.u1;:::;un/belongs toRnorCn, we denote its

Euclidean norm byjuj:

juj D? nX

1jujj2?

1=2 1 “bevbook" — 2010/12/8 — 16:35 — page 2 — #12

2Prologue

We also define the dot product of two elementsu;vofRnby u?vDnX 1u jvj:

Sets and mappings

We employ standard notation from set theory. The set inclusion sign?is interpretedin the wide sense; that is, the conditionE?Fallowsthe possi- bilitythatEDF. We denote the relative complement ofFinEbyEnF:

EnFD°x2EWx...F?:

We denote the empty set by¿. A familyfE˛g˛2Aof sets is calleddisjoint When it is understood that we are considering subsets of a fixed setX, we may speak simply of the complement of a setE(inX): E cDXnE: In this situation we haveDe Morgan"s laws: IffE˛g˛2Ais a collection of subsets ofX, then

˛2AE

c D\

˛2AE

c˛;?\

˛2AE

c D[

˛2AE

c˛: We denote the collectionof all subsets ofX(includingXand¿) byP.X/. SupposeXandYare nonemptysets. In strict set-theoretic terms, amap ormappingfromXtoYis a collectionfof ordered pairs.x;y/with x2Xandy2Y, such that for eachx2Xthere is a uniquey2Y (denoted byf.x/) with.x;y/2f. (Of course, in more informal terms, we usually think of a map as a “rule" that assigns to eachx2Xan element f.x/ofY.) A mapfWX!Yis calledinjectiveiff.x1/Df.x2/only whenx1Dx2,surjectiveifff.x/Wx2Xg DY, andbijectiveif it is both injective and surjective. When we wish to describe a map without giving it a name, we use the notationx7!yto indicate thatyis the image ofx under the map; for example, the squaring functiononRisx7!x2. Each mapfWX!Yinduces a map, still denoted byf, fromP.X/ toP.Y /, f.E/D°f.x/Wx2E?; as well as a map denoted byf?1fromP.Y /toP.X/: f ?1.E/D°xWf.x/2E?: “bevbook" — 2010/12/8 — 16:35 — page 3 — #13

Prologue3

It is an important fact that the inverse-image mapf?1WP.Y /!P.X/ preserves unions, intersections, and complements: f ?1?[

˛2AE

D[

˛2Af

?1.E˛/; f?1?\

˛2AE

D\

˛˛2Af

?1.E˛/; f ?1.Ec/DOEf?1.E/?c: (The direct-image mapfWP.X/!P.Y /preserves unions, but it fails to preserve intersections whenfis not injective, and it fails to preserve complements whenfis not bijective.) LetfX˛g˛2Abe an indexed collection of sets. TheCartesian product of the setsX˛, denoted byQ

˛2AX˛, is the set of all mapsffromAintoS

˛2AX˛such thatf.˛/2X˛for all˛:

Y

˛2AX

˛D?

fWA![

˛2AX

˛Wf.˛/2X˛for all˛2A?

IfXDQ ˛2AX˛and˛2A, the˛thcoordinate map?˛WX!X˛is defined by?˛.f /Df.˛/; we often writexandx˛instead offandf.˛/ and callx˛the˛th coordinate ofx.

Zorn"s lemma

Every so often, especially when one is working in a very general context, one needs a theorem asserting the existence of some mathematical object but has no way of producingit by explicit construction.Often the stratagem needed to resolve the question is one of a group of related principles of general set theorypertainingtopartiallyorderedsets. Hereare thenecessary definitions. Apartiallyordered setis a setXequipped witha binaryrelation?with the followingproperties: i. Ifx?yandy?zthenx?z. ii. Ifx?yandy?x, thenxDy. iii.x?xfor allx. A partially ordered setXis calledlinearly orderedif it also satisfies iv. Ifx;y2X, then eitherx?yory?x. “bevbook" — 2010/12/8 — 16:35 — page 4 — #14

4Prologue

Amaximal elementof a partially ordered set is an elementxsuch that the only elementywithx?yisxitself. For example,Ris linearly ordered by the usual ordering?, and for any setS,P.S/is partially ordered by the inclusion relation?. IfXis the collection of all proper subsets ofS, partially ordered by inclusion, its maximal elements are the subsets whose complement consistsof a single point. The collection of all finite subsets of an infinite set has no maximal elements. The general existence principle most often invoked is knownasZorn"s lemma: Zorn"s lemma.IfXis a partially ordered set and every linearly ordered subsetLofXhas an upper bound (i.e., an elementx2Xsuch thaty?x for ally2L), thenXhas a maximal element. An alternative formulation, known as theHausdorff maximal principle, is thatevery partially ordered set has a maximal linearly ordered subset. (Indeed, an upper bound for a maximal linearly ordered subset ofXis a maximal element ofX. On the other hand, an application of Zorn"s lemma to the collectionof linearly ordered subsets ofX, which is partiallyordered by inclusion, yields a maximal linearly ordered subset.) Another general existence principle is theaxiom of choice, which says that iffX˛g˛2Ais a nonempty collection of nonempty sets, one can form a new setYby picking one element from eachX˛. Since the range of any element of the Cartesian productQ

˛2AX˛is such a set, one can state the

axiom of choice as follows: The axiom of choice.The Cartesian product of any nonempty collection of nonempty sets is nonempty. Zorn"slemma impliestheaxiomofchoice. (ConsiderthecollectionFof all mappingsffrom subsets ofAintoS

˛2AX˛such thatf.˛/2X˛for

all˛in the domain off, which is partially ordered by extension:f?g if dom.g/?dom.f /andgjdom.f /Df.) One can also prove Zorn"s lemma from the axiom of choice, but the argument is more involved. (For this and related matters, a good reference is Halmos [7].) Neither of these principles can be deduced from the other standard axioms of set theory. The use of nonconstructive existence principles such as Zorn"s lemma has not been without controversy. However, most mathematicians take the attitude that they are perfectly legitimate, while recognizing that construc- tive methods tend to be more informative when they are available. “bevbook" — 2010/12/8 — 16:35 — page 5 — #15

CHAPTER1

Topology

The subject of this chapter ispoint-set topologyorgeneral topology, the abstract mathematical framework for the studyof limits,continuity,and thequotesdbs_dbs20.pdfusesText_26

[PDF] advanced regular expression in python

[PDF] advanced research methodology pdf

[PDF] advanced search app for android

[PDF] advanced segmentation adobe analytics

[PDF] advanced setting app for android

[PDF] advanced shell scripting pdf

[PDF] advanced spoken english lessons pdf

[PDF] advanced sql book reddit

[PDF] advanced sql books

[PDF] advanced sql books free download

[PDF] advanced sql commands with examples pdf

[PDF] advanced sql injection cheat sheet

[PDF] advanced sql join examples

[PDF] advanced sql notes pdf

[PDF] advanced sql pdf