Topology, Calculus and Approximation

383 Pages·2017·2.101 MB·English

by  Vilmos Komornik

Springer Undergraduate Mathematics Series
Vilmos Komornik
Topology,
Calculus and
Approximation
Springer Undergraduate Mathematics Series
AdvisoryBoard
M.A.J.Chaplain,UniversityofSt.Andrews,St.Andrews,Scotland,UK
A.MacIntyre,QueenMaryUniversityofLondon,London,England,UK
S.Scott,King’sCollegeLondon,London,England,UK
N.Snashall,UniversityofLeicester,Leicester,England,UK
E.Süli,UniversityofOxford,Oxford,England,UK
M.R.Tehranchi,UniversityofCambridge,Cambridge,England,UK
J.F.Toland,UniversityofCambridge,Cambridge,England,UK
Moreinformationaboutthisseriesathttp://www.springer.com/series/3423
Vilmos Komornik
Topology, Calculus
and Approximation
123
VilmosKomornik
DepartmentofMathematics
UniversityofStrasbourg
Strasbourg,France
TranslationfromtheFrenchlanguageedition:
Précisd’analyseréelle-Topologie-Calculdifférentiel-Méthodesd’approximations,vol-1
byVilmosKomornik
Copyright©2001EditionMarketingS.A.
www.editions-ellipses.fr/
AllRightsReserved.
ISSN1615-2085 ISSN2197-4144 (electronic)
SpringerUndergraduateMathematicsSeries
ISBN978-1-4471-7315-1 ISBN978-1-4471-7316-8 (eBook)
DOI10.1007/978-1-4471-7316-8
LibraryofCongressControlNumber:2017935974
MathematicsSubjectClassification(2010):41-01,41A05,41A10,41A15,41A50
©Springer-VerlagLondonLtd.2017
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ForGod’ssake, Ibeseech you,giveit up.
Fearitno lessthansensualpassionsbecause
ittoomaytakeallyour timeanddepriveyou
ofyour health,peaceofmindand happiness
inlife...
LetterofFarkas Bolyai tohis son,April4,
1820.
Icreated a new,differentworldout of
nothing...
LetterofJános Bolyaito hisfather,
November2, 1823.
Preface
Thistextbookcontainsthelecturenotesofthreeone-semestercoursesgivenbythe
author to third year students at the University of Strasbourg. We assume that the
readerisfamiliarwiththecalculusofonerealvariable.ThefirstpartonTopologyis
usedeverywhereinthesequel.ThefollowingtwopartsonDifferentialcalculusand
Approximationmethodsarelogicallyindependent.
We have made much effort to select the material covered by the lectures, to
formulateaestheticalandgeneralstatements,toseekshortandelegantproofs,and
to illustrate the results with simple but pertinent examples. (See also the remarks
onp.369.)OurworkisstronglyinfluencedbythebeautifullecturesofProfessors
ÁkosCsászár andLászlóCzáchduringthe1970sattheEötvösLorándUniversity
inBudapest,andmoregenerallybythemathematicaltraditioncreatedbyLeopold
Fejér,FrédéricRiesz,PaulTurán,PaulErdo˝sandothers.
Onp.337wecitemanypapersofhistoricalimportance,indicatingtheoriginof
mostofthenotionsandtheoremstreatedhere.Theyoftencontaindifferentversions
ofthetheoremswetreat,illustratingthegenesisofmathematicalinterest.
Wesuggestthat,onthefirstreading,thereadershouldskipthematerialmarked
by (cid:2). At the end of each chapter we give some exercises. However, the most
important exercises are incorporated into the text as examples and remarks, and
thereaderisexpectedtofillinthemissingdetails.
Welistonp.ixsomebooksofgeneralmathematicalinterest.
We thank Á. Besenyei, C. Baud, L. Czách, C. Disdier, D. Dumont, J. Gerner,
P. Loreti, C.-M. Marle, P. Martinez, M. Mehrenberger, P.P. Pálfy, M. Pedicini, P.
Pilibossian, J. Saint Jean Paulin, Z. Sebestyén, A. Simonovits, L. Simon, Mrs. B.
Szénássy,G.Szigeti, J. Vancostenoble,Zs.Votiskyandthe editorsofSpringerfor
theirprecioushelp.
ThisbookisdedicatedtothememoryofPaulErdo˝s.
Strasbourg,France VilmosKomornik
March26,2017
vii
Some Books of General Interest
1.M.Aigner,G.M.Ziegler,ProofsfromtheBook,4thed.,Springer,2010.
2.P.S.Alexandroff,ElementaryConceptsofTopology,Dover,NewYork,1961.
3.E.T.Bell,MenofMathematics,SimonandSchuster,NewYork,1965.
4.V.G.Boltyanskii,V.A.Efremovich,IntuitiveCombinatorialTopology,Springer,
2001.
5.J. Bolyai, Appendix. The Theory of Space. With Introduction, Comments, and
AddendaeditedbyF.Kárteszi,AkadémiaiKiadó,Budapest,1987.
6.D.Bressoud,ARadicalApproachtoRealAnalysis,TheMathematicalAssocia-
tionofAmerica,Washington,1994.
7.R. Courant, H. Robbins, What is Mathematics? An Elementary Approach to
IdeasandMethods,OxfordUniv.Press,1941.
8.H.S.M.Coxeter,S.L.Greitzer,GeometryRevisited,TheMathematicalAssocia-
tionofAmerica,Washington,1967.
9.W. Dunham, Journey Through Genius. The Great Theorems of Mathematics,
JohnWiley&Sons,NewYork,1990.
10.P.Erdo˝s,J.Surányi,TopicsintheTheoryofNumbers,Springer,2003.
11.E.Hairer,G.Wanner,AnalysisbyItsHistory,Springer,NewYork,1996.
12.G.H.Hardy,AMathematician’sApology,CambridgeUniv.Press,1940.
13.P.Hoffman,TheManWhoLovedOnlyNumbers,Hyperion,NewYork,1998.
14.M.Kac,S.M.Ulam,MathematicsandLogic,Dover,NewYork,1992.
15.A.Y.Khintchin,ThreePearlsofNumberTheory,Graylock,Rochester,1952.
16.T.W.Körner,FourierAnalysis,CambridgeUniv.Press,1988.
17.J.Kürschák,G.Hajós,G.Neukomm,J.Surányi,HungarianProblemBookI-IV,
Math.Assoc.ofAmerica,1963–2011.
18.M.Laczkovich,Conjectureandproof,CambridgeUniversityPress,2001.
19.J.Muir,OfMenandNumbers.TheStoryofGreatMathematicians,Dover,New
York,1996.
20.J.Newman(ed.),TheWorldofMathematicsI-IV,Dover,NewYork,2000.
21.B. Schechter, My Brain is Open: The Mathematical Journeys of Paul Erdo˝s,
Simon&Schuster,1998.
ix
x SomeBooksofGeneralInterest
22.H.Steinhaus,MathematicalSnapshots,Dover,NewYork,1999.
23.I.Stewart,The ProblemsofMathematics,OxfordUniversityPress, NewYork,
1992.
24.D.J.Struik,AConciseHistoryofMathematics,Dover,NewYork,1987.
Contents
PartI Topology
1 MetricSpaces ............................................................... 3
1.1 DefinitionsandExamples .......................................... 3
1.2 Convergence,LimitsandContinuity............................... 8
1.3 Completeness:AFixedPointTheorem............................ 14
1.4 Compactness........................................................ 23
1.5 Exercises ............................................................ 30
2 TopologicalSpaces.......................................................... 37
2.1 DefinitionsandExamples .......................................... 37
2.2 Neighborhoods:ContinuousFunctions............................ 43
2.3 Connectedness ...................................................... 46
2.4 *Compactness...................................................... 50
2.5 *ConvergenceofNets.............................................. 55
2.6 Exercises ............................................................ 61
3 NormedSpaces.............................................................. 65
3.1 DefinitionsandExamples .......................................... 65
3.2 MetricandTopologicalProperties................................. 73
3.3 Finite-DimensionalNormedSpaces ............................... 78
3.4 ContinuousLinearMaps ........................................... 82
3.5 ContinuousLinearFunctionals..................................... 85
3.6 ComplexNormedSpaces........................................... 88
3.7 Exercises ............................................................ 89
PartII DifferentialCalculus
4 TheDerivative .............................................................. 97
4.1 DefinitionsandElementaryProperties............................. 97
4.2 MeanValueTheorems.............................................. 105
4.3 TheFunctionsRm ,!Rn........................................... 111
4.4 Exercises ............................................................ 115
xi