A Guide to Functional Analysis

150 Pages·2013·1.405 MB·English

by  Krantz S.G.

A Guide
to
Functional Analysis
(cid:13)c 2013by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber2013935093
PrintEditionISBN978-0-88385-357-3
ElectronicEditionISBN978-1-61444-213-4
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
TheDolcianiMathematicalExpositions
NUMBERFORTY-NINE
MAAGuides#49
A Guide
to
Functional Analysis
Steven G. Krantz
Washington University in St. Louis
PublishedandDistributedby
TheMathematicalAssociationofAmerica
The DOLCIANIMATHEMATICAL EXPOSITIONS series oftheMathe-
maticalAssociationofAmericawasestablishedthroughagenerousgiftto
theAssociationfromMaryP.Dolciani,ProfessorofMathematicsatHunter
College of the City University of New York. In making the gift, Profes-
sor Dolciani,herself an exceptionallytalentedand successful expositorof
mathematics,hadthepurposeoffurtheringtheidealofexcellenceinmath-
ematicalexposition.
The Association,foritspart,wasdelightedtoaccept thegraciousges-
tureinitiatingtherevolvingfundforthisseriesfromonewhohasservedthe
Associationwithdistinction,bothas a member oftheCommitteeonPub-
licationsandasamember oftheBoardofGovernors. Itwaswithgenuine
pleasurethattheBoardchosetonametheseriesinherhonor.
Thebooksintheseriesareselectedfortheirlucidexpositorystyleand
stimulatingmathematicalcontent. Typically,theycontainanamplesupply
ofexercises, many withaccompanying solutions. They are intendedtobe
sufficientlyelementaryfortheundergraduateandeven themathematically
inclinedhigh-schoolstudentto understandandenjoy, butalso tobe inter-
estingandsometimeschallengingtothemoreadvancedmathematician.
CommitteeonBooks
FrankFarris,Chair
DolcianiMathematicalExpositionsEditorialBoard
UnderwoodDudley,Editor
JeremyS.Case
RosalieA.Dance
ChristopherDaleGoff
ThomasM.Halverson
MichaelJ.McAsey
MichaelJ.Mossinghoff
JonathanRogness
ElizabethD.Russell
RobertW.Vallin
1. MathematicalGems,RossHonsberger
2. MathematicalGemsII,RossHonsberger
3. MathematicalMorsels,RossHonsberger
4. MathematicalPlums,RossHonsberger(ed.)
5. GreatMomentsinMathematics(Before1650),HowardEves
6. MaximaandMinimawithoutCalculus,IvanNiven
7. GreatMomentsinMathematics(After1650),HowardEves
8. MapColoring,Polyhedra,andtheFour-ColorProblem,DavidBarnette
9. MathematicalGemsIII,RossHonsberger
10. MoreMathematicalMorsels,RossHonsberger
11. Old and New Unsolved Problems in Plane Geometry and Number Theory,
VictorKleeandStanWagon
12. ProblemsforMathematicians,YoungandOld,PaulR.Halmos
13. ExcursionsinCalculus:AnInterplayoftheContinuousandtheDiscrete,Robert
M.Young
14. The WohascumCountyProblem Book,GeorgeT. Gilbert, Mark Krusemeyer,
andLorenC.Larson
15. LionHuntingandOtherMathematicalPursuits:ACollectionofMathematics,
Verse,andStoriesbyRalphP.Boas,Jr.,editedbyGeraldL.Alexandersonand
DaleH.Mugler
16. LinearAlgebraProblemBook,PaulR.Halmos
17. FromErdo˝stoKiev:ProblemsofOlympiadCaliber,RossHonsberger
18. WhichWayDidtheBicycleGo?...andOtherIntriguingMathematicalMyster-
ies,JosephD.E.Konhauser,DanVelleman,andStanWagon
19. InPo´lya’sFootsteps:MiscellaneousProblemsandEssays,RossHonsberger
20. DiophantusandDiophantineEquations,I.G.Bashmakova(UpdatedbyJoseph
SilvermanandtranslatedbyAbeShenitzer)
21. LogicasAlgebra,PaulHalmosandStevenGivant
22. Euler:TheMasterofUsAll,WilliamDunham
23. TheBeginningsandEvolutionofAlgebra,I.G.BashmakovaandG.S.Smirnova
(TranslatedbyAbeShenitzer)
24. MathematicalChestnutsfromAroundtheWorld,RossHonsberger
25. CountingonFrameworks: MathematicstoAidtheDesignofRigidStructures,
JackE.Graver
26. MathematicalDiamonds,RossHonsberger
27. ProofsthatReallyCount:TheArtofCombinatorialProof,ArthurT.Benjamin
andJenniferJ.Quinn
28. MathematicalDelights,RossHonsberger
29. Conics,KeithKendig
30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J.
Simoson
31. AGardenofIntegrals,FrankE.Burk
32. AGuidetoComplexVariables(MAAGuides#1),StevenG.Krantz
33. SinkorFloat?ThoughtProblemsinMathandPhysics,KeithKendig
34. BiscuitsofNumberTheory,ArthurT.BenjaminandEzraBrown
35. Uncommon Mathematical Excursions: Polynomia and Related Realms, Dan
Kalman
36. WhenLessisMore:VisualizingBasicInequalities,ClaudiAlsinaandRogerB.
Nelsen
37. AGuidetoAdvancedRealAnalysis(MAAGuides#2),GeraldB.Folland
38. AGuidetoRealVariables(MAAGuides#3),StevenG.Krantz
39. Voltaire’s Riddle: Microme´gas and the measure of all things, Andrew J.
Simoson
40. AGuidetoTopology,(MAAGuides#4),StevenG.Krantz
41. AGuidetoElementaryNumberTheory,(MAAGuides#5),UnderwoodDudley
42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and
RogerB.Nelsen
43. MathematicsandSports,editedbyJosephA.Gallian
44. AGuidetoAdvancedLinearAlgebra,(MAAGuides#6),StevenH.Weintraub
45. IconsofMathematics:AnExplorationofTwentyKeyImages,ClaudiAlsinaand
RogerB.Nelsen
46. AGuidetoPlaneAlgebraicCurves,(MAAGuides#7),KeithKendig
47. NewHorizonsinGeometry,TomM.ApostolandMamikonA.Mnatsakanian
48. AGuidetoGroups,Rings,andFields,(MAAGuides#8),FernandoQ.Gouveˆa
49. AGuidetoFunctionalAnalysis,(MAAGuides#9),StevenG.Krantz
MAAServiceCenter
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To the memory of Stefan Banach.

Contents
Preface......................................................... xi
1 Fundamentals............................................... 1
1.1 WhatisFunctionalAnalysis? . . . . . . . . . . . . . . . . 1
1.2 NormedLinearSpaces . . . . . . . . . . . . . . . . . . . . 2
1.3 Finite-DimensionalSpaces . . . . . . . . . . . . . . . . . 5
1.4 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 TheBaireCategoryTheorem . . . . . . . . . . . . . . . . 8
1.6 TheThreeBigResults . . . . . . . . . . . . . . . . . . . . 9
1.7 ApplicationsoftheBigThree . . . . . . . . . . . . . . . . 15
2 OdetotheDualSpace....................................... 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 ConsequencesoftheHahn-BanachTheorem . . . . . . . . 29
3 HilbertSpace ............................................... 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 TheGeometryofHilbertSpace . . . . . . . . . . . . . . . 36
4 TheAlgebraofOperators.................................... 45
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 TheAlgebraofBoundedLinearOperators . . . . . . . . . 47
4.3 CompactOperators . . . . . . . . . . . . . . . . . . . . . 50
5 BanachAlgebraBasics...................................... 59
5.1 IntroductiontoBanachAlgebras . . . . . . . . . . . . . . 59
5.2 TheStructureofaBanachAlgebra . . . . . . . . . . . . . 63
5.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.4 TheWienerTauberianTheorem . . . . . . . . . . . . . . . 72
6 TopologicalVectorSpaces.................................... 75
6.1 BasicIdeas . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Fre´chetSpaces . . . . . . . . . . . . . . . . . . . . . . . . 78
ix