Functional Analysis

278 Pages·2023·2.333 MB·English

by  S. Kesavan

Texts and Readings in Mathematics
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S. Kesavan
Functional Analysis
Second Edition
S.Kesavan
Emeritus,TheInstituteofMathematical
Sciences
Chennai,India
ISSN 2366-8717 ISSN 2366-8725 (electronic)
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Preface
Functional analysis studies linear spaces provided with suitable topological struc-
turesand(continuous)lineartransformationsbetweensuchspaces.Ithasfarreaching
applicationsinseveraldisciplines.Forinstance,themoderntheoryof(partial)differ-
ential equations and the numerical approximation of their solutions rely a lot on
functionalanalytictechniques.
Functionalanalysisisnowanintegralpartofthecurriculuminanypostgraduate
courseinMathematics.Ideallyitshouldbetaughtafterhavingcovered courses in
linearalgebra,realandcomplexanalysis,andtopology.Anintroductiontothetheory
of measure and integration is also helpful since the richest examples in functional
analysiscomefromfunctionspaceswhosestudydemandssuchaknowledge.
Thepresentbookgrewoutofnotespreparedbymyselfwhilelecturingtograduate
studentsattheTataInstituteofFundamentalResearch(BangaloreCentre)andthe
InstituteofMathematicalSciences,Chennai.Thematerialpresentedinthisbookis
standardandisideallysuitedforacoursewhichcanbefollowedbymastersstudents
who have covered the necessary prerequisites mentioned earlier. While covering
all the standard material, I have also tried to illustrate the use of various theorems
viaexamplestakenfromdifferentialequationsandthecalculusofvariations,either
throughbriefsectionsorthroughtheexercises.Infact,thisbookiswellsuitedfor
students who would like to pursue a research career in the applications of mathe-
matics.Inparticular,familiaritywiththematerialpresentedinthisbookwillfacilitate
studyingmyearlierbook,publishednearlytwodecadesago,‘TopicsinFunctional
Analysis and Applications’ (Wiley Eastern, now called New Age International),
whichservesasafunctionalanalyticintroductiontothetheoryofpartialdifferential
equations.
Chapter1givesarapidrevisionoflinearalgebra,topologyandmeasuretheory.
Importantdefinitions,examplesandresultsarerecalledandnoproofsaregiven.At
theendofeachsection,thereaderisreferredtoastandardtextonthattopic.This
chapterhasbeenincludedonlyforreferencepurposesanditisnotintendedthatit
becoveredinacoursebasedonthisbook.
Chapter2introducesthenotionofanormedlinearspaceandthatofcontinuous
lineartransformationsbetweensuchspaces.
v
vi Preface
Chapter3studiestheanalyticandgeometricversionsoftheHahn-Banachtheorem
andpresentssomeapplicationsofthese.
Chapter4isdevotedtothefamous‘trinity’infunctionalanalysis—theBanach-
Steinhaus, the open mapping and the closed graph theorems, which are all conse-
quences of Baire’s theorem for complete metric spaces. Several applications are
discussed.Thenotionofan‘unboundedlinearmapping’isintroduced.
Inmyopinion,mosttextsdonotemphasizetheimportanceofweaktopologiesin
acourseonfunctionalanalysis.Thataboundedsequenceinareflexivespaceadmits
aweaklyconvergentsequenceisthecornerstoneofmanyanexistenceproofinthe
theoryof(partial)differentialequations.Thesetopologiesalsoprovidenicecounter-
examplestoshowtheinadequacyofsequencesinageneraltopologicalspace.For
instance,wewillseethattwotopologiesonasetcouldbedifferentwhilehavingthe
sameconvergentsequences.Wewillalsoseeanexampleofacompacttopological
spaceinwhichasequencedoesnothaveanyconvergentsubsequence.Chapter5deals
withweakandweak*topologiesandtheirapplicationstothenotionsofreflexivity,
separabilityanduniformconvexity.
Chapter6introducestheLebesguespacesandalsopresentsthetheoryofoneof
thesimplestclassesofSobolevspaces.Chapter7isdevotedtothestudyofHilbert
spaces.Chapter8studiescompactoperatorsandtheirspectra.
Much of the fun in learning mathematics comes from actually doing it! Every
chapter from Chapters 2 through 8 has a fairly large collection of exercises at the
end.Theseillustratetheresultsofthetext,showtheoptimalityofthehypothesesof
the various theorems via various examples or counterexamples, or develop simple
versionsoftheoriesnotelaborateduponinthetext.Theyareofvaryingdegreesof
difficulty.Occasionally,somehintsforthesolutionareprovided.Itishopedthatthe
studentswillbenefitbysolvingthem.
Since this is meant to be a first course on functional analysis, I have kept the
bibliographic references to a minimum, merely citing important texts where the
readermayfindfurtherdetailsofthetopicscoveredinthisbook.
No claim of originality is made with respect to the material presented in this
book. My treatment of this subject has been influenced by the writings of authors
like Simmons (whose book was my first introduction to Functional Analysis) and
Rudin. These works figure in the bibliography of this book. I would also like to
mention a charming book, hardly known outside the francophonic mathematical
community,viz.‘AnalyseFonctionnelle’byH.Brézis.
The preparation of this manuscript would not have been possible but for the
excellentfacilitiesprovidedbytheInstituteofMathematicalSciences,Chennai,and
I wish to place on record my gratitude. I also thank the Hindustan Book Agency
and the editor of the TRIM Series, Prof. R. Bhatia, for their kind cooperation in
bringing out this volume. I must thank the anonymous referees who painstakingly
went through the first draft of the manuscript. I have tried to incorporate many of
theirconstructivesuggestionsinthecurrentversion.
Preface vii
Finally,Ithankmyfamilyforitsconstantsupportandencouragement.
Chennai,India S.Kesavan
May2008
Preface to the Second Edition
ItisnowmorethanadecadesincethefirsteditionofthisbookappearedandIam
gratified by the reception accorded to it by mathematics students and teachers in
thiscountryandalsoinmanyplacesabroad.Thebookhasbeenusedasatextfor
FunctionalAnalysiscoursesinmanyinstitutionsandalsoasthemainreferencetext
forseveralrefreshercoursesonthissubject.
ImyselfhaveusedittoteachacourseonFunctionalAnalysisininstitutionslike
the Chennai Mathematical Institute (CMI) and the Indian Institute of Technology
Madras (IITM). I have also used it to deliver a course on this subject under the
NPTELbanner.Whilegivingtheselectures,Ihavefacedquestionsfromstudents,
preparedexercisesforweeklyassignmentsandexaminationsandhavealsoimproved
myownappreciationofthefinerpointsofthesubject.Thesefactorsencouragedme
to prepare this revised version of the book, and I hope it will be more useful and
user-friendlythanthefirstedition.
In the present (second) edition, I have completely overhauled the presentation
withoutchangingthebasicstructureofthebook.Thestatementsofresults,definitions
and remarks have been modified wherever necessary, and many proofs have been
rewritten, in view of greater clarity of the exposition. Some examples have been
addedtoillustratetheresultsprovedinthetext.Severalexerciseshavebeenaddedto
theexistingcollectionineachchapter.Itishopedthatstudentswillhavefunsolving
them.
Section 3.1, which originally dealt with the Hahn-Banach theorem and intro-
duced the notion of reflexivity, has now been bifurcated into two sections. In the
newframework,Sect.3.1dealswiththeHahn-Banachtheorem(extensionversion)
anditsimmediateconsequences.Section3.2isdevotedtoreflexivity.Severalexam-
ples of continuous linear functionals, which do not realize their norm on the unit
sphere,arepresented,therebyprovingthenon-reflexivityofmanyfamiliarBanach
spaces.Similarly,Sect.8.3,whichoriginallyintroducedthenotionofthespectrum
of an operator and described that of a compact operator, has been bifurcated into
two sections. Section 8.3 introduces the notion of the spectrum. Several examples
are presented, and the spectra of some special operators on a Hilbert space are
characterized.Section8.4dealswiththespectrumofacompactoperator.
ix
x PrefacetotheSecondEdition
The final preparation of the manuscript was done while I was working from
home,partlyduetotheCOVID-19pandemic,andthiswaspossiblebecauseofthe
help rendered by the system administration team of the Institute of Mathematical
Sciences(IMSc).Inparticular,IthankMr.G.S.Vasanforhishelp.Ialsothankmy
colleaguesProfs.AmritanshuPrasadandS.Viswanathforintroducingmetothejoys
ofteachingontheNPTELplatform.IusethisopportunitytothanktheNPTELteam
atIITMadras,headedbyMs.BharathiandablyassistedbyMs.LakshmiPriya,for
theirsupportandhelpwhenpreparingmyNPTELcourses.Finally,IthankShriJ.
K.JainoftheHindustanBookAgencyandProf.RajendraBhatia,thechiefeditor
oftheTRIMSeries,forencouragingmetobringoutthisnewedition.
Chennai,India S.Kesavan
April2022