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Cornerstones Michel Willem Functional Analysis Fundamentals and Applications Second Edition Cornerstones SeriesEditor StevenG.Krantz,WashingtonUniversity,St.Louis,MO,USA Michel Willem Functional Analysis Fundamentals and Applications Second Edition MichelWillem DepartmentofMathematics UniversitécatholiquedeLouvain Louvain-la-Neuve,Belgium ISSN2197-182X ISSN2197-1838 (electronic) Cornerstones ISBN978-3-031-09148-3 ISBN978-3-031-09149-0 (eBook) https://doi.org/10.1007/978-3-031-09149-0 MathematicsSubjectClassification:46-XX,46Nxx 1stedition:©SpringerScience+BusinessMedia,LLC2013 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Tothememoryofmyfather,RobertWillem, andtomymother,Gilberte Willem-Groeninckx Preface to the Second Edition Inthissecondedition,someimprovementshavebeencarriedoutandsupplementary materialhasbeeninserted. In particular, the section on distribution theory and the chapter on “Topics in Calculus”havebeencompletelyrewrittenandextended.Newproofsofthedensity theoreminthespaceoffunctionsofboundedvariationsandofthecoareaformula aregiven. In this book, the abstract integration theory depends only on Daniell’s axioms and, when it is necessary, on Stone axiom, without any other assumption. In this general framework, we have added in Chap.3 a proof of Vitali’s characterization of convergence in L1((cid:2),μ) in terms of equi-integrability and convergence in measure.Inthesamechapter,wehaveaddedZabreiko’stheoremonthecontinuity of seminorms and its applications to the closed graph theorem and to the open mappingtheorem. I want to thank my colleagues Jacques Boël, Augusto Ponce, and Jean Van Schaftingen for their suggestions, and I am particularly obliged to Cathy Brichard forherhelpintherealizationofthissecondedition. Louvain-la-Neuve,Belgium MichelWillem vii Preface to the First Edition L’inductionpeutêtreutilementemployêeenAnalysecommeun moyendedêcouvertes.Maislesformulesgênêralesainsi obtenuesdoiventtreensuitevêrifiêesàl’aidededêmonstrations rigoureusesetpropresàfaireconnaîtrelesconditionssous lesquellessubsistentcesmêmesformules. AugustinLouisCauchy Mathematical analysis leads to exact results by approximate computations. It is basedonthenotionsofapproximationandlimitprocess.Forinstance,thederivative isthelimitofdifferentialquotients,andtheintegralisthelimitofRiemannsums. Howtocomputedoublelimits?Insomecases, (cid:2) (cid:2) lim u dx = lim u dx, (cid:2)n→∞ n n→∞ (cid:2) n ∂ ∂ lim u = lim u . ∂xk n→∞ n n→∞∂xk n In the preceding formulas, three functional limits and one numerical limit appear. The first equality leads to the Lebesgue integral (1901), and the second to the distributiontheoryofSobolev(1935)andSchwartz(1945). In1906,Fréchetinventedanabstractframeworkforthelimitingprocess:metric spaces.AmetricspaceisasetXwithadistance d :X×X →R:(u,v)(cid:4)→d(u,v) satisfyingsomeaxioms.IftherealvectorspaceXisprovidedwithanorm X →R:u(cid:4)→||u||, thentheformula d(u,v)=||u−v|| ix x PrefacetotheFirstEdition definesadistanceonX.Finally,iftherealvectorspaceXisprovidedwithascalar product X×X →R:(u,v)(cid:4)→(u|v), thentheformula (cid:3) ||u||= (u|u) definesanormonX. In1915,Fréchetdefinedadditivefunctionsofsets,ormeasures.Heextendedthe Lebesgueintegraltoabstractsets.In1918,Daniellproposedafunctionaldefinition oftheabstractintegral.Theelementaryintegral (cid:2) L→R:u(cid:4)→ udμ, (cid:2) definedonavectorspaceLofelementaryfunctionson(cid:2)satisfiescertainaxioms. When u is a nonnegative μ-integrable function, its integral is given by the Cavalieriprinciple: (cid:2) (cid:2) ∞ udμ= μ({x ∈(cid:2):u(x)>t})dt. (cid:2) 0 Tomeasureasetistointegrateitscharacteristicfunction: (cid:2) μ(A)= χ dμ. A (cid:2) Inparticular,thevolumeofaLebesgue-measurablesubsetAofRN isdefinedby (cid:2) m(A)= χ dx. A RN A function space is a space whose points are functions. Let 1 ≤ p < ∞. The realLebesguespaceLp((cid:2),μ)withthenorm (cid:4)(cid:2) (cid:5) 1/p ||u|| = |u|pdμ p (cid:2) isacompletenormedspace,orBanachspace.ThespaceL2((cid:2),μ),withthescalar product (cid:2) (u|v)= uvdμ, (cid:2) PrefacetotheFirstEdition xi isacompletepre-Hilbertspace,orHilbertspace. Dualityplaysabasicroleinfunctionalanalysis.Thedualofanormedspaceis thesetofcontinuouslinearfunctionalsonthisspace.Let1<p <∞anddefinep(cid:7), theconjugateexponentofp,by1/p+1/p(cid:7) =1.ThedualofLp((cid:2),μ)isidentified withLp(cid:7)((cid:2),μ). Weak derivatives are also defined by duality. Let f be a continuously differen- tiablefunctiononanopensubset(cid:2)ofRN.Multiplying ∂f =gbythetestfunction u∈D((cid:2))andintegratingbyparts,weobtain ∂xk (cid:2) (cid:2) ∂u f dx =− gudx. ∂x (cid:2) k (cid:2) Theprecedingrelationretainsitsmeaningiff andgarelocallyintegrablefunctions on(cid:2).Ifthisrelationisvalidforeverytestfunctionu ∈ D((cid:2)),thenbydefinition, g is the weak derivative of f with respect to x . Like the Lebesgue integral, the k weakderivativessatisfysomesimpledouble-limitrulesandareusedtodefinesome completenormedspaces,theSobolevspacesWk,p((cid:2)). A distribution is a continuous linear functional on the space of test functions D((cid:2)).Everylocallyintegrablefunctionf on(cid:2)ischaracterizedbythedistribution (cid:2) D((cid:2))→R:u(cid:4)→ fudx. (cid:2) Thederivativesofthedistributionf aredefinedby ∂f ∂u (cid:8) ,u(cid:9)=−(cid:8)f, (cid:9). ∂x ∂x k k Whereasweakderivativesmaynotexist,distributionalderivativesalwaysexist!In thisframework,Poisson’stheoreminelectrostaticsbecomes (cid:4) (cid:5) 1 −(cid:5) =4πδ, |x| whereδistheDiracmeasureonR3. Theperimeter ofaLebesgue-measurablesubsetAofRN,definedbyduality,is thevariationofitscharacteristicfunction: (cid:6)(cid:2) (cid:7) p(A)=sup divvdx :v ∈D(RN;RN),(cid:10)v(cid:10)∞ ≤1 . A ThespaceoffunctionsofboundedvariationBV(RN)containstheSobolevspace W1,1(RN).