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A Course in Functional Analysis and Measure Theory
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Library of Congress Cataloging‑in‑Publication Data
Names: Shalit, Orr Moshe.
Title: A first course in functional analysis / Orr Moshe Shalit.
Description: Boca Raton : CRC Press, [2016] | Includes bibliographical references and index.
Identifiers: LCCN 2016045930| ISBN 9781498771610 (hardback : alk. paper) | ISBN 9781315367132 (ebook) | ISBN 9781498771627 (ebook) | ISBN 9781498771641 (ebook) | ISBN 9781315319933 (ebook)
Subjects: LCSH: Functional analysis--Textbooks.
C lassification: LCC QA320 .S45927 2016 | DDC 515/.7--dc23
LC record available at https://lccn.loc.gov/2016045930
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AsIpreparedformyfirsttimeteachingsuchacourse,Ifoundnothing amongthecountlessexcellenttextbooksinfunctionalanalysisavailablethat perfectlysuitedmyneeds.Iendedupwritingmyownlecturenotes, which evolvedintothisbook(anearlierversionappearedonmyblog[31]).
Myloveforthesubjectandmypointofviewonitwerestronglyshapedby myteachers,andinparticularbyBorisPaneah(myMaster’sthesisadvisor) andBaruchSolel(myPh.D.thesisadvisor).Ifthisbookisanygood, then thesemendeservemuchcredit.
ThisequationwasconsideredbyI.Fredholmin1903,and,likemanyimportant mathematicalproblemsofthetime,itarosefrommathematicalphysics.In theequationabove,thefunctions g ∈ CR ([a,b])and k ∈ CR([a,b] × [a,b])are givencontinuousfunctions,and f isanunknownfunction(hereandbelow CR(X)denotesthespaceofcontinuous,real-valuedfunctionsonatopological space X).Fixingthefunction k,therearethreebasicquestionsonecanask aboutsuchequations:
1. Solvability. Doesthereexistacontinuoussolution f tothisequation given g?Forwhat g doesasolutionexist?
2. Uniqueness. Isthesolutionunique(whenitexists)?Givenaparticular solutiontotheequation,canwedescribethespaceofallsolutions, or atleastcanwetellhow“big”itis?
3. Methodofsolution. Whatisthesolution?Inotherwords,given g canwewritedownaformulaforthesolution f ,oratleastdescribea methodofobtaining f approximately?
whenconsideringasystemoflinearequations Ax = b,where x and b are vectorsin Rn,and A isan n × n matrix.Therearemanynontrivialthings tosayregardingthesolvabilityoftheequation Ax = b whichdonotrequire knowingthespecificmatrix A,forexample:iftheequation Ax =0hasa uniquesolution(namely, x =0),then Ax = b hasauniquesolutionfor any b ∈ Rn.Inthesamevein,oneisinterestednotonlyinansweringtheabove threequestionsfortheintegralequation(1.1)givenaparticular k;itisalsoof interesttounderstandtheunifyingcharacteristicsofequations ofthistype.
AstandardapplicationofZorn’slemmashowsthateveryvectorspace V hasabasis—aset {ui}i∈I ⊂ V suchthatforallnonzero v ∈ V ,thereis auniquechoiceoffinitelymanydistinctindices i1,...,ik ∈ I andnonzero scalars c1,...,ck satisfying
Theorem1.2.2 (Weierstrass’sapproximationtheorem). Let f :[a,b] → R beacontinuousfunction.Forevery ǫ> 0,thereexistsapolynomial p with realcoefficients,suchthatforall t ∈ [a,b], |p(t) f (t)| <ǫ.
Arethereanyothersequencesofnicefunctionsthatgenerate CR ([a,b])?To giveanotherexample,weneedacoupleofdefinitions.A Z-periodicfunction isafunction f : R → R suchthat f (x + n)= f (x)forall x ∈ R andall n ∈ Z. A trigonometricpolynomial isafunction q oftheform
Theorem1.2.3 (Trigonometricapproximationtheorem). Let f : R → R bea continuous Z-periodicfunction.Forevery ǫ> 0,thereexistsatrigonometric polynomial q,suchthatforall t ∈ R, |q(t) f (t)| <ǫ.
Let X beacompactHausdorfftopologicalspace(the appendix containsall thematerialintopologicalandmetricspacesthatisrequiredforthisbook).
Wewilllet CR (X)denotethespaceofcontinuous,real-valuedfunctionson X;likewise, C(X)denotesthespaceofcontinuous,complex-valuedfunctions on X.Onbothofthesespaceswedefinethe supremumnorm ofafunction f tobe
∞ =sup x∈X |f (x)|
Thequantity d(f,g)= f g ∞ definesametricon CR (X)(andalsoon C(X))andisconsideredtobethedistancebetweenthetwofunctions f and g.Thisdistancemakesboth CR (X)and C(X)intocompletemetricspaces. Boththesespacesarevectorspacesovertheappropriatefield, withtheusual operationsofpointwiseadditionoffunctionsandscalarmultiplication.Infact, if f,g ∈ CR(X),thenthepointwiseproduct fg isalsoin CR(X),andtogether withthevectorspaceoperations,thisgives CR(X)thestructureofan algebra
Definition1.3.1. If A isasubspaceof CR(X)orof C(X),thenitissaidto bea subalgebra ifforall f,g ∈ A, fg isalsoin A.
Definition1.3.2. Asubalgebra A ⊆ CR(X)issaidto separatepoints if foreverypairofdistinctpoints x,y ∈ X thereexistssome f ∈ A suchthat f (x) = f (y).
Theorem1.3.3 (Stone-Weierstrasstheorem(realversion)). Let A beaclosed subalgebraof CR (X) whichcontainstheconstantfunctionsandseparates points.Then A = CR(X)
OneobtainsTheorem1.2.2immediatelybyletting X =[a,b]andtaking A tobetheclosureofthealgebraofpolynomialswithrespecttothesupremum norm,notingthatthenormclosureofanalgebraisanalgebra.
Exercise1.3.4. Let A ⊆ CR(X)beasubalgebra,andlet A beitsclosure. Then A isalsoasubalgebra.
Exercise1.3.5. ProvethatTheorem1.3.3isequivalenttothefollowingstatement: If A isasubalgebraof CR(X) whichcontainstheconstantfunctionsand separatespoints,then A = CR (X).
Nowput X = T,andtake A ⊆ CR(T)tobetheinverseimageofall trigonometricpolynomialsunderΦ(ifweidentify CR(T)and CR,per (R),then underthisidentification A issimplythealgebraofalltrigonometricpolynomials,butwehavechosentomakethisidentificationexplicitwiththeuseof themapΦ).Bythepreviousparagraph,inordertoproveTheorem 1.2.3it sufficestoshowthat A isdensein CR (X).Itiselementarytocheckthat A containstheconstantsandseparatespointson X = T.Applyingtheversionof theStone-WeierstrasstheoremgiveninExercise1.3.5,wefindthat A isdense in CR (T),thereforeΦ(A)isdensein CR,per (R).Thatconcludestheproofof thetrigonometricapproximationtheorem.
Exercise1.3.6. FillinthedetailsintheproofofTheorem1.2.3.Inparticular, provethat A isanalgebrathatseparatespoints,andprovethatΦissurjective
ProofoftheStone-Weierstrasstheorem
Let A ⊆ CR (X)beasinthestatementofTheorem1.3.3.Weisolateafew lemmasbeforereachingthemainargumentoftheproof.
Exercise1.3.8. Fillinthedetailsoftheaboveproof;inparticularprovethe uniformconvergenceoftheMaclaurinseriesof h on[ 1, 1].(Hint: usethe integralformoftheremainderfortheTaylorpolynomialapproximation.)
Foreveryfunction f ,welet |f | denotethefunction |f | : x →|f (x)|.
Lemma1.3.9. If f ∈ A,thenthefunction |f | isalsoin A
Proof. Let ǫ> 0begiven.Wewillfindafunction g ∈ A suchthat g−|f | ∞ < ǫ.Since A isclosedandsince ǫ isarbitrary,thiswillshowthat |f |∈ A.
Let I =[− f ∞, f ∞].Bythepreviouslemmathereexistsapolynomial p suchthatsupt∈I |p(t) −|t|| <ǫ.Put g = p ◦ f .Since A isanalgebraand p isapolynomial, g ∈ A.Thus
asrequired.
Foranytwofunctions f,g,welet f ∧ g and f ∨ g denotethefunctions f ∧ g : x → min{f (x),g(x)} and f ∨ g : x → max{f (x),g(x)}.
Lemma1.3.10. If f,g ∈ A,thenthefunctions f ∧ g and f ∨ g arealsoin A.
Proof. ThisfollowsimmediatelyfromLemma1.3.9togetherwiththeformulas min{a,b} = a+b−|a b| 2 andmax{a,b} = a+b+|a b| 2 ,whichholdtrueforallreal a and b
Lemma1.3.11. Foreverypairofdistinctpoints x,y ∈ X,andevery a,b ∈ R, thereexistsafunction g ∈ A suchthat g(x)= a and g(y)= b.
Proof. Exercise.
CompletionoftheproofoftheStone-Weierstrasstheorem. Let f ∈ CR(X).Wemustshowthat f ∈ A.Itsuffices,forafixed ǫ> 0,tofind h ∈ A suchthat f h ∞ <ǫ Westartbychoosing,forevery x,y ∈ X,afunction fxy ∈ A suchthat fxy(x)= f (x)and fxy(y)= f (y).ThisispossiblethankstoLemma1.3.11. Nextweproduce,forevery x ∈ X,afunction gx ∈ A suchthat gx(x)= f (x)and gx(y) <f (y)+ ǫ forall y ∈ X.Thisisdoneasfollows.Forevery y ∈ X,let Uy beaneighborhoodof y inwhich fxy <f + ǫ.Thecompactnessof X ensuresthattherearefinitelymanyoftheseneighborhoods,say Uy1 ,...,Uym , thatcover X.Then gx = fxy1 ∧ ... ∧ fxym doesthejob(gx isin A,thanksto Lemma1.3.10).
Finally,wefind h ∈ A suchthat |h(x) f (x)| <ǫ forall x ∈ X.Forevery x ∈ X let Vx beaneighborhoodof x where gx >f ǫ.Againwefinda finitecover Vx1 ,...,Vxn andthendefine h = gx1 ∨ ... ∨ gxn .Thisfunctionlies between f + ǫ and f ǫ,soitsatisfies |h(x) f (x)| <ǫ forall x ∈ X,andthe proofiscomplete.
Exercise1.3.12. Didweusetheassumptionthat X isHausdorff?Explain.
Example1.4.2. Let D denotetheopenunitdiscin C,andlet D denoteits closure.Let A(D)denotethe discalgebra,whichisdefinedtobetheclosure ofcomplexpolynomialsin C(D),thatis
Itisworthstressingthatintheaboveexamplewemeanpolynomialsin onecomplexvariable z.We donotmean polynomialsinthetwovariables x and y,where z = x + iy =Rez + iImz.
Definition1.4.4. Asubspace S ⊆ C(X)issaidtobe self-adjoint iffor every f ∈ S,thecomplexconjugateof f (i.e.,thefunction f : x → f (x))is alsoin S
Theorem1.4.5 (Stone-Weierstrasstheorem(complexversion)). Let A bea closedandself-adjointsubalgebraof C(X) whichcontainstheconstantfunctionsandseparatespoints.Then A = C(X).
Proof. ConsidertherealvectorspaceReA = {Ref : f ∈ A}.SinceRef = f +f 2 and A isself-adjoint,itfollowsthatReA ⊆ A.Because A isasubalgebraof C(X),ReA isasubalgebraof CR(X).Fromclosednessof A itfollowsthat ReA isclosed,too.
Fromtheassumptionthat A isasubspacethatseparatespoints,itfollows thatReA alsoseparatespoints.Indeed,given x,y ∈ X,let f ∈ A suchthat f (x) = f (y).Then,eitherRef (x) =Ref (y),orImf (x) =Imf (y).ButImf = Re( if ) ∈ ReA,soReA separatespoints.
Thus,ReA isaclosed,realsubalgebraof CR(X)thatcontainstheconstants andseparatespoints.Bythe(real)Stone-Weierstrasstheorem,ReA = CR (X). Itfollowsthateveryreal-valuedcontinuousfunctionon X isin A.Symmetrically,everyimaginaryvaluedcontinuousfunctionon X isin A.ByExercise 1.4.1weconcludethat C(X)= A.
TheelegantandabstractproofoftheWeierstrasstheorempresentedabove isquitesofterthanWeierstrass’soriginalproof,whichinvolvedintegrationon thereallineandtheuseofpowerseries.However,wecouldnotdowithout somepieceof“hardanalysis”:werequiredthefactthattheTaylor series of h(x)= √1 x convergesuniformlyin[ 1, 1].Thisisaninstanceofthe followingmaxim(tobetakenwithagrainofsalt): thereisnoanalysiswithout “hardanalysis”.
Whatismeantbythisisthatonewillhardlyevergetaninterestingtheoremofsubstance,whichappliestoconcretecasesinanalysis,thatdoesnot involvesomekindofdifficultcalculation,delicateestimation,oratleast a clevertrick.Inourexample,thehardanalysispartisshowingthatthepower seriesof √1 x convergesuniformlyin[ 1, 1].TheStone-Weierstrasstheorem,whichhasaverysoftproof,thensparesustheuseofhardtechniques whenitgivesusthepolynomialandthetrigonometricapproximationtheoremsforfree.Butthehardanalysiscannotbeeliminatedentirely.Itisgood tokeepthisinmind.
1.6Additionalexercises
Exercise1.6.1. Let f ∈ CR([ 1, 1])suchthat f (0)=0.Provethat f canbe approximatedinuniformnormbypolynomialswithzeroconstantcoefficient.
Exercise1.6.2. Let C0(R)bethealgebraofallcontinuousfunctionsthat vanishatinfinity,thatis C0(R)= {f ∈ C(R):lim x→∞ f (x)=lim x→−∞ f (x)=0}.
Let A beaself-adjointsubalgebraof C0(R)thatseparatespoints,andassume thatforall x ∈ R,thereexists f ∈ A suchthat f (x) =0.Provethat A = C0(R).
1.Forevery f ∈ CR([a,b]),all x1,x2,...,xn ∈ [a,b],andevery ǫ> 0,there existsatrigonometricpolynomial p suchthatsupx∈[a,b] |p(x) f (x)| <ǫ and p(xi)= f (xi)for i =1, 2,...,n
2.Forevery f ∈ CR([a,b])whichhas n continuousderivativeson[a,b], andevery ǫ> 0,thereexistsatrigonometricpolynomial p suchthat supx∈[a,b] |p(k)(x) f (k)(x)| <ǫ forall k =0, 1,...,n.(Here f (k) denotes the kthderivativeof f .)
Exercise1.6.4. 1.Let f beacontinuousreal-valuedfunctionon[a,b] suchthat b a f (x)xndx =0forall n =0, 1, 2,....Provethat f ≡ 0.
2.Let f beacontinuousreal-valuedfunctionon[ 1, 1]suchthat 1 1 f (x)xndx =0forallodd n.Whatcanyousayabout f ?
Exercise1.6.5. Let X beacompactmetricspace.Provethat CR(X)isa separablemetricspace.
Exercise1.6.6. Let X and Y becompactmetricspaces.Let X × Y bethe productspace(seeDefinitionA.3.10)andfixafunction h ∈ C(X × Y ).Prove thatforevery ǫ> 0,thereexist f1,...,fn ∈ C(X)and g1,...,gn ∈ C(Y ) suchthatforall(x,y) ∈ X × Y , |h(x,y) n i=1 fi(x)gi(y)| <ǫ.
Exercise1.6.7. A Zk-periodicfunction isafunction f : Rk → R suchthat f (x + n)= f (x)forall x ∈ Rk and n ∈ Zk.A trigonometricpolynomial on Rk isafunction q oftheform q(x)= n a
cos(2πn · x)+ bn sin(2πn · x), wherethesumaboveisfinite,andfor x =(x1,...,xk)and n =(n1,...,nk) wewrite n · x = nixi.Provethefollowingtheorem.
Theorem (Trigonometricapproximationtheorem). Let f : Rk → R bea continuous Zk-periodicfunction.Forevery ǫ> 0 thereexistsatrigonometric polynomial q suchthatforall x ∈ Rk , |q(x) f (x)| <ǫ.
Exercise1.6.8 (Holladay1). Let X beacompactHausdorffspaceandlet CH (X)denotethealgebraofcontinuousfunctionstakingquaternionicvalues.
Here, H denotesthequaternions,thatis,thefour-dimensionalrealvector space
H = {a + bi + cj + dk : a,b,c,d ∈ R}, withabasisconsistingof1 ∈ R andthreeadditionalelements i,j,k,and withmultiplicationdeterminedbythemultiplicationoftherealsandbythe identities
i2 = j2 = k2 = ijk = 1
(Thetopologyon H —requiredforthenotionofcontinuousfunctionsfrom X into H —istheonegivenbyidentifying H with R4.)Provethatif A isa closedsubalgebraof CH (X)thatseparatespointsandcontainsalltheconstant H-valuedfunctions,then A = CH (X).
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