Rudin (1991) Functional Analysis - WordPress.com - Get a Free Blog

441 Pages·2011·5.04 MB·English

FUNCTIONAL
ANALYSIS
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FUNCTIONAL ANALYSIS
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thper oducstuipoenr visLoerrA o.wyY a osu ng.
Thec ovwearsd esigbnyHe edr maSntnr ohbach.
LibraorfyC ongreCsast aloging-in-PuDbaltiac ation
RudiWna,l t(edra,t e).
Functiaonnaally sisR/uWdailnt.ee-rd2 .n d
p. em-.( internatsieorniianepl su raen adp plmiaetdh ematics)
Inclubdiebsl iogrraepfheir(cepan.lc e)s.
ISBN 0-07-054236-8
Functiaonnaally sIi.Ts i.t le.S eries.
I. II.
QA320.R83 1991
51'5.7-dc20 90-5677
When orderitnhgit si tle,I SuBsN0e - 07-100944-2
PrinitnSe idn gapore
ABOUT THE AUTHOR
FunctAinaolnyasli s,
In addititoon SeconEdd itioWna,l teRru dini st he
PrinocfMi paltehse mAanatliycsaiRlse ala nd
authoorf t woo thebro oks:
anCdo mpAlneaxl ysis,
whosew idespreuasdei si llustrbaytt ehdef actth at
13 Principles
theyh aveb eent ranslaitnetdao t otaolf languagHees w.r ote
ofM athemAantailcyasli s
whilhee wasa C.L.EM.o oreI nstrucattot rh e
MassachuseItntsst itouftT ee chnology-jtuwsoyt e arasf terre ceivhiinsg
Ph.Da.t D ukeU niversLiattye.rh e,t aughattt heU niversoiftR yo chester,
and inso w a VilaRse searcPhr ofessaottr h eU niversoiftW yi sconsin­
MadisoInn.t hep asth,eh ass pent leaYvaelseU naitv erstihteyU ,n iversity
ofC aliforinnLi aaJ ollaan,d t heU niversoiftH ya waii.
Dr.R udin'rse searhcahs d ealmta inlwyi thh armoniacn alysainsd
withc omplevxa riabHlee s.h awsr itttehnr eree searmcohn ograpohnst hese
FourAinearlo ynGs riosu Fpusn,c Tthieooinrn Py o lydainsdc s,
topics:
FuncTthieoiontnr h yUe n Biatol fl
C .
n
vii

CONTENTS
Preface Xlll
ParIt G eneral Theory
TopologiVceaclt oSrp aces 3
1
Introduction 3
Separpartoipoenr ties 10
Linmeaaprp ings 14
Finite-dsiapmceenss ional 16
Metrization 18
Boundednceosnst iannudi ty 23
Seminaonrdm scl ooncvaelx ity 25
Quotsipeancte s 30
Examples 33
Exercises 38
Completeness 42
2
Baicraet egory 42
The Bha-nSatcetihnehoaruesm 43
Thoep emna pptihnego rem 47
Thcel ogsreadtp hhe orem 50
Bilimnaepapri ngs 52
Exercises 53
Convexity 56
3
ThHea hBna-natchhe orems 56
Weatko pologies 62
Compcaocntsv eetxs 68
Vectori-nvtaelgureadt ion 77
Holomofruptnhiciocn s 82
Exercises 85
ix
CONTENTS
X
DualiitnyB anacShp aces 92
4
Thneo rmdeudoa fln oar msepda ce 92
Adj onits 97
Compoapcetr ators 130
Exercises 1 1 1
SomeA pplications 161
5
A conttihneuoirteym 161
Clossuepbdas coefIs f -spaces 171
Threa nogfev eac torm-evaasluureed 102
Ag enerSatloinzee-dW eierstrass the1or2e1m
Twoi nterptohleaotrieomns 142
Kauktafinxiep'dos i nt theorem 126
Haamre asouncr oem pgarcotu ps 182
Uncomplesmupebanscteesd 132
SumosfP oiskseornn els 138
Two mfioxrepedo itnhte orems 139
Exercises 144
ParItI D istribauntdiF oonusr iTerra nsforms
TesFtu nctioannsdD istributions 149
6
Introduction 149
Tefsutin costnp aces 151
Calcwuildtuihss uttriiobns 157
Localization 162
Suppoofdr itsst ributions 164
Distrnisab sdu etriiovatives 176
Convolutions 107
Exercises 177
FouriTerra nsforms 182
7
Baspirco perties 182
Tempedriesdt ributions 198
PalWeiye-ntehre orems 196
Sobolleemvm'as 202
Exercises
204
ApplicattiooD nisff erentEiqaula tions 210
8
Fundamseonlotunatsli
210
Elleiqputaitci ons
215
Exercises
222
CONTENTS
Xi
TauberiTahne ory 226
9
Wientehre'osr em 226
Thper ime tnhuemobreerm 230
Three neewqaula tion 236
Exercises 239
ParItI IB anacAhl gebarnadsS pectTrhaelo ry
BanacAhl gebras 245
10
Introduction 245
Comphloemxo morphisms 249
Baspirco peorfstp ieecst ra 252
Symbcoalliccu lus 258
Thger ooufip n veerlteimbelnet s 267
Lomonoisnovva'srsui basntptha ecoer em 269
Exercises 271
CommutatiBvaen acAhl gebras 275
11
Idealhso maonmdo rphisms 275
Gelftarnadn sforms 280
Involutions 287
Applictaont oinocnosm maultgaetbirvaes 292
Posiftutinivcoen als 296
Exercises 301
BoundeOdp eratoornsa HilbeSrpta ce 306
12
Basfciatcs 306
Bounded operators 309
Ac ommuttahteiovrietmy 315
Resonlsou tfth iieod entity 316
The sptehcetorraelm 321
Eigenovnfao lrumoeapsle rators 327
Posiotpievreaa ntsdoq rusra oroet s 330
Thger oouifpn veorpteirbalteo rs 333
Ac haractoefBr* -iazlagteiborna s 336
Ane rgotdhieco rem 339
Exercises 341
UnboundeOdp erators 347
13
Introduction
347
Grapahnssdy mmeotpreirca tors
351
ThCea ytlreayn sform
356
Resonlsou tfth iieod entity 360
The sptehcetorraelm 368
Semigorofop ueprsa tors 375
Exercises 385
CONTENTS
XII
Appendix AC ompactnaensdsC ontinuity 391
AppendiBx Noteasn dC omments 397
Bibliography 412
Lisotf S peciaSly mbols 414
Index 417
PREFACE
Functioannaall ysiists h es tudy coefr tationp ological-asltgreubcrtauirce s
ando ft hem ethodbsy w hickh nowledogfet hesset ructucraenbs e a pplied
toa nalytpirco blems.
A goodi ntroductteoxrotyn this subject shao purleds einntcaltuidoen
ofi tasx iomat(iic.soe f.t ,h e gentehreaolr oyft opologivceaclt osrp aceist) ,
shoultdr eaattl easat f ewt opiicnss omed epth, asnhdo uiltcd o ntasionm e
interesatpipnlgi cattiooo ntsh ebrr acnheso fm athematiIc hso.p et hatth e
presebnoto km eettsh escer iteria.
The subjeicsth ugea nd isg rowinrga pidl(yT.h eb ibliograipnh y
96 1597.)
volumIeo f[ 4]c ontainsp ageasn dg oeso nlyt o Ino rdetro w rite
a booko fm oderatsei ziet, wtahse refnoercee ssatrosy e leccetr taairne as
andt o ignore Iof tuhlelrryes a.l ize thata nayel xmpoesrwtth ol ookast t he
tabloef c ontenwtisl l fitnhda sto meo fh iso rh er( andm y)f avorittopei cs
arem issinbgu,tt hisse emusn avoidaIbtlw ea.sn otm y intenttioow nr itaen
encyclopterdeiact iIws ae.n tetdo wrai tbeo okt hawto uldo pent he way to
furtheexrp loration.
Thisi st her easofno ro mittimnagn y of mtohreee sotetroipci tchsa t
mighhta veb eeni ncludientd h ep resentaotfit ohneg enertahle oroyft opo­
logicvaelc tosrp aces. inFsotra ntchee,ri esn od iscussoifuo nni forsmp aces,
ofM oore-Smictohn vergenocfne e,t so,ro f filtersn.o tioTnho efc omplete­
nesso ccurosn lyi nt hec onteoxftm etriscp aces. Bornoslpoagcieacsra el
notm entionendo,ra reb arreloende sD.u alitiyso f couprrsees entbeudt,
noti ni tust mosgte neralIinttye.g ratoifvo enc tor-vafluunecdt iiosnt sr eated
stricatsla y t oola;t tentiisoc no nfinetdoc ontinuionutse grawnidtsvh,a lues
ina Frechsepta ce.
Nevertheltehsems a,t erioaflP artI is fualdleyq uaftoer a lmosatl l
applicattiooc nosn creptreo blemAsn.d t hiissw hato ughtto b es tressine d
sucha coursTeh:e c losien terpbleatyw eetnh ea bstraacntdt hec oncreitse
xiii