Descriptive Topology in Selected Topics of Functional Analysis PDF
Preview Descriptive Topology in Selected Topics of Functional Analysis
Developments in Mathematics VOLUME 24 SeriesEditors: KrishnaswamiAlladi,UniversityofFlorida HershelM.Farkas, HebrewUniversityofJerusalem RobertGuralnick,UniversityofSouthern California Forfurthervolumes: www.springer.com/series/5834 Jerzy Ka˛kol (cid:2) Wiesław Kubis´ (cid:2) Manuel López-Pellicer Descriptive Topology in Selected Topics of Functional Analysis JerzyKa˛kol WiesławKubis´ FacultyofMathematicsandInformatics InstituteofMathematics A.MickiewiczUniversity JanKochanowskiUniversity 61-614Poznan 25-406Kielce Poland Poland [email protected] and InstituteofMathematics ManuelLópez-Pellicer AcademyofSciencesoftheCzechRepublic IUMPA 11567Praha1 UniversitatPoltècnicadeValència CzechRepublic 46022Valencia [email protected] Spain and RoyalAcademyofSciences 28004Madrid Spain [email protected] ISSN1389-2177 ISBN978-1-4614-0528-3 e-ISBN978-1-4614-0529-0 DOI10.1007/978-1-4614-0529-0 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011936698 MathematicsSubjectClassification(2010): 46-02,54-02 ©SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To ourFriend andTeacher Prof. Dr.ManuelValdivia Preface We invoke (descriptive) topology recently applied to (functional) analysis of infinite-dimensional topological vector spaces, including Fréchet spaces, (LF)- spaces and their duals, Banach spaces C(X) over compact spaces X, and spaces C (X), C (X) of continuous real-valued functions on a completely regular Haus- p c dorffspaceX endowedwithpointwiseandcompact–opentopologies,respectively. The(LF)-spacesanddualsparticularlyappearinmanyfieldsoffunctionalanalysis anditsapplications:distributiontheory,differentialequationsandcomplexanalysis, tonameafew. Ourmaterial,muchofitinbookformforthefirsttime,carriesforwardtherich legacy of Köthe’s Topologische lineare Räume (1960), Jarchow’s Locally Convex Spaces(1981),Valdivia’sTopicsinLocallyConvexSpaces(1982),andPérezCar- rerasandBonet’sBarrelledLocallyConvexSpaces(1987).Weassumetheir(stan- dardEnglish)terminology.Atopologicalvectorspace(tvs)mustbeHausdorffand havearealorcomplexscalarfield.A locallyconvexspace (lcs)is atvsthatislo- callyconvex.Engelking’sGeneralTopology(1989)servesasadefaultreferencefor generaltopology. TheauthorswishtothankProfessorB.Cascales,ProfessorM.Fabian,Professor V.Montesinos,andProfessorS.Saxonfortheirvaluablecommentsandsuggestions, whichmadethismaterialmuchmorereadable. TheresearchofJ.Ka˛kolwaspartiallysupportedbytheMinistryofScienceand HigherEducation,Poland,undergrantno.NN201274033. W.Kubis´ wassupportedinpartbygrantIAA100190901,bytheInstitutional ResearchPlanoftheAcademyofSciencesoftheCzechRepublicundergrantno. AVOZ 101 905 03, and by an internal research grant from Jan Kochanowski Uni- versityinKielce,Poland. The research of J. Ka˛kol and M. López-Pellicer was partially supported by the SpanishMinistryofScienceandInnovation,underprojectno.MTM2008-01502. Poznan,Poland JerzyKa˛kol Kielce,Poland WiesławKubis´ Valencia,Spain ManuelLópez-Pellicer vii Contents 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Generalcommentsandhistoricalfacts . . . . . . . . . . . . . . . 7 2 ElementaryFactsaboutBaireandBaire-TypeSpaces . . . . . . . . 13 2.1 BairespacesandPolishspaces . . . . . . . . . . . . . . . . . . . 13 2.2 AcharacterizationofBairetopologicalvectorspaces. . . . . . . . 18 2.3 AriasdeReyna–Valdivia–Saxontheorem . . . . . . . . . . . . . . 20 2.4 LocallyconvexspaceswithsomeBaire-typeconditions . . . . . . 24 2.5 StronglyrealcompactspacesXandspacesC (X) . . . . . . . . . 36 c 2.6 Pseudocompactspaces,WarnerboundednessandspacesC (X) . . 46 c 2.7 SequentialconditionsforlocallyconvexBaire-typespaces . . . . . 56 3 K-analyticandQuasi-SuslinSpaces. . . . . . . . . . . . . . . . . . . 63 3.1 Elementaryfacts . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 ResolutionsandK-analyticity . . . . . . . . . . . . . . . . . . . . 71 3.3 Quasi-(LB)-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4 Suslinschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.5 ApplicationsofSuslinschemestoseparablemetrizablespaces. . . 93 3.6 Calbrix–Hurewicztheorem . . . . . . . . . . . . . . . . . . . . . 101 4 Web-CompactSpacesandAngelicTheorems . . . . . . . . . . . . . 109 4.1 Angeliclemmaandangelicity . . . . . . . . . . . . . . . . . . . . 109 4.2 Orihuela’sangelictheorem . . . . . . . . . . . . . . . . . . . . . 111 4.3 Web-compactspaces . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Subspacesofweb-compactspaces. . . . . . . . . . . . . . . . . . 116 4.5 AngelicdualsofspacesC(X) . . . . . . . . . . . . . . . . . . . . 118 4.6 AboutcompactnessviadistancestofunctionspacesC(K) . . . . . 120 5 StronglyWeb-CompactSpacesandaClosedGraphTheorem . . . . 137 5.1 Stronglyweb-compactspaces . . . . . . . . . . . . . . . . . . . . 137 5.2 Productsofstronglyweb-compactspaces . . . . . . . . . . . . . . 138 5.3 Aclosedgraphtheoremforstronglyweb-compactspaces . . . . . 140 ix x Contents 6 WeaklyAnalyticSpaces . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.1 Afewfactsaboutanalyticspaces . . . . . . . . . . . . . . . . . . 143 6.2 Christensen’stheorem . . . . . . . . . . . . . . . . . . . . . . . . 149 6.3 Subspacesofanalyticspaces . . . . . . . . . . . . . . . . . . . . 155 6.4 Trans-separabletopologicalspaces . . . . . . . . . . . . . . . . . 157 6.5 Weaklyanalyticspacesneednotbeanalytic . . . . . . . . . . . . 164 6.6 Moreaboutanalyticlocallyconvexspaces . . . . . . . . . . . . . 167 6.7 Weaklycompactdensitycondition . . . . . . . . . . . . . . . . . 168 6.8 Moreexamplesofnonseparableweaklyanalytictvs . . . . . . . . 174 7 K-analyticBaireSpaces . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1 Bairetvswithaboundedresolution . . . . . . . . . . . . . . . . . 183 7.2 Continuousmapsonspaceswithresolutions . . . . . . . . . . . . 187 8 AThree-SpacePropertyforAnalyticSpaces . . . . . . . . . . . . . . 193 8.1 AnexampleofCorson . . . . . . . . . . . . . . . . . . . . . . . . 193 8.2 Apositiveresultandacounterexample . . . . . . . . . . . . . . . 196 9 K-analyticandAnalyticSpacesC (X) . . . . . . . . . . . . . . . . . 201 p 9.1 AtheoremofTalagrandforspacesC (X) . . . . . . . . . . . . . 201 p 9.2 TheoremsofChristensenandCalbrixforC (X) . . . . . . . . . . 204 p 9.3 BoundedresolutionsforC (X) . . . . . . . . . . . . . . . . . . . 215 p 9.4 SomeexamplesofK-analyticspacesC (X)andC (X,E) . . . . 230 p p 9.5 K-analyticspacesC (X)overalocallycompactgroupX . . . . . 231 p ∧ 9.6 K-analyticgroupX ofhomomorphisms . . . . . . . . . . . . . . 234 p 10 PrecompactSetsin(LM)-SpacesandDualMetricSpaces . . . . . . 239 10.1 Thecaseof(LM)-spaces:elementaryapproach . . . . . . . . . . 239 10.2 Thecaseofdualmetricspaces:elementaryapproach . . . . . . . . 241 11 MetrizabilityofCompactSetsintheClassG . . . . . . . . . . . . . 243 11.1 TheclassG:examples . . . . . . . . . . . . . . . . . . . . . . . . 243 11.2 Cascales–Orihuelatheoremandapplications . . . . . . . . . . . . 245 12 WeaklyRealcompactLocallyConvexSpaces. . . . . . . . . . . . . . 251 12.1 Tightnessandquasi-Suslinweakduals . . . . . . . . . . . . . . . 251 12.2 AKaplansky-typetheoremabouttightness . . . . . . . . . . . . . 254 12.3 K-analyticspacesintheclassG . . . . . . . . . . . . . . . . . . . 258 12.4 EveryWCGFréchetspaceisweaklyK-analytic . . . . . . . . . . 260 12.5 Amir–Lindenstrausstheorem . . . . . . . . . . . . . . . . . . . . 266 12.6 AnexampleofPol . . . . . . . . . . . . . . . . . . . . . . . . . . 271 12.7 MoreaboutBanachspacesC(X)overcompactscatteredX . . . . 276 13 Corson’sProperty(C)andTightness . . . . . . . . . . . . . . . . . . 279 13.1 Property(C)andweaklyLindelöfBanachspaces . . . . . . . . . 279 13.2 Theproperty(C)forBanachspacesC(X) . . . . . . . . . . . . . 284 Contents xi 14 Fréchet–UrysohnSpacesandGroups . . . . . . . . . . . . . . . . . . 289 14.1 Fréchet–Urysohntopologicalspaces . . . . . . . . . . . . . . . . 289 14.2 AfewfactsaboutFréchet–Urysohntopologicalgroups . . . . . . 291 14.3 SequentiallycompleteFréchet–UrysohnspacesareBaire . . . . . 296 14.4 Three-spacepropertyforFréchet–Urysohnspaces . . . . . . . . 299 14.5 Topologicalvectorspaceswithboundedtightness . . . . . . . . . 302 15 SequentialPropertiesintheClassG . . . . . . . . . . . . . . . . . . 305 15.1 Fréchet–UrysohnspacesaremetrizableintheclassG . . . . . . 305 15.2 Sequential(LM)-spacesandthedualmetricspaces. . . . . . . . 311 − 15.3 (LF)-spaceswiththepropertyC . . . . . . . . . . . . . . . . 320 3 16 TightnessandDistinguishedFréchetSpaces . . . . . . . . . . . . . . 327 16.1 Acharacterizationofdistinguishedspaces . . . . . . . . . . . . . 327 16.2 G-basesandtightness . . . . . . . . . . . . . . . . . . . . . . . 334 16.3 G-bases,bounding,dominatingcardinals,andtightness . . . . . 338 16.4 MoreabouttheWulbert–MorrisspaceC (ω ) . . . . . . . . . . 349 c 1 17 BanachSpaceswithManyProjections . . . . . . . . . . . . . . . . . 355 17.1 Preliminaries,model-theoretictools . . . . . . . . . . . . . . . . 355 17.2 Projectionsfromelementarysubmodels . . . . . . . . . . . . . . 361 17.3 Lindelöfpropertyofweaktopologies . . . . . . . . . . . . . . . 364 17.4 Separablecomplementationproperty . . . . . . . . . . . . . . . 365 17.5 Projectionalskeletons . . . . . . . . . . . . . . . . . . . . . . . 369 17.6 Normingsubspacesinducedbyaprojectionalskeleton . . . . . . 375 17.7 Sigma-products . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 17.8 Markushevichbases,PlichkospacesandPlichkopairs . . . . . . 383 17.9 PreservationofPlichkospaces . . . . . . . . . . . . . . . . . . . 388 18 SpacesofContinuousFunctionsoverCompactLines . . . . . . . . . 395 18.1 Generalfacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 18.2 Nakhmanson’stheorem . . . . . . . . . . . . . . . . . . . . . . 398 18.3 Separablecomplementation . . . . . . . . . . . . . . . . . . . . 399 19 CompactSpacesGeneratedbyRetractions . . . . . . . . . . . . . . 405 19.1 Retractiveinversesystems . . . . . . . . . . . . . . . . . . . . . 405 19.2 Monolithicsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 19.3 ClassesR andRC . . . . . . . . . . . . . . . . . . . . . . . . . 411 19.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 19.5 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 19.6 Thefirstcohomologyfunctor . . . . . . . . . . . . . . . . . . . 418 19.7 Compactlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 19.8 ValdiviaandCorsoncompactspaces . . . . . . . . . . . . . . . . 425 19.9 Preservationtheorem . . . . . . . . . . . . . . . . . . . . . . . . 432 19.10 Retractionalskeletons . . . . . . . . . . . . . . . . . . . . . . . 434 19.11 PrimarilyLindelöfspaces . . . . . . . . . . . . . . . . . . . . . 438 19.12 CorsoncompactspacesandWLDspaces . . . . . . . . . . . . . 440 19.13 Adichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 xii Contents 19.14 Alexandrovduplications . . . . . . . . . . . . . . . . . . . . . . 446 19.15 Valdiviacompactgroups . . . . . . . . . . . . . . . . . . . . . . 448 19.16 CompactlinesinclassR . . . . . . . . . . . . . . . . . . . . . . 451 19.17 MoreonEberleincompactspaces . . . . . . . . . . . . . . . . . 456 20 ComplementablyUniversalBanachSpaces . . . . . . . . . . . . . . 467 20.1 Amalgamationlemma . . . . . . . . . . . . . . . . . . . . . . . 467 20.2 Embedding-projectionpairs . . . . . . . . . . . . . . . . . . . . 469 20.3 AcomplementablyuniversalBanachspace . . . . . . . . . . . . 471 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Chapter 1 Overview Letusbrieflydescribetheorganizationofthebook. Chapter2,essentialtothesequel,containsclassicalresultsaboutBaire-typecon- ditions(Baire-like,b-Baire-like,CS-barrelled,s-barrelled)ontvs.Weincludeappli- cationstoclosedgraphtheoremsandC(X)spaces.Wealsoprovidethefirstproofin bookformofaremarkableresultofSaxon[355](extendingearlierresultsofArias deReynaandValdivia)thatstatesthat,underMartin’saxiom,everylcscontaining a dense hyperplane contains a dense non-Baire hyperplane. Ours, then, is the first booktosolvethefirstproblemformallyposedinPérezCarrerasandBonet’sexcel- lentmonograph.Chapter2alsocontainsanalyticcharacterizationsofcertaincom- pletely regular Hausdorff spaces X. For example, we show that X is pseudocom- pact,isWarnerbounded,orC (X)isa(df)-spaceifandonlyifforeachsequence c (μ ) in the dual C (X)(cid:2) there exists a sequence (t ) ⊂(0,1] such that (t μ ) n n c n n n n n isweaklybounded,stronglybounded,orequicontinuous,respectively([231,232]). These characterizations help us produce a (df)-space C (X) that is not a (DF)- c space[232],solvingabasicandlong-standingopenquestion.Thethirdcharacteri- zationisjoinedbyninemorethatsupplytenfoldanimpliedJarchowrequest.These forgeastronglinkwehappilyclaimbetweenhisbookandours. Chapter 3 deals with the K-analyticity of a topological space E and the con- cept of a(cid:2)resolution generated on E (i.e., a family of sets {Kα : α ∈ NN} such that E = K and K ⊂K if α ≤β). Compact resolutions (i.e., resolutions α α α β {K :α∈NN} whose members are compact sets) naturally appear in many situa- α tions in topology and functional analysis. Any K-analytic space admits a compact resolution[388],andformanytopologicalspacesXtheexistenceofsucharesolu- tionisenoughforXtobeK-analytic;(see[80],[82]).Manyoftheideasinthebook arerelatedtotheconceptofcompactresolutionandarealreadyinorhavebeenin- spiredbypapers[388],[80],[82].Itisaneasyandelementaryexercisetoobserve thatanyseparablemetricandcompletespaceE admitsacompactresolution,even swallowingcompactsets.InChapter3,wegathersomeresults,mostlyduetoVal- divia [421], about lcs (called quasi-(LB)-spaces) admitting resolutions consisting ofBanachdiscsandtheirrelationswiththeclosedgraphtheorems. J.Ka˛koletal.,DescriptiveTopologyinSelectedTopicsofFunctionalAnalysis, 1 DevelopmentsinMathematics24, DOI10.1007/978-1-4614-0529-0_1,©SpringerScience+BusinessMedia,LLC2011
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