Descriptive Topology in Selected Topics of Functional Analysis

493 Pages·2011·2.394 MB·English

by  Jerzy Kąkol, Wiesław Kubiś, Manuel López-Pellicer (auth.)

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
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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