Table Of ContentCompact Textbooks in Mathematics
Jürgen Voigt
A Course
on Topological
Vector Spaces
CompactTextbooksinMathematics
CompactTextbooksinMathematics
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Ju¨rgenVoigt
A Course on
Topological Vector
Spaces
Ju¨rgenVoigt
InstitutfürAnalysis
FakultätMathematik
TechnischeUniversitätDresden
Dresden,Germany
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-030-32944-0 ISBN978-3-030-32945-7 (eBook)
https://doi.org/10.1007/978-3-030-32945-7
MathematicsSubjectClassification(2010):46A03,46-01,46A20,46A08,46A13
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v
Preface
Thetheoryoftopologicalvectorspaces–asabranchoffunctionalanalysis,
motivated by applications and pushed forward also for abstract reasons –
was developed over a long period of time, say, starting in the 1940s, and
adoptedinuniversitiesforteachinginthe1960s,whenitbecamemandatory
foradvancedstudentsinanalysistoacquireknowledgeinthistopic.Itwas
indeedinthelate1960sthatIattendedacourse,givenbythelateProf.W.
Roelcke, University of Munich, on topological vector spaces and received
myfundamentaleducationinthisarea.Whenworkinginpartialdifferential
equations, operator theory, or some other topics in functional analysis, I
alwaysappreciatedmyknowledgeinthisabstractpartoffunctionalanalysis,
mainlyasasomehowalwayspresentbackground.
It was in 2011, during discussions with some graduate students and
youngcolleagues,thatIdiscoveredthattheydidnothave,andmissed,this
kindofbackground–andinfactaskedhowtheyshouldhaveacquiredit,due
tothelackofofferedcourses.ItwasthenthatIdecidedtoteachacourseon
thistopic.
Duetothenatureofthecourse,thebookiscertainlynotintendedtogive
anexhaustivetreatmentofthetopic.Thebackgroundthereadershouldhave
isthematerialpresentedinabasiccourseinfunctionalanalysis.Infact,the
firsttwochaptersofthebookcontaintopicswhichmostlyhadbeentreated
alreadyinmybasiccourseonfunctionalanalysis.Also,asseenimmediately
fromthetableofcontents,thecourseisdirectedtowardthetheoryoflocally
convexspaces.
Theintendedmainobjectiveofthecoursewasthetreatmentoftopologies
fordualpairs,fundamentalpropertiesofwhicharecontainedinChapters3
to 6 – but unavoidably dual pairs are always present in the theory of
locally convex spaces. In particular, the introduction of polar topologies
and the Mackey–Arens theorem can be considered as a minimal kernel
for the treatment of dual pairs. The topics of Chapters 8 to 11, reflexivity,
completeness, locally convex final topology including applications, and
compactness,arestillprettystandardforthecourse.
Having covered these basic topics, I decided to present a choice of
resultswhichareofinterestevenforBanachspacesbutneedthetheoryof
locallyconvexspaces.ThesearetheKrein–ŠmuliantheoreminChapter12,
the Eberlein–Šmulian theorem in Chapter 13 and the theorem of Krein
in Chapter 14. Having talked so much about weakly compact sets in
Chapters 13 and 14, I used Chapter 15 to present an important nontrivial
example, in the form of weakly compact sets in L -spaces. Finally, in
1
Chapter16,Ithoughtitofinteresttopresentanexamplewhereitispossible
todeterminethebidualofalocallyconvexspaceforwhichonedoesnothave
anexplicitrepresentationofthedual.
vi Preface
ThetopicsofChapter7,onFréchetspaces,andChapter17,theKrein–
Milmantheorem,werenotpartoftheactualcourse.
Clearly,inapresentationofresultsthathavebeendevelopedoveralong
time,onecannotexpectmuchoriginality.Nevertheless,desiringtoproceed
to interesting topics as fast as possible, I tried to present a streamlined
approach, omitting many sidelines which might be interesting but not
directly contributing to the aim I had in mind. According to my personal
tastesinreading,Ipreferredaconcisestylewhereallneededingredientsare
mentioned,butsomeactivecollaborationofthereaderisrequired.
It may seem somewhat strange that I delegated the Hahn–Banach
theorem and the uniform boundedness theorem to appendices. The reason
isthatIconsideredthemasbelongingtotheprerequisitescoveredinabasic
courseonfunctionalanalysis(andinfactinthecourseitself,theywerenot
included).
Atthebeginningofeachchapter,Igiveabriefoutlineoftopicstreated
therein.InthenotesattheendofeachchapterItrytomentionthesources
forthemainresults,sometimesaddingfurthercomments.
Inanindexofnotationandanindex,thereadercanfindtheexplanation
ofthesymbolsandoftheterminologyusedinthetext.
Finally, I want to add acknowledgementsof various kinds. First of all,
I want to thank the late Prof. Walter Roelcke for his introduction to the
topic. Next, it is a pleasure to thank my colleagues and friends for many
years from Munich times, Peter Dierolf and the late Susanne Dierolf, for
many discussions and exchanges on various topics in the area as well as
for the collaboration with Peter Dierolf. Finally, to come to more recent
times,mythanksgotothelateProf.JohnHorváthforcommunicationonthe
manuscriptandforencouragement.ItisapleasuretothankSaschaTrostorff,
HendrikVogt,andMarcusWaurickformanydiscussionsonvarioustopics
inthebook,andtoSaschaTrostorffandMarcusWaurickforreadingthefirst
manuscript and discovering gaps, errors, and misprints. Also, I am much
obliged to Dirk Werner for various comments on contents, examples, and
misprints;inparticular,itwashissuggestiontoincludeachapteronFréchet
spaces,becauseoftheirimportanceinanalysis.
Andlastbutnotleast,alonganotherline,IthankmywifeMariannefor
lifelongsupportandpatience.
Dresden,Germany JürgenVoigt
August2019
vii
Contents
1 InitialTopology,TopologicalVectorSpaces,
WeakTopology................................................... 1
2 Convexity,SeparationTheorems,LocallyConvexSpaces.... 11
3 Polars,BipolarTheorem,PolarTopologies .................... 23
4 TheTikhonovandAlaoglu–BourbakiTheorems.............. 29
5 TheMackey–ArensTheorem.................................... 37
6 TopologiesonE(cid:2)(cid:2),Quasi-barrelledandBarrelledSpaces.... 45
7 FréchetSpacesandDF-Spaces.................................. 53
8 Reflexivity ........................................................ 63
9 Completeness .................................................... 71
10 LocallyConvexFinalTopology,TopologyofD((cid:2))............ 81
11 Precompact–Compact–Complete ............................ 93
12 TheBanach–DieudonnéandKrein–ŠmulianTheorems ...... 97
13 TheEberlein–ŠmulianandEberlein–Grothendieck
Theorems ......................................................... 103
14 Krein’sTheorem.................................................. 113
15 WeaklyCompactSetsinL1(μ).................................. 119
16 B(cid:2)(cid:2) =B............................................................ 125
0
17 TheKrein–MilmanTheorem..................................... 131
viii Contents
A TheHahn–BanachTheorem..................................... 139
B Baire’sTheoremandtheUniformBoundednessTheorem.... 143
References................................................................. 147
IndexofNotation......................................................... 151
Index....................................................................... 153
1
1
Initial Topology, Topological Vector
Spaces, Weak Topology
Themainobjectiveofthischapteristopresentthedefinitionoftopologicalvectorspaces
andtoderivesomefundamentalproperties.Wewillalsointroducedualpairsofvector
spaces and the weak topology. We start the chapter by briefly recalling concepts of
topologyandcontinuity,therebyalsofixingnotation.
LetX beaset,τ ⊆ P(X)(thepowersetofX). Then τ iscalled atopology,and
(X,τ)iscalledatopologicalspace,if
(cid:2)
foranyS ⊆τ onehas S∈(cid:3)τ,
foranyfiniteF ⊆τ onehas F∈τ.
(cid:2) (cid:3)
(This definition is with the understanding that ∅ = ∅, ∅ = X, with the
consequencethatalways∅,X∈τ.)Concerningnotation,wecouldalsowrite
(cid:4) (cid:4) (cid:5) (cid:5)
S = U, F = A.
U∈S A∈F
IfS = (Uι)ι∈I orF = (An)n∈N arefamilies ofsets, withN finite, thenonecan also
write
(cid:4)(cid:6) (cid:7) (cid:4) (cid:5)(cid:6) (cid:7) (cid:5)
U ; ι∈I = U, A ; n∈N = A .
ι ι n n
ι∈I n∈N
ThesetsU ∈τ arecalledopen,whereasasetA ⊆ X iscalledclosedifX\Ais
open.ForasetB ⊆Xwedefine
(cid:2)
◦
B (=intB):= {U; U∈τ, U ⊆B},theinteriorofB (anopenset),
(cid:3)
B (=clB):= {A; A⊇B, Aclosed},theclosureofB (aclosedset).
©SpringerNatureSwitzerlandAG2020
J.Voigt,ACourseonTopologicalVectorSpaces,CompactTextbooksinMathematics,
https://doi.org/10.1007/978-3-030-32945-7_1
