Topological Vector Spaces and Their Applications

466 Pages·2017·2.4 MB·English

by  V.I. Bogachev

Springer Monographs in Mathematics
V.I. Bogachev
O.G. Smolyanov
Topological Vector
Spaces and Their
Applications
Springer Monographs in Mathematics
Editors-in-Chief
Isabelle Gallagher, Paris, France
Minhyong Kim, Oxford, UK
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V.I. Bogachev · O.G. Smolyanov
Topological Vector Spaces
and Their Applications
1 3
V.I. Bogachev O.G. Smolyanov
Department of Mechanics Department of Mechanics
and Mathematics and Mathematics
Moscow State University Moscow State University
Moscow Moscow
Russia Russia
and
Higher School of Economics
National Research University
Moscow
Russia
ISSN 1439-7382 ISSN 2196-9922 (electronic)
Springer Monographs in Mathematics
ISBN 978-3-319-57116-4 ISBN 978-3-319-57117-1 (eBook)
DOI 10.1007/978-3-319-57117-1
Library of Congress Control Number: 2017939903
Mathematics Subject Classification: 46A03, 58C20, 28C20
The book is an expanded and revised version of the Russian edition under the same title, published in
Regular and Chaotic Dynamics, Moscow-Izhevsk, 2012.
© Springer International Publishing AG 2017
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Contents
Preface ix
Chapter1. Introductiontothetheoryoftopological
vectorspaces 1
1.1. Linearspacesandtopology 1
1.2. Basicdefinitions 11
1.3. Examples 17
1.4. Convexsets 29
1.5. Finite-dimensionalandnormablespaces 35
1.6. Metrizability 41
1.7. Completenessandcompletions 45
1.8. Compactandprecompactsets 53
1.9. Linearoperators 59
1.10. TheHahn–Banachtheorem: geometricform 63
1.11. TheHahn–Banachtheorem: analyticform 71
1.12. Complementsandexercises 81
Uniformspaces(81).Convexcompactsets(84).Fixedpointtheorems(86).
Sequencespaces(89).DualstoBanachspaces(90).Separabilityproperties(91).
Continuousselectionsandextensions(93).Exercises(94).
Chapter2. Methodsofconstructingtopological
vectorspaces 101
2.1. Projectivetopologies 101
2.2. Examplesofprojectivelimits 104
2.3. Inductivetopologies 109
2.4. Examplesofinductivelimits 113
2.5. Grothendieck’sconstruction 119
2.6. Strictinductivelimits 125
2.7. Inductivelimitswithcompactembeddings 127
2.8. Tensorproducts 130
2.9. Nuclearspaces 134
2.10. Complementsandexercises 139
PropertiesofthespacesDandD(cid:2)(139).Absolutelysummingoperators(143).
Localcompleteness(145).Exercises(147).
v
vi Contents
Chapter3. Duality 153
3.1. Polars 153
3.2. Topologiescompatiblewithduality 158
3.3. Adjointoperators 162
3.4. Weakcompactness 164
3.5. Barrelledspaces 170
3.6. Bornologicalspaces 175
3.7. Thestrongtopologyandreflexivity 180
3.8. Criteriaforcompleteness 186
3.9. Theclosedgraphtheorem 193
3.10. Compactoperators 199
3.11. TheFredholmalternative 205
3.12. Complementsandexercises 208
Bairespaces(208).TheBorelgraphtheorem(211).Boundingsets(212).
TheJamestheorem(213).Topologicalpropertiesoflocallyconvexspaces(214).
Eberlein–Sˇmulianproperties(218).Schauderbases(219).Minimalspaces
andpowersoftherealline(221).Exercises(224).
Chapter4. Differentialcalculus 243
4.1. Differentiabilitywithrespecttosystemsofsets 244
4.2. Examples 251
4.3. Differentiabilityandcontinuity 257
4.4. Differentiabilityandcontinuityalongasubspace 261
4.5. Thederivativeofacomposition 263
4.6. Themeanvaluetheorem 273
4.7. Taylor’sformula 275
4.8. Partialderivatives 278
4.9. TheinversionofTaylor’sformulaandthechainrule 279
4.10. Complementsandexercises 289
Theinversefunctiontheorem(289).Polynomials(291).Ordinarydifferential
equationsinlocallyconvexspaces(294).Passagetothelimitinderivatives(297).
Completenessofspacesofsmoothmappings(300).Differentiabilityvia
pseudotopologies(305).SmoothfunctionsonBanachspaces(307).Exercises(308).
Chapter5. Measuresonlinearspaces 311
5.1. Cylindricalsets 311
5.2. Measuresontopologicalspaces 313
5.3. Transformationsandconvergenceofmeasures 321
5.4. Cylindricalmeasures 327
5.5. TheFouriertransform 333
5.6. Covarianceoperatorsandmeansofmeasures 337
5.7. Gaussianmeasures 345
5.8. Quasi-measures 354
5.9. Sufficienttopologies 357
5.10. TheSazonovandGross–Sazonovtopologies 359
Contents vii
5.11. Conditionsforcountableadditivity 366
5.12. Complementsandexercises 372
Convolution(372).0–1laws(376).Convexmeasures(378).Thecentrallimit
theorem(381).Infinitelydivisibleandstablemeasures(383).Banachsupports
ofmeasures(391).Infinite-dimensionalWienerprocesses(393).Prohorovlocally
convexspaces(394).Measurablelinearandpolylinearfunctions(399).Relations
betweendifferentσ-algebras(409).Radonifyingoperators(412).Measurable
norms(412).Exercises(413).
Comments 419
Bibliography 425
Authorindex 447
Subjectindex 451
Preface
Thisbookgivesaconciseexpositionofthefundamentalsofthetheoryoftopo-
logicalvectorspaces,complementedbyasurveyofthemostimportantresultsofa
moresubtlenature,whichcannotbequalifiedasbasic,butknowledgeofwhichis
usefulforapplications,and,finally,someofsuchapplicationsconnectedwithdif-
ferential calculus in infinite-dimensional spaces and measure theory. Almost half
of the book is devoted to these applications, which makes it very different from
thewholeseriesofknowntextsontopologicalvectorspaces. Anothernotabledif-
ferencebetweenthisbookandknowntreatiseslikeBourbaki[87],Edwards[150],
Grothendieck [207], Jarchow [237], Kelley, Namioka [270], Ko¨the [292], Nar-
ici, Beckenstein [365], Pe´rez Carreras, Bonet [385], Robertson, Robertson [420],
Schaefer[436],Tre`ves[530],andWilansky[567]isthatwedecidedtoincludealso
some results without proofs (this does not concern the fundamentals, of course)
with references instead, which enables us to inform the reader about many rela-
tively recent achievements; some of them are disguised as exercises (with refer-
encestotheliterature),suchexercisesshouldnotbeconfusedwithusualexercises
markedbythesymbol◦. Thus,withrespecttothepresentedinformation,ourbook
is not covered by any other book on this subject (though, we cannot claim that it
coversanysuchbook).
Chapter 1 contains the fundamentals of the theory, including a large list of
concrete examples, some general concepts (convex sets, seminorms, linear map-
pings) and a number of facts, the most important of which is the Hahn–Banach
theoremonextensionsoffunctionalsinitsdiverseversions.
The main material of Chapter 2 is connected with projective and inductive
limits (including strict inductive limits and inductive limits with compact em-
beddings, which is not sufficiently discussed in the existing literature), and also
Grothendieck’smethodofconstructingBanachspacesembeddedintolocallycon-
vexspaces.
Chapter3containstheclassicalmaterialrelatedtotheso-calleddualitytheory,
i.e.,introductionofdifferentlocallyconvextopologiesonagivenspacegivingthe
samesetofcontinuouslinearfunctionals. ThecentraltopicsherearetheMackey–
Arens theorem on topologies compatible with duality, the results on weak com-
pactness, including the Eberlein–Sˇmulian and Krein–Sˇmulian theorems, and also
someconceptsandfactsconnectedwithcompletenessoflocallyconvexspaces.
ix
x Preface
Chapter 4 is devoted to the fundamentals of the differentiation theory in lo-
cally convex spaces. It presents a general scheme of differentiability with respect
to a system of sets (partial cases of which are Gaˆteaux, Hadamard and Fre´chet
differentiabilities) and a thorough discussion of important for applications differ-
entiabilitieswithrespecttosystemsofboundedandcompactsets.
Chapter 5 gives a concise introduction to measure theory on locally convex
spaces. Here we discuss extensions of cylindrical measures, the Fourier trans-
form and conditions for the countable additivity in its terms (in particular, the
Minlos and Sazonov theorems and their generalizations), covariance operators,
measurable linear functionals and operators, measurable polynomials, and some
importantclassesofmeasures(suchasGaussian,stable,andconvex).
Each chapter opens with a brief synopsis of its content. All chapters contain
many additional subsections with some more specialized information related to
the main themes of the chapter, and also many exercises are given (more difficult
ones are provided with hints or references). The book ends with the historic-
bibliographiccomments,thelistofreferences(withindicationofpagenumbersof
citingtheincludedworks),andtheauthorandsubjectindices.
The prerequisites for the first chapter of this book are just a grasp knowledge
of calculus and linear algebra and some experience with basic concepts of topol-
ogy, but for a thorough study it is advisable to be acquainted with a university
course of functional analysis (following any text, e.g., Kolmogorov, Fomin [284]
orRudin[425]).
We are very grateful to T.O. Banakh, E.D. Kosov, I. Marshall, S.N. Popova,
A.V. Shaposhnikov, A.S. Tregubov and E.V. Yurova for useful remarks and cor-
rections.
Our work on this book began 25 years ago by the initiative of Vladimir
Ivanovich Sobolev (1913–1995), the author of a series of widely known texts on
functional analysis (including one of the first Russian texts, published as early as
in1951),anditscompletionisatributetothememoryofthisremarkablescientist
andteacher.