From vector spaces to function spaces: Introduction to functional analysis with applications

267 Pages·2012·1.34 MB·English

by  Yutaka Yamamoto

Yutaka Yamamoto
Kyoto University
Kyoto, Japan
From Vector Spaces
to Function Spaces
Introduction to Functional
Analysis with Applications
Society for Industrial and Applied Mathematics
Philadelphia
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Copyright © 2012 by the Society for Industrial and Applied Mathematics
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may
be reproduced, stored, or transmitted in any manner without the written permission of
the publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.
Library of Congress Cataloging-in-Publication Data
Yamamoto, Yutaka, 1950-
[Shisutemu to seigyo no sugaku. English]
From vector spaces to function spaces : introduction to functional analysis with
applications / Yutaka Yamamoto.
p. cm. -- (Other titles in applied mathematics)
Includes bibliographical references and index.
ISBN 978-1-611972-30-6 (alk. paper)
1. Functional analysis. 2. Engineering mathematics. I. Title.
TA347.F86Y3613 2012
515’.7--dc23
2012010732
is a registered trademark.
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Contents
Preface ix
GlossaryofNotation xiii
1 VectorSpacesRevisited 1
1.1 Finite-DimensionalVectorSpaces . . . . . . . . . . . . . . . . . . . . 1
1.2 LinearMappingsandMatrices . . . . . . . . . . . . . . . . . . . . . 14
1.3 SubspacesandQuotientSpaces . . . . . . . . . . . . . . . . . . . . . 19
1.4 DualityandDualSpaces. . . . . . . . . . . . . . . . . . . . . . . . . 30
2 NormedLinearSpacesandBanachSpaces 39
2.1 NormedLinearSpaces. . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 BanachSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 ClosedLinearSubspacesandQuotientSpaces . . . . . . . . . . . . . 51
2.4 Banach’sOpenMappingandClosedGraphTheorems . . . . . . . . . 54
2.5 Baire’sCategoryTheorem . . . . . . . . . . . . . . . . . . . . . . . . 55
3 InnerProductandHilbertSpaces 59
3.1 InnerProductSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 HilbertSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 ProjectionTheoremandBestApproximation . . . . . . . . . . . . . . 73
4 DualSpaces 77
4.1 DualSpacesandTheirNorms . . . . . . . . . . . . . . . . . . . . . . 77
4.2 TheRiesz–FréchetTheorem . . . . . . . . . . . . . . . . . . . . . . . 81
∗
4.3 WeakandWeak Topologies . . . . . . . . . . . . . . . . . . . . . . 82
4.4 DualityBetweenSubspacesandQuotientSpaces . . . . . . . . . . . . 85
5 TheSpaceL(X,Y)ofLinearOperators 89
5.1 TheSpaceL(X,Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 DualMappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 InverseOperators,Spectra,andResolvents . . . . . . . . . . . . . . . 92
5.4 AdjointOperatorsinHilbertSpace . . . . . . . . . . . . . . . . . . . 94
5.5 ExamplesofAdjointOperators . . . . . . . . . . . . . . . . . . . . . 96
5.6 HermitianOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 CompactOperatorsandSpectralResolution . . . . . . . . . . . . . . 101
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vi Contents
6 SchwartzDistributions 107
6.1 WhatAreDistributions? . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 TheSpaceofDistributions . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 DifferentiationofDistributions . . . . . . . . . . . . . . . . . . . . . 116
6.4 SupportofDistributions . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 ConvergenceofDistributions . . . . . . . . . . . . . . . . . . . . . . 119
6.6 ConvolutionofDistributions . . . . . . . . . . . . . . . . . . . . . . 126
6.7 SystemTheoreticInterpretationofConvolution . . . . . . . . . . . . 131
6.8 ApplicationofConvolution . . . . . . . . . . . . . . . . . . . . . . . 132
7 FourierSeriesandFourierTransform 141
7.1 FourierSeriesExpansioninL2[−π,π]orofPeriodicFunctions. . . . 141
7.2 FourierSeriesExpansionofDistributions. . . . . . . . . . . . . . . . 147
7.3 FourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4 SpaceS ofRapidlyDecreasingFunctions . . . . . . . . . . . . . . . 152
7.5 FourierTransformandConvolution . . . . . . . . . . . . . . . . . . . 156
7.6 ApplicationtotheSamplingTheorem . . . . . . . . . . . . . . . . . . 159
8 LaplaceTransform 165
8.1 LaplaceTransformforDistributions . . . . . . . . . . . . . . . . . . 165
8.2 InverseLaplaceTransforms . . . . . . . . . . . . . . . . . . . . . . . 170
8.3 Final-ValueTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9 HardySpaces 175
9.1 HardySpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.2 PoissonKernelandBoundaryValues . . . . . . . . . . . . . . . . . . 178
9.3 CanonicalFactorization . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.4 ShiftOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.5 NehariApproximationandGeneralizedInterpolation . . . . . . . . . 190
9.6 ApplicationofSarason’sTheorem. . . . . . . . . . . . . . . . . . . . 196
9.7 Nehari’sTheorem—Supplements . . . . . . . . . . . . . . . . . . . . 201
10 ApplicationstoSystemsandControl 203
10.1 LinearSystemsandControl . . . . . . . . . . . . . . . . . . . . . . . 203
10.2 ControlandFeedback . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.3 ControllabilityandObservability . . . . . . . . . . . . . . . . . . . . 207
10.4 Input/OutputCorrespondence . . . . . . . . . . . . . . . . . . . . . . 215
10.5 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
∞
10.6 H Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
10.7 SolutiontotheSensitivityMinimizationProblem . . . . . . . . . . . 224
10.8 GeneralSolutionforDistributedParameterSystems . . . . . . . . . . 229
10.9 SupplementaryRemarks . . . . . . . . . . . . . . . . . . . . . . . . . 230
A SomeBackgroundonSets,Mappings,andTopology 233
A.1 SetsandMappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.2 Reals,UpperBounds,etc. . . . . . . . . . . . . . . . . . . . . . . . . 235
A.3 TopologicalSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.4 ProductTopologicalSpaces . . . . . . . . . . . . . . . . . . . . . . . 239
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A.5 CompactSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
A.6 NormsandSeminorms . . . . . . . . . . . . . . . . . . . . . . . . . 241
A.7 ProofoftheHahn–BanachTheorem . . . . . . . . . . . . . . . . . . 241
A.8 TheHölder–MinkowskiInequalities . . . . . . . . . . . . . . . . . . 245
B TableofLaplaceTransforms 247
C Solutions 249
C.1 Chapter1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
C.2 Chapter2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
C.3 Chapter3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C.4 Chapter4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C.5 Chapter5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C.6 Chapter6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
C.7 Chapter7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
C.8 Chapter8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
C.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
D BibliographicalNotes 259
Bibliography 261
Index 265
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Preface
This book intends to give an accessible account of applied mathematics, mainly of
analysis subjects, with emphasis on functional analysis. The intended readers are senior
orgraduatestudentswhowishtostudyanalyticalmethodsinscienceandengineeringand
researcherswhoareinterestedinfunctionalanalyticmethods.
Needlesstosay, scientistsandengineerscanbenefitfrommathematics. Thisisnot
confinedtoameremeansofcomputationalaid,andindeedthebenefitcanbegreaterand
far-reaching if one becomes more familiar with advanced topics such as function spaces,
operators, and generalized functions. This book aims at giving an accessible account of
elementary real analysis, from normed spaces to Hilbert and Banach spaces, with some
extendedtreatmentofdistributiontheory,FourierandLaplaceanalyses,andHardyspaces,
accompanied by some applications to linear systems and control theory. In short, it is a
modernizedversionofwhathasbeentaughtasappliedanalysisinscienceandengineering
schools.
Tothisend,amoreconceptualunderstandingisrequired. Infact,conceptualunder-
standingisnotonlyindispensablebutalsoagreatadvantageeveninmanipulatingcompu-
tational tools. Unfortunately, it is not always accomplished, and indeed often left aside.
Mathematicsisoftenlearnedbymanypeopleasacollectionofmeretechniquesandswal-
lowedasveryformalprocedures.
This is deplorable, but from my own experience of teaching, its cure seems quite
difficult. Forstudentsandnovices,definitionsareoftendifficulttounderstand,andmathe-
maticalstructuresarehardtopenetrate,letalonethebackgroundmotivationastohowand
whytheyareformulatedandstudied.
This book has a dual purpose: one is to provide young students with an accessible
accountofaconceptualunderstandingoffundamentaltoolsinappliedmathematics. The
otheristogivethosewhoalreadyhavesomeexposuretoappliedmathematics,butwishto
acquireamoreunifiedandstreamlinedcomprehensionofthissubject,adeeperunderstand-
ingthroughbackgroundmotivations.
Toaccomplishthis,Ihaveattemptedto
• elaborateupontheunderlyingmotivationoftheconceptsthatarebeingdiscussedand
• describehowonecangetanideaforaproofandhowoneshouldformalizetheproof.
I emphasized more verbal, often informal, explanations rather than streams of logically
complete yet rather formal and detached arguments that are often difficult to follow for
nonexperts.
ix
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x Preface
Thetopicsthataredealtwithherearequitestandardandincludefundamentalnotions
of vector spaces, normed, Banach, and Hilbert spaces, and the operators acting on them.
They are in one way or another related to linearity, and understanding linear structures
formsacoreofthetreatmentofthisbook. Ihavetriedtogiveaunifiedapproachtoreal
analysis,suchasFourieranalysisandtheLaplacetransforms. Tothisend,distributionthe-
ory gives an optimal platform. The second half of this book thus starts with distribution
theory. Witharigoroustreatmentofthistheory,thereaderwillseethatvariousfundamental
results in Fourier and Laplace analyses can be understood in a unified way. Chapter 9 is
devoted to a treatment of Hardy spaces. This is followed by a treatment of a remarkably
successfulapplicationinmoderncontroltheoryinChapter10.
Letusgiveamoredetailedoverviewofthecontentsofthebook.Asanintroduction,
we start with some basics in vector spaces in Chapter 1. This chapter intends to give a
conceptualoverviewandreviewofvectorspaces. Thistopicisoften,quiteunfortunately,
ahiddenstumblingpointforstudentslackinganin-depthunderstandingofwhatlinearity
isallabout. Ihavetriedtoilluminatetheconceptualsidesofthenotionofvectorspacesin
thischapter. Sometimes,Ihaveattemptedtoshowtheideaofaprooffirstandthenshow
that a complete proof is a “realization” of making such an idea logically more complete.
Ihavealsochosentogivedetailedtreatmentsofdualandquotientspacesinthischapter.
Theyareofteneitherverylightlytouchedonorneglectedcompletelyinstandardcourses
inappliedmathematics. Iexpectthatthereaderwillbecomemoreaccustomedtostandard
mathematicalthinkingasaresultofthischapter.
FromChapter2on,weproceedtomoreadvancedtreatmentsofinfinite-dimensional
spaces. Amongthem, normedlinearspacesaremostfundamentalandallowratherdirect
generalizations of finite-dimensional spaces. Anew element here is the notion of norms,
whichintroducestheconceptoftopology. Topologyplaysacentralroleinstudyinginfinite-
dimensionalspacesandlinearmapsactingonthem. Normedspacesgivethefirststeptoward
suchstudies.
Aproblem is that limits cannot, generally, be taken freely for those sequences that
may appear to converge (i.e., so-called Cauchy sequences). The crux of analysis lies in
limiting processes, and to take full advantage of them, the space in question has to be
“closed” under such operations. In other words, the space must be complete. Complete
normedlinearspacesarecalledBanachspaces,andtherearemanyinterestingandpowerful
theorems derived for them. If, further, the norm is derived from an inner product, the
space is called a Hilbert space. Hilbert spaces possess a number of important properties
due to the very nature of inner products—for example, the notion of orthogonality. The
Rieszrepresentationtheoremforcontinuouslinearfunctionals,aswellasitsoutcomeofthe
orthogonalprojectiontheorem,isatypicalconsequenceofaninnerproductstructureand
completeness. Hilbertspaceappearsveryfrequentlyinmeasuringsignalsduetoitsaffinity
with such concepts as energy, and hence in many optimization problems in science and
engineeringapplications. Theproblemofbestapproximationisnaturallystudiedthrough
theprojectiontheoremintheframeworkofHilbertspace. ThisisalsoatopicofChapter3.
Discussing properties of spaces on their own will give only half the story. What is
equallyimportantistheirinterrelationship,andthisexhibitsitselfthroughlinearoperators.
In this connection, dual spaces play crucial roles in studying Banach and Hilbert spaces.
We give in Chapter 5 a basic treatment of them and prove the spectral resolution theo-
remforcompactself-adjointoperators—whatisknownastheHilbert–Schmidtexpansion
theorem.
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Preface xi
We turn our attention to Schwartz distributions in Chapter 6. This theory makes
transparent the treatments of many problems in analysis such as differential equations,
Fourieranalysis(Chapter7),Laplacetransforms(Chapter8),andPoissonintegrals(Chap-
ter9),anditishighlyvaluableinmanyareasofappliedmathematics,bothtechnicallyand
conceptually. Inspiteofthisfact,thistheoryisoftenveryinformallytreatedinintroduc-
tory books and thereby hardly appreciated by engineers. I have strived to explain why it
is important and how some more rigorous treatments are necessary, attempting an easily
accessibleaccountforthistheorywhilenotsacrificingmathematicalrigortoomuch.
The usefulness of distributions hinges largely on the notion of the delta function
(distribution). This is the unity element with respect to convolution, and this is why it
appears so frequently in many situations of applied mathematics. Many basic results in
applied mathematics are indeed understood from this viewpoint. For example, a Fourier
series or Poisson’s integral formula is the convolution of an objective function with the
Dirichlet or the Poisson kernel, and its convergence to such a target function is a result
ofthefactthattherespectivekernelconvergestothedeltadistribution. Wetakethisasa
leadingprincipleofthesecondhalfofthisbook,andIattemptedtoclarifythestructureof
thislineofideasinthetreatmentsofsuchconvergenceresultsinChapter6,andsubsequent
Chapters7and8dealingwithFourierandLaplacetransforms.
Chapter9givesabasictreatmentofHardyspaces,whichinturnplayedafundamental
∞
roleinmoderncontroltheory. So-calledH controltheoryiswhatweareconcernedwith.
Of particular interest is generalized interpolation theory given here, which also plays a
fundamental role in this new control theory. We will prove Nehari’s theorem as well as
Sarason’s theorem, along with the applications to the Nevanlinna–Pick interpolation and
theCarathéodory–Fejértheorem. Wewillalsodiscusstherelationshipwithboundaryvalues
and the inner-outer factorization theorem. I have tried to give an easy entry point to this
theory.
Chapter10isdevotedtobasiclinearcontrolsystemtheory. Startingwithaninverted
pendulumexample,wewillseesuchbasicconceptsaslinearsystemmodels,theconceptof
feedback, controllability, and observability, a realization problem, an input/output frame-
∞
work,andtransferfunctions,leadingtothesimplestcaseofH controltheory. Wewillsee
solutionsviatheNevanlinna–Pickinterpolation,Nehari’stheorem,andSarason’stheorem,
applyingtheresultsofChapter9. Fourieranalysis(Chapter7)andtheLaplacetransforms
(Chapter8)alsoplaykeyroleshere. Thereaderwillnodoubtseepartofabeautifulappli-
cationofHardyspacetheorytocontrolandsystems. Itishopedthatthischaptercanserve
asaconciseintroductiontothosewhoarenotnecessarilyfamiliarwiththesubject.
Itisalwaysadifficultquestionhowmuchpreliminaryknowledgeoneshouldassume
andhowself-containedthebookshouldbe. Ihavemadethefollowingassumptions:
• As prerequisite, I assumed that the reader has taken an elementary course in linear
algebraandbasiccalculus. Roughlyspeaking,Iassumedthereaderisatthejunior
orahigherlevelinscienceandengineeringschools.
• Ididnotassumemuchadvancedknowledgebeyondthelevelabove. Thetheoryof
integration(Lebesgueintegral)isdesirable,butIchosenottorelyonit.
• However, if applied very rigorously, the above principles can lead to a logical dif-
ficulty. Forexample, Fubini’stheoreminLebesgueintegrationtheory, variousthe-
orems in general topology, etc., can be an obstacle for self-contained treatments.
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xii Preface
Itriedtogiveprecisereferencesinsuchcasesandnotoverloadthereaderwithsuch
concepts.
• SomefundamentalnotionsinsetsandtopologyareexplainedinAppendixA. Itried
tomaketheexpositionaselementaryandintuitiveastobebeneficialtostudentswho
arenotwellversedinsuchnotions. Someadvancedbackgroundmaterial, e.g., the
Hahn–Banachtheorem,isalsogivenhere.
This book is based on the Japanese version published by the Asakura Publishing
Company,Ltd.,in1998. ThepresentEnglishversiondiffersfromthepredecessorinmany
respects. Particularly, it now contains Chapter 10 for application to system and control
theory,whichwasnotpresentintheJapaneseversion. Ihavealsomadeseveraladditions,
butittookmuchlongerthanexpectedtocompletethisversion. Partofthereasonliesin
itsdualpurpose—tomakethebookaccessibletothosewhofirststudytheabovesubjects
andsimultaneouslyworthwhileforreferencepurposesonadvancedtopics.Agoodbalance
wasnoteasytofind,butIhopethatthereaderfindsthebookhelpfulinbothrespects.
ItisapleasuretoacknowledgetheprecioushelpIhavereceivedfrommanycolleagues
andfriends,towhomIamsomuchindebted,notonlyduringthecourseofthepreparation
ofthisbookbutalsooverthecourseofalong-rangefriendshipfromwhichIhavelearned
somuch. Particularly,JanWillemsreadthroughthewholemanuscriptandgaveextensive
andconstructivecomments. IamalsomostgratefulforhisfriendshipandwhatIlearned
from numerous discussions with him. Likewise, I have greatly benefited from the com-
mentsandcorrectionsmadebyThanosAntoulas,BrianAnderson,BruceFrancis,Tryphon
Georgiou, Nobuyuki Higashimori, Pramod Khargonekar, Hitay Özbay, Eduardo Sontag,
andMathukumalliVidyasagar. IwouldalsoliketoacknowledgethehelpofMasaakiNaga-
haraandNaokiHayashiforpolishingsomeproofsandalsohelpingmeinpreparingsome
figures. I wish to acknowledge the great help I received from the editors at SIAM in
publishingthepresentbook.
I would like to conclude this preface by thanking my family, particularly my wife
Mamikoforhersupportineveryrespectinthepast30years. Withoutherhelpitwouldnot
havebeenpossibletocompletethiswork.
YutakaYamamoto
Kyoto
February,2012
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Chapter 10
Applications to Systems and
Control
Moderncontrolandsystemtheoryisbuiltuponasolidmathematicalbasis.Advancedmath-
ematical concepts developed in this book are indispensable for further in-depth study of
elaboratetheoryofsystemsandcontrol. Weherepresentsomefundamentalsofthistheory.
10.1 Linear Systems and Control
Many physical systems are described by differential equations, ordinary or partial. We
oftenwishtocontrolsuchsystems,naturalorartificial. Inotherwords,givenasystem,we
wantittobehaveinsuchawaythatisinsomesensedesirableforus. Normally,asystem
may contain three types of variables: an input variable that drives the system, an output
variable that can either be observed by some devices or affect the external environment,
and a state variable that describes the internal behavior of the system which is not nec-
essarily observable from the external environment. Summarizing, a differential equation
descriptionmaytakethefollowingform:
dx
(t)=f(x(t),u(t)), (10.1)
dt
y(t)=g(x(t)), (10.2)
wherex(t)∈Rn,u(t)∈Rm,andy(t)∈Rpdenote,respectively,thestate,input,andoutput
variables. Thesetwoequationshavethestructurethat
1. starting with an initial state x at some time t , and for a given input u(t), t ≥0,
0 0
(10.1)describeshowthesystemstatex(t)evolvesintime,and
2. theoutputy(t)isdeterminedbythepresentvalueofthestatex(t)accordingto(10.2).
The fundamental objective of control is to design or synthesize u(t) so as to make the
abovesystembehavedesirably. Herethecontrolinputu(t)issomethingwecanmaneuver
tocontrolthesystemstatex(t),andtheoutputy(t)isthencontrolledaccordingly. Ingeneral,
it is necessary to make x(t) behave nicely rather than merely controlling the behavior of
y(t)alone,astherecanbesomehiddenbehaviorinx(t)thatdoesnotexplicitlyappearin
thatofy(t).
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