Table Of ContentCarlos A. de Moura
Functional
Analysis Tools
for Practical Use
in Sciences and
Engineering
Functional Analysis Tools for Practical Use
in Sciences and Engineering
Carlos A. de Moura
Functional Analysis Tools for
Practical Use in Sciences and
Engineering
CarlosA.deMoura
IME–InstitutodeMatemáticaeEstatística
RiodeJaneiroStateUniversity-UERJ
RiodeJaneiro,RJ,Brazil
ThisworkwassupportedbyFAPERJ-FundaçãoCarlosChagasFilhodeAmparoàPesquisa
doEstadodoRiodeJaneiroE-26/202.500/2019
ISBN978-3-031-10597-5 ISBN978-3-031-10598-2 (eBook)
https://doi.org/10.1007/978-3-031-10598-2
MathematicsSubjectClassification:40A30,46-XX,46-01,46B40,46B50,46F12,46Nxx,46N40
TranslationfromthePortugueselanguageedition:“AnáliseFuncionalparaAplicações:Posologia”by
CarlosA.deMoura,©CarlosA.deMoura2002.PublishedbyCiênciaModerna.AllRightsReserved.
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Preface (to the Portuguese Version)
MathematicaltopicslabelledasFunctionalAnalysis–concepts,definitions,results
– compose a deep example of a mathematical structure aimed at applications –
problems originated from Physics, from Engineering Sciences, even from other
areas within Mathematics. That is also the purpose carried by this book: to bring
familiarity to the reader who has been dwelling on topics where these tools are
needed, but definitely who does not wish to develop research on the associated
theory.Instead,aslongassafetyandprecisionarestronglyrequired,lookforward
to be kept safe of fallacious slips and misconceptions that a more shallow contact
oftenleadsto.
Thetextemphasizesmotivation,justificationforthechoicesmade,therightway
to employ discussed techniques, but in a large number of spots, proofs are not
shown.Insuchcases,thepreferredauthor’sreferencesarepointed.Completeproofs
– or sometimes only their sketch – get exposed whenever fared their knowledge
andfamiliarityasimperativeforasafeuseoftheresultstheyclaim.Orelsewhen
they indicate a technique which itself will bring its presence in any of these very
applications.
A main thread may be found throughout this exposition: to link the ideas
inside the completion of a metric space to those on the continuous extension
of operators. That is the tool that lets us, for example, to introduce in a quick
way Lebesgue integral, Sobolev spaces, and that leads to the almost ubiquitous
regularizationtechnique,orthemollifiers,inK.O.Friedrichs’favoriteterminology.
Thisbookisthethirdversionof[22],writtenforashorttimeschoolandwhich,
due to the lack in Brazil, at that time, of texts with this approach, was later used
in different academic environments, until a second version [23] was published.
Thisonetookonthesameroad,andwiththesuggestionsofmanycolleaguesthat
have used it, I faced the adventure of the present write-up, more careful and with
some additions. The previous versions fail to include applications because these
were part of other mini-courses presented at the schools – this and those other
ones have been written for. Now this version includes additional applications, all
related to Numerical Analysis – finite elements, stability for numerical solutions
for differential equations. Besides, the presented results about Lebesgue integral
vii
viii Preface(tothePortugueseVersion)
havebeenputtogetherandaremorecomplete.TheFourierandLaplacetransforms,
togetherwiththetempereddistributions,arenowincluded.
Thesharpnessofthemathematicalremarksaswellassuggestionsaboutwriting
language and exposition view, thanks to the previous careful reading carried on
by Professor Dinamérico Pombo – a very particular friend of mine, then at UFF
MathematicsInstitute,wasofoutofsightvalue.
To Professor David Isaacson, from Rensselaer Polytechnic Institute, I owe a
dialogue with such a convincing power, which I always brought it back to my
memorywhenthegivingupoptionwaspullingmeout.
A large portion of this book was written when the author was visiting the
Mathematics and Computation Institutes of the Fluminense Federal University –
UFF.Iwouldliketoexpressmythankfulwordsfortheirsupport.
The book was finished and published while Iwas a visitoratthe cozy environ-
mentfromIME-UERJ,theMathematicsandStatisticsInstituteoftheRiodeJaneiro
StateUniversity.Thevaluablesupportmustalsobeacknowledged.
Conventions
Our notation follows the mathematical text standards, either theoretical or from
applications; exceptions are described whenever introduced, or in the paragraphs
thatfollow.Wemakeuseofthefollowingconventions,allusual.
N :thepositiveintegers
Z :theintegers
Q :therationals
R :therealline
RN :theEuclideanspacewithN dimensions
C :thecomplexplane
FortwosetsAandB,wedenotebyA\B thethirdcombiningset–countingA∪B
andA∩B –ofelementsfromAthatdonotbelongtoB.
Given the reals a,b, with −∞ < a ≤ b < ∞, we denote by [a,b] the closed
interval {x ∈ IR;a ≤ x ≤ b}, while for the open interval {x ∈ IR;a < x < b}
we shall use either ]a,b[ or (a,b), with analogous convention for the semi-open
intervals.
Sequences will be denoted by {xn}n∈IN, (xn)n∈IN or, in a simpler way, by {xn}
or (x ), even using upper indices. Not much strictness either with the notation for
n
functions:f(x),f(·),orf.
The symbol := in a given expression means that the “right-hand side” defines
whateveroccupiesthe“left-handside.”Asregardstothesymbol•,topointtothe
end of some topic, its use was quite stingy, it only shows up if its absence would
bringdoubtstothereader.Observethatarelativeofthisisusedtoindicatewhichis
themainargumentinsomeexpressions,like f(·), (cid:7)·(cid:7) oranalogousones.
Preface(tothePortugueseVersion) ix
Tofinish,itiswisenottotakethenotationforgranted,aslongaswehaveallowed
itbeing“unadonnamobile”.Indeed,itmayexhibitchangesfromchaptertochapter,
wheneverconvenient,asforexamplein:
(cid:2)
(cid:7)f(cid:7) = (cid:7)f(cid:7) = (cid:7)f(cid:7) = (cid:7)f(cid:7) = { f2}1/2,
L2 0 2 0,2
(cid:2) (cid:2)
(cid:7)f(cid:7) = (cid:7)f(cid:7) = (cid:7)f(cid:7) = { f2 + (f(cid:8))2}1/2.
H1 1 1,2
Itinerary
With Chap. 1 we look to motivate and, to some extent, to justify the developed
theory being announced. Chapter 2 introduces or recalls definitions and notation
for abstract spaces as well as for specific function spaces. This endeavor requires
Lebesgueintegral,whosedevelopmentforthereallinethroughanalmostgeodesic
way,wediscuss.Andtrytoleaveclearthechangesneededtoapplytheresultsand
proofsfor IRN .
TheaiminChap.3istodiscusssomepropertiesofdualspaces,whicharefirmly
gluedtothoseparticularspacespresentedinthepreviouschapter;inparticular,the
classical–andindispensable–identificationsthatallowtooperatewiththem.They
turnouttobetoolsfrequentlyusedinChap.4,whereideal,orgeneralized,functions
are introduced. They allow us to refer with the needed rigor to, for example, the
Dirac delta. Thereby, we also present the construction for Sobolev spaces besides
introducingtheFourierandLaplacetransforms.
The so-called Linear Functional Analysis basic triplet is the core of Chap. 5,
some applications being discussed therein. Chapter 6 introduces the concept of
compactness,fundamentalfortheconstructionofapproximationsequences,aslong
asitallowstoguaranteeconvergence,inmanycontexts.
Up to this point, this text reading is assumed to be made sequentially. Now,
Chap. 71 uses only concepts introduced on Chap. 2, while the last one depends
on concepts from Chaps. 2 and 3. Chapter 7 generalizes the concept of derivative
for functions defined on normed spaces. Such a generalization allows to use, on
these spaces, approximation algorithms, like Newton-Raphson’s, besides several
optimizationmethods.
In the last chapter, one finds basic results about Hilbert spaces that turn out to
be of heavy importance to numerical approximations as well as to applications
on Physics, e.g., Quantum Mechanics. Within this framework, spectral theory is
discussedonlyforcompactoperators,sothatwesuggestformoregeneralbrowsing
1Portugueseversionhaditasthelastchapter.Weallowedourselvesheretoadaptthetranslation
dataasthisversionhasexchangeditwiththechapterpreviouslytakenastheseventh.
x Preface(tothePortugueseVersion)
[6,72].Theseones arealsoreference forthesemi-grouptheory which,despiteits
importanceinseveralapplications,see[36,49],welackedtobringintodiscussion.
RiodeJaneiro,Brazil CarlosA.deMoura
October24th2002
P.S.:contactswouldbewelcomeat
[email protected]
Preface to the English Version
Taken the decision to write myself this translation of [25], I thought at once that,
forthistask,nosensetoapplytheoldcliché“traduttore=traditore”.ButquicklyI
realizedthatitwouldbehardtorestrainfromscatteredminorchanges,besidestwo
sensible ones: the addition of the last section to the last chapter as well as, in the
firstchapter,the5thencountertothefirstsection.
Despitealotoftimegonesincethepreviousversion,Ifailtofindmuchtoadd
totheabovetranslatedpreface.Nevertheless,Ioughttorecognizethefourcorners
ofasquare.Eachonelodgesanunforgettablehelpwell.
ThefirstisoccupiedbymywifeSandra.Impossibletobelievehowsheactedto
bringjoytothesepandemicdays(andthatisasshealwaysacts!).Withoutherpush,
noendforthistask...
Along-timefriend–lifehasturnedhimmybrother–,JerryA.Goldstein,made
me so happy to have him installed at the second corner, testifying the decades we
haveshared.Hedefinitelyconvincedmeitwouldbeworthtoputhoursandthoughts
onthisversion.
ThesupportfromUERJ,theuniversityIworkforsincethismillenniumstarted,
particularlyfromIME,itsmathinstitute,withfriendsandfriendlycolleagues,fills
upthenextcorner.
Thelastonesettlesallsine-qua-nonscientific/financialhelpfromFAPERJ–the
CarlosChagasFilhoRiodeJaneiroStateResearchSupportFoundation–whichhas
providedagrantthatbecameessential.Itisapleasuretoacknowledgethatsupport
filedby:
E-26/202.500/2019,ProgramforVisitingProfessorPV-2019.
RiodeJaneiro,Brazil CarlosA.deMoura
September8th,2020
[email protected]
xi
