Calculus without derivatives

541 Pages·6.95 MB·English

by  Penot, Jean-Paul

Graduate Texts in Mathematics 266
Graduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Graduate Texts in Mathematics bridge the gap between passive study and
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textbooksingraduatecourses,theyarealsosuitableforindividualstudy.
Forfurthervolumes:
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Jean-Paul Penot
Calculus Without Derivatives
123
Jean-PaulPenot
LaboratoireJacques-LouisLions
Universite´PierreetMarieCurie
Paris,France
ISSN0072-5285
ISBN978-1-4614-4537-1 ISBN978-1-4614-4538-8(eBook)
DOI10.1007/978-1-4614-4538-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012945592
MathematicsSubjectClassification(2010):49J52,49J53,58C20,54C60,52A41,90C30
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Preface
AfamousIndiansayingcanbeapproximativelyphrasedinthefollowingway:“Our
earthisnotjustalegacyfromourparents;itisaloanfromourchildren.”
In mathematical analysis, a precious legacy has been given to us: differential
calculusandintegralcalculusaretoolsthatplayanimportantroleinthepresentstate
ofknowledgeandtechnology.Theyevengaverisetoaphilosophicalopinion,often
calleddeterminism,thatamountstosayingthatanyphenomenoncanbepredicted,
providedoneknowsitsrulesandtheinitialconditions.Suchatriumphantclaimhas
beenmitigatedbymoderntheoriessuchasquantummechanics.The“fuzziness”one
meetsinthisbookpresentssomeanalogywithmodernmechanics.Insomesense,
itisthebestwecanleavetoourchildrenincasetheyhavetodealwithroughdata.
Inthemiddleofthenineteenthcentury,Weierstrassmadeclearthefactthatnot
allfunctionsaredifferentiable.Heevenprovedthattherearecontinuousfunctionsof
onerealvariablethatarenowheredifferentiable.Althoughsuch“exotic”functions
arenotnegligible,itappearsthatmostnonsmoothfunctionsthataremetinconcrete
mathematicalproblemshaveabehaviorthatisnotbeyondthereachofanalysis.
It is the purpose of the present book to show that an organized bundle of
knowledgecanbeappliedtosituationsinwhichdifferentiabilityisnotpresent.
Infavorablecases,suchaspointwisemaximaoffinitefamiliesofdifferentiable
functionsorsumsofconvexfunctionswithdifferentiablefunctions,arathersimple
apparatus allows us to extend in a unified way the rules known in the realms
of convex analysis and differentiable analysis. The pioneers in this restricted
framework were Pshenichnii, Ioffe, and Tikhomirov (and later on, Demy’anov,
Janin, among others). For general functions, more subtle constructions must be
devised.
Already at this elementary stage, a combination of geometrical and analytical
viewpoints gives greater and more incisive insight. Such a unified viewpoint is
one of the revolutionary characteristics of nonsmooth analysis: functions, sets,
mappings,andmultimappings(orcorrespondences)canbeconsideredtobeequally
important, and the links between them allow us to detect fruitful consequences.
Historically, geometrical concepts (tangent and normal cones with Bouligand,
Severi,Choquet,Dubovitskii-Milyutin,...)appearedearlierthananalyticalnotions
vii
viii Preface
(generalized directional derivatives, subdifferentials with Clarke, Ioffe, Kruger,
Mordukhovich,...).
On the other hand, the variety of situations and needs has led to different
approaches. In our opinion, it would not be sensible to leave the reader with the
impressionthatasingletypeofanswerorconstructioncanmeetalltheneedsone
mayencounter(itisnoteventhecasewithsmoothcalculus).Itisourpurposetogive
thereadertheabilitytochooseanappropriateschemedependingonthespecificities
oftheproblemathand.Quiteoften,theproblemitselfleadstoanadaptedspace.In
turn, the space often commandsthe choice of the subdifferentialas a manageable
substituteforthederivative.
In this book we endeavorto present a balancedpicture of the most elementary
attempts to replace a derivative with a one-sided generalized derivative called a
subdifferential.Thismeansthatinsteadofassociatingalinearformtoafunctionat
somereferencepointinordertosummarizesomeinformationaboutthebehaviorof
thefunctionaroundthatpoint,oneassociatesabunchoflinearforms.Ofcourse,the
usefulnessofsuchaprocessreliesontheaccuratenessoftheinformationprovided
by such a set of linear forms. It also relies on the calculus rules one can design.
Thesetworequirementsappearto besomewhatantagonistic.Therefore,itmaybe
worthwhile to dispose of various approaches satisfying at least one of these two
requirements.
Inspiteofthevarietyofapproaches,wehopethatourpresentationherewillgive
an impression of unity. We do not consider the topic as a field full of disorder.
On the contrary, it has its own methods, and its various achievements justify a
comprehensive approach that has not yet been presented. Still, we do not look
for completeness; we rather prefer to present significant tools and methods. The
references,notes,exercises,andsupplementswepresentwillhelpthereadertoget
amorethoroughinsightintothesubject.
Inwritingabook,onehastofaceadelicatechallenge:eitherfollowatraditionor
prepareforamorerigoroususe.Ourexperiencewithtextsthatwerewrittenabout
a lifetime ago showed us that the need for rigor and precision has increased and
is likely to increase more. Thus, we have avoided some common abuses such as
confusingafunctionwithitsvalue,a sequencewithitsgeneralterm,a spacewith
its dual, the gradient of a function with its derivative, the adjoint of a continuous
linearmapwithitstranspose.Thatchoicemayleadtounusualexpressions.Butin
general, we have made efforts to reach as much simplicity as possible in proofs,
terminology and notation, even if some proofs remain long. Moreover, we have
preferred suggestive names (such as allied, coherence, gap, soft) to complicated
expressions or acronyms, and we have avoided a heavy use of multiple indices,
of Greek letters (and also of Cyrillic, Gothic, Hebrew fonts). It appearsto us that
sophisticated notation blossoms when the concepts are fresh and still obscure; as
soonastheconceptsappearasnaturalandsimple,thenotationtendstogetsimpler
too. Of course, besides mathematicians who are attached to traditions, there are
someotherswhoimplicitlypresentthemselvesasmagiciansorlearnedpeopleand
liketokeepsophisticatednotation.
Preface ix
Let us present in greater detail the analysis that served as a guideline for this
book.
Thefieldofmathematicsoffersa numberoftopicspresentingbeautifulresults.
However, many of them are rather remote from practical applications. This fact
makes them not too attractive to many students. Still, they are proposed in many
coursesbecausetheyareconsideredeitherasimportantfromatheoreticalviewpoint
orpreciousfortheformationofminds.
Itisthepurposeofthisbooktopresentfundamentalaspectsofanalysisthathave
close connections with applications. There is no need to insist on the success of
analysis. So manyachievementsof moderntechnologyrely on methodsor results
from mathematical analysis that it is difficult to imagine what our lives would be
like if the consequencesofthe so-calledinfinitesimalanalysisof Fermat,Leibniz,
Newton,Eulerandmanyotherswouldbewithdrawnfromus.
However, the classical differential calculus is unable to handle a number of
problems in which order plays a key role; J.-J. Moreau called them “unilateral
problems,” i.e., one-sided problems. Usually, they are caused by constraints or
obstacles.
A few decades ago, some tools were designed to study such problems. They
are applied in a variety of fields, such as economics, mechanics, optimization,
numericalanalysis,partialdifferentialequations.Webelievethatthisrichspectrum
of applications can be attractive for the reader and deserves a sequel to this book
with complementary references, since here we do not consider applications as
importantas those in optimal control theory and mathematicalanalysis. Also, we
do not consider special classes of functions or sets, and we do not even evoke
higher-ordernotions,althoughconsideringsecond-ordergeneralizedderivativesof
nondifferentiablefunctionscanbeconsideredafeat!
Besides some elements of topology and functional analysis oriented to our
needs,wegatherherethreeapproaches:differentialcalculus,convexanalysis,and
nonsmooth analysis. The third of these is the most recent, but it is becoming a
classicaltopicencompassingthefirsttwo.
The novelty of a joint presentation of these topics is justified by several
arguments.Firstofall,sincenonsmoothanalysisencompassesbothconvexanalysis
anddifferentialcalculus,itisnaturaltopresentthesetwosubjectsasthetwobasic
elementsonwhichnonsmoothanalysisisbuilt.Theybothserveasanintroduction
to the newest topic. Moreover, they are both used as ingredients in the proofs of
calculusrulesinthenonsmoothframework.Ontheotherhand,nonsmoothanalysis
representsan incentive to enrich convexanalysis (and maybe differentialcalculus
too, as shown here by the novelty of incorporating directional smoothness in the
approach).Asanexample,wementiontherelationshipbetweenthesubdifferential
of the distance function to a closed convex set C at some point z out of C and
the normalcone to the setC at pointsofC that almostminimize the distance to z
(Exercises6and7ofSect.7.1ofthechapteronconvexanalysis).Anotherexample
isthefuzzycalculusthatiscommontoconvexanalysisandnonsmoothanalysisand
waspromptedbythelastdomain.