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I
Introduction
OnethingthereadersmayexpectfromtheIntroductionisthatweare
goingtoexplainwhatthisbookisabout.Thisissurprisinglydifficult.
Sometimesonemaygettheimpressionthatflnonlinearanalysis”isan
umbrellatermforstuffthatdidnotfitanywhereelse.Hereisourtake
onthemeaningoftheterm(notethatthisisavastoversimplification
forillustrativepurposesonly).Incalculus,welearn(amongotheren-
tities)aboutfunctions,theirproperties(likecontinuity,integrability,
differentiability,periodicity,monotonicity...)andoperationsonthem
(likecomposition,differentiation,integration...).Also,zerosoffunc-
tionsarejustsolutionstoequations.Functionalanalysisgoesonestep
furtherandstartswithexaminingwholespacesoffunctions.Someof
thepropertieswementionednowcorrespondtospecificspaces(of
continuousfunctions,ofintegrablefunctions,ofcontinuouslydiffer-
entiablefunctions...),andsomeoperations(likeintegration)corre-
spondtolinearoperatorsonthesespaces.Further,someequations
(likecertaindifferentialorintegralequations)alsogiverisetolinear
operators.Soonitturnsoutthatthisisnotenough,bothformathe-
maticians,alwayshungryforinterestingpropertiesofabstractnotions,
andscientists,whoneedtodescribenaturalworldphenomenawith
theapparatusofmathematics.Therearebothnaturalsetsoffunctions
whichdonotpossessthestructureofalinearspace(liketheperiodic
ormonotonefunctions),naturaloperatorswhicharenotlinear(like
superposition),andequationswhichdonothavecorrespondinglinear
operators(butcanbeexpressedintermsoffixedpointsofnonlinear
5
