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ones).Thus,thereisaneedfornonlinearanalysis,inwhichsuchideas
arestudied.
TheconceptforthebookcamefromProfessorDariuszBugajewski
about2015,andthefirstvisionforitwastobeaseriesoflecture
notestoaccompanyhisclassonnonlinearanalysis.Wewouldliketo
expressourgratitudetohimbothfortheideaandfortheopportunity
totakepartinpreparationofthislecture.Soonitbecameobviousthat
properlypresentingthetopicsinvolvedinwrittenformcallsforamore
ambitiousproject.Itturnsoutthatapparentlyyoucannotdecidewhat
thebookisgoingtobelikeinadvance–somethingsbecomevisible
onlywellafteryouwroteseveraldozen(orevenhundred)pages...
Settingouttowriteabookislikegoingoutsideyourfrontdoorto
greetawanderingwizard–youhaveabsolutelynoideawhatliesin
storeforyou.Toparaphrasetheadviceoneoldchapgavehisyounger
cousin:flIt’sadangerousbusiness,writingabook.Youtakeablank
sheetofpaper,andifyoudon’tkeepyourpen,thereisnoknowing
whereyoumightbesweptoffto.”
Earlyonwesetafewgoalswhichwehopedwouldmakeour
bookdifferent–notjustyourbasic,average,everyday,ordinary,run-
of-the-mill,ho-hummathstextbook.Ofcourse,thereaderistheone
competenttoassessifwesucceeded.
Oneofourgoalswastomakethisbookreallyaccessible.Inorder
toachieveit,wedecidedtoincludeproofsasdetailedaswecould
(althoughthelevelofdetailslowlydecreasesinlaterchapters).We
wereespeciallycarefultoprovemanythingsshruggedasideinmost
booksasflobvious”orfltrivial”,butbeingquitedifficultforstudents
orjuniorresearchersencounteringthemforthefirsttime.(Having
learnedaboutourapproach,afriendofourssaidthatweshouldtitle
thebookAnIntroductiontoFoundationsofElementsofNonlinearAnalysis
orsomethingtothateffect.Wedidnotwanttoknowifhewasjoking.)
Thatdoesnotmean,however,thatweshiedawayfrommoredifficult
subjects.Whilewesometimesomittedcertainproofs(especiallywhen
theyarenotreallyaboutnonlinearanalysisbutsomeotherbranchof
mathematics,andwejustneededthecorrespondingresulttoprove
something),inthecaseofreallyhardtheoremslyingwithinthescope
