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enoughtofillintheoccasionalgapsinthereasonings.(Eventhere,we
tendedtoerronthesideofmoreverbosity.)
Thefirstchapterisaveryconciseintroductiontothetheoryof
measuresofnon-compactness.Althoughthenotionofacompactsetis
probablyknowntomostreaders,wedecidedtobeginwithintroducing
itsdefinitionandbasicproperties.Compactnessisoneofthetopicsso
crucialtononlinearanalysis(andinfact,mathematicalanalysis–and
mathematics–ingeneral)thatstartingthebookwithitseemedveryfit-
ting.Thechapterisfairlyshort,though,sincewearemainlyinterested
inthedefinitionsandpropertiesneededdowntheroad.Nevertheless,
weincludeasectionshowinganexampleofapplyingtheapparatus
ofmeasuresofnon-compactnesstoabstractdifferentialequations.
Thesecondchapterdealswithfixedpointtheory.Thisisaclassical
topic,andmanybookshavebeenwrittenonit.Outofthevastarrayof
fixedpointresultsofallshapesandsizeswechosetwobroadclasses
oftheorems.Theformerone,representedfirstandforemostbythe
Banachcontractionprinciple,isaboutLipschitzmappings,animpor-
tantclassoffunctionsencounteredinnonlinearanalysis.Thelatter
one,startingwiththeBrouwertheorem,promptlyfollowedbythe
Schaudertheoremanditsgeneralizations,isaboutcompactnessand
fixedpoints.Wefinishthechapterwithtwoexamplesofapplications–
onetointegralequationsandonetogametheory.
Inthethirdchapterwedeviatefromthewell-chartedterritory,
andpresentthebasicsofthetheoryofhyperconvexmetricspaces.The
studyofthemarisedfroma(successful)attempttoproveametric
versionoftheHahn–Banachextensiontheorem.Westartwiththe
definitions,followedbysomeexamples.Wethendevoteasection
tosomefactsabouthyperconvexityofBanachspaces.Here,wein-
cludeaproofofonespecialcaseofaninterestingtheoremrelating
hyperconvexityofatwo-dimensionalnormwiththesetofextremal
pointsoftheunitball.Thisillustratesthesomewhatsurprisingfact
thatreasoningsreminiscentofhigh-schoolgeometryhavetheirplace
intwenty-firstcenturymathematicalresearch.Afterall,assomeone
oncesaid,ancientGreeksareinasensenotourpredecessors,butolder
colleagues.Then,weexaminetheraisond’êtreofhyperconvexspaces–
