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non-compactness1
Measuresof
1
Inmanybranchesofmathematicsthenotionofcompactnessplays
acrucialrole.Compactsetsaresomewhatflsmaller”orflsimpler”than
non-compactsets.Itturnsout,however,thatinthecontextofinfinite-
dimensionalnormedspacestheclassificationofsetsintocompactand
non-compactonesisverycrude.Aswewilllearninthenextchapter,
therearecaseswhenwemightwanttosaythatsomesetisflmore”
orflless”compactthananother.Thequestionis,whatcouldthateven
mean?
Inthischapter,weintroducethenotionofameasureofnon-compact-
ness,whichispreciselyawaytoanswertheabovequestion.Before
weseehowwecanapplythisconcepttofixedpointtheory(oneof
themanyofitsapplications),letuslookcloseratthefundamental
notionofcompactness,definethetwomostknownmeasuresofnon-
compactness,andexaminetheirproperties.
101Basicpropertiesandexamples
Oneofthelastthingswewouldliketodoistoscarethereaderon
thefirstpageofthefirstchapter(flNow’snotthetimeforfear.That
comeslater!”).So,wearegoingtostarteasy.Wewillfirstintroduce
threepropertiesofmetricspaces,whichturnouttobeequivalent
(andimportantenoughtodeserveaname).Eventhoughthereader
probablyalreadyknownswhatacompactmetricspaceis,wewill
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