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44
Measuresofnon-compactness
discussedinmoredetailinthenextchapter.)Itrequireslittleeffortto
checkthatsuchasetWisequicontinuous.
1
Notation0
Givenanon-emptyset
V⊆C([a
,
b]
;
E)
andapoint
t∈[a
,
b]
,
let
V(t):={x(t)|x∈V}
;wewillcallsuchasetasectionof
V
at
t
.In
asimilarvein,if
V⊆C([a
,
b]
;
E)
and
J⊆[a
,
b]
,weput
V(J):=U
V(t)
.
t∈J
Moreover,sometimes–inordertoavoidmisunderstandings–wewill
useanadditionalsubscripttodistinguishbetweenmeasuresofnon-
compactnessindifferentBanachspaces;forexample,theKuratowski
measureofnon-compactnessin
E
and
C([a
,
b]
;
E)
willbedenoted
byDEandDC,respectively.
ThedefinitionoftheKuratowskimeasureofnon-compactnessis
relativelysimple,butactuallycomputingitforagivensubsetisnot.
Wehaveseenthateventhecaseofaunitballrequiressomeingenuity
andanon-triviallemma.(Therearewaystocomputemeasuresofnon-
compactnessinsomespecificmetricspaces–seee.g.thepaper[57]–
butthetaskseemshopelessingeneral.)Itisthereforenowonderthat
anyresultwhichreducesfindingthemeasureofnon-compactness
ofsomecomplicatedsettosomethingthatatleastlookssimplerisof
interest.Ambrosetti’slemmaisanotableexampleofsucharesult.
Here,theproblemoffindingthemeasureofnon-compactnessof
asubsetofthespaceofvector-valuedcontinuousfunctionsonsome
realintervalisreducedtocomputingmeasuresofnon-compactness
ofitssections.
102080Lemma
(Ambrosetti)
0
Let
E
beaBanachspaceandlet
V
beanon-
empty,equicontinuousandboundedsubsetofC([a,b];E).Then,
(a)
thefunction
v:[a
,
b]→[
0,
+∞)
definedforeach
t∈[a
,
b]
bythe
formulav(t):=DE(V(t))isuniformlycontinuous,and
(b)DC(V)=DE(V([a,b]))=max
DE(V(t)).
t∈[a,b]
Proof.
Letusbeginwithfourremarks.Forthesakeofsimplicity,
throughouttheproof,theinterval
[a
,
b]
willbedenotedby
J
.Fur-
thermore,letusnotethattheboundednessof
V
in
C(J
;
E)
implies
thatthesets
V(J)
and
V(t)
,where
t∈J
,areboundedin
E
.Hence,the
