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16
Measuresofnon-compactness
1
spendsometimediscussingthisnotion(andbrieflymentionitstwo
cousins–relativecompactnessandprecompactness).Whenwearedone
withthat,wearegoingtodefinetwomeasuresofnon-compactness
anddiscusstheirproperties.Forreaderswithcalculophobia(thisisnot
reallyaword,butitdescribesaverycommonfearoflengthycalcula-
tions),thatwillbethefllater”fromthequotationabove.Nevertheless,
weaskyoutostaywithus–therewillbebeautifulproofs,too,and
whilemathematicsingeneral–andnonlinearanalysisinparticular–
sometimesinvolvesboringcalculations,itisalsoaboutelegantthe-
orems,surprisingexamplesandawe-inspiringreasonings.(Wewill
encounterrepresentativesofallfourinthisbook.)
So,letusdivein.
101010Theorem0
Inametricspace
X
,thefollowingconditionsareequivalent:
(a)
everysequenceofpointsin
X
hasaconvergentsubsequence,inother
words,aclusterpoint,
(b)X
iscompleteandforany
8>
0thereexistsafinite
8
-net,thatis,aset
n
{a1,...,an}⊆XsuchthatX⊆
U
BX(ak,8),
k=1
(c)everyopencoveringofXhasafinitesubcovering.
Proof.
Wewillprovethateachofthefirsttwoconditionsimpliesthe
nextone,andthethirdoneimpliesthefirstagain.
(a)⇒(b)
.Thecompletenessofthespace
X
followseasilyfromthe
factthataCauchysequencehavingaconvergentsubsequenceisitself
convergenttothesamelimit.Indeed,if
(xn)n∈N
isaCauchysequence
and
(xn
k)k∈N
isitssubsequenceconvergentto
x∈X
,thengiven
8>
0thereis
N∈N
suchthat
d(x
,
xn
k)<1
28
and
d(xn
,
xm)<1
28
for
n
,
m
,
k>N
.As
nN>N
,fromthisweimmediatelyseethat
d(x
,
xn)<d(x
,
xn
N)+d(xn
N
,
xn)<8
forall
n>N
,whichmeans
thatthesequence(xn)n∈Nconvergestox.
Now,letusassumethatforsome
8>
0thereisnofinite
8
-net.
Wewillconstructasequence
(xn)n∈N
whichhasnoconvergentsub-
sequence.Let
x1
beanarbitrarypointin
X
.Forany
n>
2,inview
ofourassumption,itispossibletochooseanelement
xn
suchthat
