Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
Basicpropertiesandexamples
37
Therefore,
D(SE(
0,1
))>
2.This,togetherwiththeoppositeinequality,
provesthatD(SE(0,1))=2.
1010290Corollary0
TheKuratowskimeasureofnon-compactnessoftheclosed
unitballinaninfinite-dimensionalnormedspaceequals2.
Proof.
Denotetheclosedunitballby
B(
0,1
)
andtheunitsphereby
S(
0,1
)
.Usingthemonotonicityof
D
,weobtain2
=D(S(
0,1
))<
D(B(0,1))<diamB(0,1)<2.
1010300Corollary0
TheHausdorffmeasureofnon-compactnessofboththe
unitsphere
S(
0,1
)
andtheclosedunitball
B(
0,1
)
inaninfinite-dimensional
normedspaceequals1.
Proof.
Itisenoughtoobservethat1
=1
2D(S(
0,1
))<β(S(
0,1
))<
β(B(
0,1
))<
1,wherewefirstusedTheorem1.1.28,thenProposi-
tion1.1.14,thenthemonotonicityof
β
andfinallytheunsurprising
factthattheclosedunitballisincludedintheunionofthefamily
consistingofitself.
1010310Remark0
ItisnowclearthattheHausdorffmeasureofnon-
compactnessofanyballorsphereinaninfinite-dimensionalnormed
spaceisequaltoitsradius,andthattheKuratowskimeasureofaball
orsphereinsuchaspaceisequaltotwiceitsradius.
Wewillfinishthissectionwiththetheoremthatwastheactualmo-
tivationforKuratowskitoconsiderhismeasureofnon-compactness.
ItisageneralizationofCantor’stheorem,whichwestateasCorol-
lary1.1.33.
1010320Theorem
(Kuratowski)
0
Assumethat
(An)n∈N
isanon-increasing
sequenceofnon-emptyclosedsetsinacompletemetricspace
X
suchthat
∞
n→∞
lim
D(An)=
0.Then,
A=
n=1
Π
An
isnon-emptyandcompact.Moreover,
ifeverysetAnisconnected,thenthesetAisconnectedaswell.
Proof.
Foreach
n∈N
let
xn
beanarbitrary(butfixed)elementof
theset
An
.Put
B:={xn|n∈N}
and
Bk:={xn|n>k}
.Wehave
1
