﻿
542 Chapter8
MetricSpaces
Theorem2.2.8impliesthatforeachf inCŒa;bthereisaconstantM
f
whichdepends
onf,suchthat
jf.x/jM
f
if axb;
andTheorem2.2.12impliesthatthereisaconstantı
f
whichdependsonfandsuchthat
jf.x
1
/f.x
2
/j if x
1
;x
2
2Œa;b and jx
1
x
2
j<ı
f
:
ThedifferenceinDeﬁnition8.2.11isthatthesameMandıapplytoallfinT.
Theorem8.2.11
AnonemptysubsetTofCŒa;biscompactifandonlyifitisclosed;
uniformlybounded;andequicontinuous.
Proof
Fornecessity, supposethatT T iscompact. . ThenT T isclosed(Theorem8.2.6)
andtotallybounded(Theorem8.2.8). Therefore,if  >0,thereisaﬁnitesubsetT
D
fg
1
;g
2
;:::;g
k
gofCŒa;bsuchthatiff 2T,thenkfg
i
kforsomeiinf1;2;:::;kg.
IfwetemporarilyletD1,thisimpliesthat
kfkDk.fg
i
/Cg
i
kkfg
i
kCkg
i
k1Ckg
i
k;
whichimplies(8.2.6)with
MD1Cmax
˚
kg
i
k
ˇ
ˇ
1ik
:
For(8.2.7),weagainletbearbitary,andwrite
jf.x
1
/f.x
2
/jjf.x
1
/g
i
.x
1
/jCjg
i
.x
1
/g
i
.x
2
/jCjg
i
.x
2
/f.x
2
/j
jg
i
.x
1
/g
i
.x
2
/jC2kfg
i
k
<jg
i
.x
1
/g
i
.x
2
/jC2:
(8.2.8)
Sinceeachoftheﬁnitelymanyfunctionsg
1
,g
2
,...,g
k
isuniformlycontinuousonŒa;b
(Theorem2.2.12),thereisaı>0suchthat
jg
i
.x
1
/g
i
.x
2
/j< if jx
1
x
2
j<ı; 1ik:
Thisand(8.2.8)imply(8.2.7)with replacedby3. Sincethisreplacementisofno
consequence,thisprovesnecessity.
Forsufﬁciency,wewillshowthatTistotallybounded.SinceTisclosedbyassumption
andCŒa;biscomplete,Theorem8.2.9willthenimplythatT iscompact.
Letmandnbepositiveintegersandlet
r
DaC
r
m
.ba/; 0rm; and
s
D
sM
n
; nsnI
thatis,a D
0
< 
1
< <
m
D bisapartitionofŒa;bintosubintervalsoflength
.ba/=m,andM D D 
n
<
nC1
< < 
n1
< 
n
DM isapartitionofthe
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Section8.2
CompactSetsinaMetricSpace
543
segmentofthey-axisbetweeny DMandyDM intosubsegmentsoflengthM=n.
LetS
mn
bethesubsetofCŒa;bconsistingoffunctionsgsuchthat
fg.
0
/;g.
1
/;:::;g.
m
/gf
n
;
nC1
:::;
n1
;
n
g
andgislinearonŒ
i1
;
i
,1i m.Sincethereareonly.mC1/.2nC1/pointsofthe
form.
r
;
s
/,S
mn
isaﬁnitesubsetofCŒa;b.
Nowsupposethat>0,andchooseı>0tosatisfy(8.2.7). Choosemandnsothat
.ba/=m<ıand2M=n<.IffisanarbitrarymemberofT,thereisaginS
mn
such
that
jg.
i
/f.
i
/j<; 0i i m:
(8.2.9)
If0im1,
jg.
i
/g.
iC1
/jDjg.
i
/f.
i
/jCjf.
i
/f.
iC1
/jCjf.
iC1
/g.
iC1
/j: (8.2.10)
Since
iC1

i
<ı,(8.2.7),(8.2.9),and(8.2.10)implythat
jg.
i
/g.
iC1
/j<3:
Therefore,
jg.
i
/g.x/j<3; 
i
x
iC1
;
(8.2.11)
sincegislinearonŒ
i
;
iC1
.
NowletxbeanarbitrarypointinŒa;b,andchooseisothatx2Œ
i
;
iC1
.Then
jf.x/g.x/jjf.x/f.
i
/jCjf.
i
/g.
i
/jCjg.
i
/g.x/j;
so(8.2.7),(8.2.9),and(8.2.11)implythatjf.x/g.x/j<5,axb.Therefore,S
mn
isaﬁnite5-netforT,soTistotallybounded.
Theorem8.2.12(AscoliArzelaTheorem)
SupposethatF isaninﬁniteuni-
formlyboundedandequicontinuousfamilyoffunctionsonŒa;b:Thenthereisasequence
ff
n
ginF thatconvergesuniformlytoacontinuousfunctiononŒa;b:
Proof
LetT betheclosureofF;thatis,f f 2 2 T T ifandonlyifeitherf f 2 2 T T orf
istheuniformlimitofasequenceofmembersofF. ThenT T isalsouniformlybounded
andequicontinuous(verify),andT isclosed. . Hence,T T iscompact,byTheorem8.2.12.
Therefore,F hasalimitpointinT. (Inthiscontext,thelimitpointisafunctionf f in
T.)Sincef isalimitpointofF,thereisforeachintegernafunctionf
n
inF suchthat
kf
n
fk<1=n;thatisff
n
gconvergesuniformlytof onŒa;b.
8.2Exercises
1.
SupposethatT
1
,T
2
,...,T
k
arecompactsetsinametricspace.A;/. Showthat
[k
jD1
T
j
iscompact.
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544 Chapter8
MetricSpaces
2. (a)
Showthataclosedsubsetofacompactsetiscompact.
(b)
SupposethatT isanycollectionofclosedsubsetsofametricspace.A;/,
andsome
b
T inT T iscompact.Showthat\
˚
T
ˇ
ˇ
T 2T
iscompact.
(c)
ShowthatifT isacollectionofcompactsubsetsofametricspace.A;/,
then\
˚
T
ˇ
ˇ
T 2T
iscompact.
3.
IfS andT arenonemptysubsetsofametricspace.A;/,wedeﬁnethedistance
fromStoT by
dist.S;T/Dinf
˚
.s;t/
ˇ
ˇ
s2S;t2T
:
ShowthatifSandT arecompact,thendist.S;T/D.s;t/forsomesinS and
sometinT.
4. (a)
Showthateverytotallyboundedsetisbounded.
(b)
Let
ı
ir
D
1
ifiDr;
0
ifi¤r;
andletT bethesubsetof`
1
consistingofthesequencesX
r
D fı
ir
g1
iD1
,
r1.ShowthatTisbounded,butnottotallybounded.
5.
LetT beacompactsubsetofametricspace.A;/.Showthattherearemembers
s
and
tofTsuchthatd.
s;
t/Dd.T/.
6.
LetTbethesubsetof`
1
suchthatjx
i
j
i
,i 1,where
P
1
iD1
i
<1.Show
thatT iscompact.
7.
LetT bethesubsetof`
2
suchthatjx
i
j
i
,i 1,where
P
1
i
2
i
<1. Show
thatT iscompact.
8.
LetS beanonemptysubsetofametricspace.A;/andletu
0
beanarbitrary
memberofA. ShowthatS S isboundedifandonlyifD D
˚
.u;u
0
/
ˇ
ˇ
u2S
is
bounded.
9.
Let.A;/beametricspace.
(a)
Prove:IfSisaboundedsubsetofA,then
S(closureofS)isbounded.Find
d.
S/.
(b)
Prove: IfeveryboundedclosedsubsetofAiscompact,then.A;/iscom-
plete.
10.
Let.A;/bethemetricspacedeﬁnedinExercise8.1.16Let
T DT
1
T
2
T
k
;
whereT
i
A
i
andT
i
¤;,1i k. ShowthatT T iscompactifandonlyT
i
is
compactfor1i k.
11.
Let.A;/bethemetricspacedeﬁnedinExercise8.1.17.Let
T DT
1
T
2
T
n
;
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Section8.3
ContinuousFunctionsonMetricSpaces
545
whereT
i
A
i
andT
i
¤;,i 1.ShowthatifT iscompact,thenT
i
iscompact
foralli1.
12.
LetfT
n
g1
nD1
beasequenceofnonemptyclosedsetsofametricspacesuchthat
(a)
T
1
iscompact;
(b)
T
nC1
T
n
,n1;and
(c)
lim
n!1
d.T
n
/ D0. . Showthat
\
1
nD1
T
n
containsexactlyonemember.
8.3CONTINUOUS FUNCTIONSONMETRICSPACES
InChapter westudiedreal-valuedfunctionsdeﬁnedonsubsetsofR
n
,andinChapter6.4.
westudiedfunctionsdeﬁnedonsubsetsofR
n
withvaluesinR
m
. Theseareexamplesof
functionsdeﬁnedononemetricspacewithvaluesinanothermetricspace.(Ofcourse,the
twospacesarethesameifnDm.)
Inthissectionwebrieﬂyconsiderfunctionsdeﬁnedonsubsetsofametricspace.A;/
withvaluesinametricspace.B;/.Weindicatethatf issuchafunctionbywriting
f W.A;/!.B;/:
Thedomainandrangeoff arethesets
D
f
D
˚
u2A
ˇ
ˇ
f.u/isdeﬁned
and
R
f
D
˚
v2B
ˇ
ˇ
vDf.u/forsomeuinD
f
:
Deﬁnition8.3.1
Wesaythat
lim
u!
b
u
f.u/Dbv
ifbu2
D
f
andforeach>0thereisaı>0suchthat
.f.u/;bv/< if u2D
f
and
0<.u;bu/<ı:
(8.3.1)
Deﬁnition8.3.2
Wesaythatfiscontinuousatbuifbu2D
f
andforeach>0there
isaı>0suchthat
.f.u/;f.bu//< if u2D
f
\N
ı
.bu/:
(8.3.2)
Iff iscontinuousateverypointofasetS,thenfiscontinuousonS.
Notethat(8.3.2)canbewrittenas
f.D
f
\N
ı
.bu//N
.f.bu//:
Also,f isautomaticallycontinuousateveryisolatedpointofD
f
.(Why?)
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546 Chapter8
MetricSpaces
Example8.3.1
If.A;kk/isanormedvectorspace,thenTheorem8.3.5impliesthat
f.u/Dkukisacontinuousfunctionfrom.A;/toR,since
jkukkbukjkubuk:
HereweareapplyingDeﬁnition8.3.2with.u;bu/Dkubukand.v;bv/Djvbvj.
Theorem8.3.3
Supposethatbu2
D
f
:Then
lim
u!
b
u
f.u/Dbv
(8.3.3)
ifandonlyif
lim
n!1
f.u
n
/Dbv
(8.3.4)
foreverysequencefu
n
ginD
f
suchthat
lim
n!1
u
n
Dbu:
(8.3.5)
Proof
Supposethat(8.3.3)is true, , andletfu
n
g beasequence inD
f
thatsatisﬁes
(8.3.5). Let > > 0andchooseı ı > > 0tosatisfy(8.3.1). . From(8.3.5), , thereisaninte-
gerN suchthat.u
n
;bu/<ıifn N. . Therefore,.f.u
n
/;bv/ <ifn N,which
implies(8.3.4).
Fortheconverse,supposethat(8.3.3)isfalse. Thenthereisan
0
>0andasequence
fu
n
ginD
f
suchthat.u
n
;bu/<1=nand.f.u
n
/;bv/
0
,so(8.3.4)isfalse.
Weleavetheproofofthenexttwotheoremstoyou.
Theorem8.3.4
Afunctionf iscontinuousatbuifandonlyif
lim
u!
b
u
f.u/Df.bu/:
Theorem8.3.5
Afunctionf iscontinuousatbuifandonlyif
lim
n!1
f.u
n
/Df.bu/
wheneverfu
n
gisasequenceinD
f
thatconvergestobu.
Theorem8.3.6
Iff iscontinuousonacompactsetT;thenf.T/iscompact.
Proof
Letfv
n
gbeaninﬁnitesequenceinf.T/.Foreachn,v
n
Df.u
n
/forsomeu
n
2
T. SinceT T iscompact,fu
n
ghasasubsequencefu
n
j
gsuchthatlim
j!1
u
n
j
Dbu2 T
(Theorem8.2.4). FromTheorem8.3.5,lim
j!1
f.u
n
j
/Df.bu/;thatis,lim
j!1
v
n
j
D
f.bu/.Therefore,f.T/iscompact,againbyTheorem8.2.4.
Deﬁnition8.3.7
Afunctionf isuniformlycontinuousonasubsetSofD
f
ifforeach
>0thereisaı>0suchthat
.f.u/;f.v//< whenever .u;v/<ı and
u;v2S:
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Section8.3
ContinuousFunctionsonMetricSpaces
547
Theorem8.3.8
IffiscontinuousonacompactsetT;thenf isuniformlycontinuous
onT.
Proof
Iff isnotuniformlycontinuousonT,thenforsome
0
>0therearesequences
fu
n
gandfv
n
ginTsuchthat.u
n
;v
n
/<1=nand
.f.u
n
/;f.v
n
//
0
:
(8.3.6)
SinceTiscompact,fu
n
ghasasubsequencefu
n
k
gthatconvergestoalimitbuinT (Theo-
rem8.2.4).Since.u
n
k
;v
n
k
/<1=n
k
,lim
k!1
v
n
k
Dbualso.Then
lim
k!1
f.u
n
k
/D lim
k!1
f.v
n
k
/Df.bu/
Deﬁnition8.3.9
Iff W.A;/!.A;/isdeﬁnedonallofAandthereisaconstant˛
in.0;1/suchthat
.f.u/;f.v//˛.u;v/ forall .u;v/2AA;
(8.3.7)
thenf isacontractionof.A;/.
Wenotethatacontractionof.A;/isuniformlycontinuousonA.
Theorem8.3.10(Contraction MappingTheorem)
Iff isacontraction
ofacompletemetricspace.A;/;thentheequation
f.u/Du
(8.3.8)
hasauniquesolution:
Proof
Toseethat(8.3.8)cannothavemorethanonesolution,supposethatuDf.u/
andvDf.v/.Then
.u;v/D.f.u/;f.v//:
(8.3.9)
However,(8.3.7)impliesthat
.f.u/;f.v//˛.u;v/:
(8.3.10)
Since(8.3.9)and(8.3.10)implythat
.u;v/˛.u;v/
and˛<1,itfollowsthat.u;v/D0.HenceuDv.
Wewillnowshowthat(8.3.8)hasasolution.Withu
0
arbitrary,deﬁne
u
n
Df.u
n1
/; n1:
(8.3.11)
Wewillshowthatfu
n
gconverges.From(8.3.7)and(8.3.11),
.u
nC1
;u
n
/D.f.u
n
/;f.u
n1
//˛.u
n
;u
n1
/:
(8.3.12)
548 Chapter8
MetricSpaces
Theinequality
.u
nC1
;u
n
/˛
n
.u
1
;u
0
/; n0;
(8.3.13)
followsbyinductionfrom(8.3.12).Ifn>m,repeatedapplicationofthetriangleinequality
yields
.u
n
;u
m
/.u
n
;u
n1
/C.u
n1
;u
n2
/CC.u
mC1
;u
m
/;
and(8.3.13)yields
.u
n
;u
m
/.u
1
;u
0
m
.1C˛CC˛
nm1
/<
˛
m
1˛
:
Nowitfollowsthat
.u
n
;u
m
/<
.u
1
;u
0
/
1˛
˛
N
if n;m>N;
and,sincelim
N!1
˛D0,fu
n
gisaCauchysequence.SinceAiscomplete,fu
n
ghasa
limitbu.Sincef iscontinuousatbu,
f.bu/D lim
n!1
f.u
n1
/D lim
n!1
u
n
Dbu;
whereTheorem 8.3.5impliestheﬁrstequalityand(8.3.11)impliesthesecond.
Example8.3.2
SupposethathDh.x/iscontinuousonŒa;b,KDK.x;y/iscon-
tinuousonŒa;bŒa;b,andjK.x;y/j  M ifa  x;y  b. . Showthatifjj j <
1=M.ba/thereisauniqueuinCŒa;bsuchthat
u.x/Dh.x/C
Z
b
a
K.x;y/u.y/dy; axb:
(8.3.14)
(ThisisFredholm’sintegralequation.)
Solution
LetAbeCŒa;b,whichiscomplete.Ifu2CŒa;b,letf.u/Dv,where
v.x/Dh.x/C
Z
b
a
K.x;y/u.y/dy; axb:
Sincev2CŒa;b,f WCŒa;b!CŒa;b.Ifu
1
,u
2
2CŒa;b,then
jv
1
.x/v
2
.x/jjj
Z
b
a
jK.x;y/jju
1
.y/v
1
.y/jdy;
so
kv
1
v
2
kjjM.ba/ku
1
u
2
k:
SincejjM.ba/<1,f isacontraction.Hence,thereisauniqueuinCŒa;bsuchthat
f.u/Du.Thisusatisﬁes(8.3.14).
Section8.3
ContinuousFunctionsonMetricSpaces
549
8.3Exercises
1.
Supposethatf W.A;/!.B;/andD
f
DA. Showthatthefollowingstate-
mentsareequivalent.
(a)
f iscontinuousonA.
(b)
IfV isanyopensetin.B;/,thenf1.V/isopenin.A;/.
(c)
IfVisanyclosedsetin.B;/,thenf
1
.V/isclosedin.A;/.
2.
1
[A
2
,where
A
1
andA
2
arenonemptydisjointopensets. Supposethat.A;/isconnectedand
f W.A;/!.B;/,whereD
f
DA,R
f
DB,andf iscontinuousonA.Show
that.B;/isconnected.
3.
Letf beacontinuousreal-valuedfunctiononacompactsubsetSofametricspace
.A;/.LetbetheusualmetriconR;thatis,.x;y/Djxyj.
(a)
Showthatf isboundedonS.
(b)
Let˛Dinf
u2S
f.u/andˇ Dsup
u2S
f.u/. Showthattherearepointsu
1
andu
2
inŒa;bsuchthatf.u
1
/D˛andf.u
2
/Dˇ.
4.
Letf W.A;/ / !.B;/becontinuousonasubsetU U ofA. . Let
ubeinU and
deﬁnethereal-valuedfunctiongW.A;/!Rby
g.u/D.f.u/;f.
u//; u2U:
(a)
ShowthatgiscontinuousonU.
(b)
ShowthatifU iscompact,thengisuniformlycontinuousonU.
(c)
ShowthatifU iscompact, , thenthereisabu2 2 U suchthatg.u/   g.bu/,
u2U.
5.
Supposethat.A;/,.B;/,and.C;/aremetricspaces,andlet
f W.A;/!.B;/ and gW.B;/!.C;/;
whereD
f
DA,R
f
DD
g
DB,andf andgarecontinuous.DeﬁnehW.A;/!
.C;/byh.u/Dg.f.u//.ShowthathiscontinuousonA.
6.
Let.A;/bethesetofallboundedreal-valuedfunctionsonanonemptysetS,
with.u;v/ D D sup
s2S
ju.s/v.s/j. Lets
1
,s
2
, ..., s
k
bemembersofS, and
f.u/Dg.u.s
1
/;u.s
2
/;:::;u.s
k
//,wheregisreal-valuedandcontinuousonRk.
Showthatf isacontinuousfunctionfrom.A;/toR.
7.
Let.A;/bethesetofallboundedreal-valuedfunctionsonanonemptysetS,
with.u;v/Dsup
s2S
ju.s/v.s/j. Showthatf.u/Dinf
s2S
u.s/andg.u/ D
sup
s2S
u.s/areuniformlycontinuousfunctionsfrom.A;/toR.
8.
LetIŒa;bbethesetofallreal-valuedfunctionsthatareRiemannintegrableon
Œa;b,with.u;v/Dsup
axb
ju.x/v.x/j.Showthatf.u/D
Z
b
a
u.x/dxisa
uniformlycontinuousfunctionfromIŒa;btoR.
550
Exercises
Section1.1 pp. 910
1:1:1
(p.9)(a)
2max.a;b/
(b)
2min.a;b/
(c)
4max.a;b;c/
(d)
4min.a;b;c/
1:1:5
(p. 9)(a)
1(no);1(yes)
(b)
3(no);3(no)
(c)
p
7(yes);
p
7(yes)
(d)
2(no);3(no)
(e)
1(no);1(no)
(f)
p
7(no);
p
7(no)
Section1.2 pp. 1519
1:2:9
(p.16)(a)
2
n
=.2n/Š
(b)
23
n
=.2nC1/Š
(c)
2
n
.2n/Š=.nŠ/
2
(d)
n
n
=nŠ
1:2:10
(p. 16)(b)
no 1:2:11
(p. 16)(b)
no
1:2:20
(p. 18)
A
n
D
x
n
0
@
lnx
Xn
jD1
1
j
1
A
1:2:21
(p. 18)
f
n
.x
1
;x
2
;:::;x
n
/ D 2n1max.x
1
;x
2
;:::;x
n
/, g
n
.x
1
;x
2
;:::;x
n
/ D
2n1min.x
1
;x
2
;:::;x
n
/
Section1.3 pp. 2729
1:3:1
(p.27)(a)
Œ
1
2
;1/;.1;
1
2
/[Œ1;1/;.1;0[.
3
2
;1/;.0;
3
2
;.1;0[.
3
2
;1/;
.1;
1
2
[Œ1;1/
(b)
.3;2/[.2;3/;.1;3[Œ2;2[Œ3;1/;;;.1;1/; ;;
.1;3[Œ2;2[Œ3;1/
(c)
;;.1;1/;;;.1;1/;;;.1;1/
(d)
;;.1;1/;Œ1;1;.1;1/[.1;1/;Œ1;1;.1;1/
1:3:2
(p. 27)(a)
.0;3
(b)
Œ0;2
(c)
.1;1/[.2;1/
(d)
.1;0[.3;1/
1:3:4
(p. 27)(a)
1
4
(b)
1
6
(c)
6
(d)
1