pdfsharp c# : Add hyperlinks pdf file software Library project winforms asp.net html UWP TRENCH_REAL_ANALYSIS54-part275

532 Chapter8
MetricSpaces
Theorem8.1.23
SupposethatandareequivalentmetricsonA:Then
(a)
Asequencefu
n
gconvergestouin.A;/ifandonlyifitconvergestouin.A;/:
(b)
Asequencefu
n
gisaCauchysequencein.A;/ifandonlyifitisaCauchysequence
in.A;/:
(c)
.A;/iscompleteifandonlyif.A;/iscomplete:
8.1Exercises
1.
Showthat
(a)
,
(b)
,and
(c)
ofDefinition8.1.1areequivalentto
(i)
.u;v/D0ifandonlyifuDv;
(ii)
.u;v/.w;u/C.w;v/.
2.
Prove:Ifx,y,u,andvarearbitrarymembersofametricspace.A;/,then
j.x;y/.u;v/j.x;u/C.v;y/:
3. (a)
Supposethat.A;/isametricspace,anddefine
1
.u;v/D
.u;v/
1C.u;v/
:
Showthat.A;
1
/isametricspace.
(b)
ShowthatinfinitelymanymetricscanbedefinedonanysetAwithmorethan
onemember.
4.
Let.A;/beametricspace,andlet
.u;v/D
.u;v/
1C.u;v/
:
ShowthatasubsetofAisopenin.A;/ifandonlyifitisopenin.A;/.
5.
ShowthatifAisanarbitrarynonemptyset,then
.u;v/D
0 ifvDu;
1 ifv¤u;
isametriconA.
6.
Supposethat.A;/isametricspace,u
0
2A,andr>0.
(a)
Showthat
S
r
.u
0
/
˚
u
ˇ
ˇ
.u;u
0
/r
ifAcontainsmorethanonepoint.
(b)
Verifythatifisthediscretemetric,then
S
1
.u
0
˚
u
ˇ
ˇ
.u;u
0
/1
.
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Section8.1
IntroductiontoMetricSpaces
533
7.
Prove:
(a)
Theintersectionoffinitelymanyopensetsisopen.
(b)
Theunionoffinitelymanyclosedsetsisclosed.
8.
Prove:
(a)
IfU isaneighborhoodofu
0
andUV,thenV isaneighborhoodofu
0
.
(b)
IfU
1
,U
2
,...,U
n
areneighborhoodsofu
0
,sois\
n
iD1
U
i
.
9.
Prove:AlimitpointofasetSiseitheraninteriorpointoraboundarypointofS.
10.
Prove:AnisolatedpointofSisaboundarypointofSc.
11.
Prove:
(a)
AboundarypointofasetSiseitheralimitpointoranisolatedpointofS.
(b)
AsetSisclosedifandonlyifSD
S.
12.
LetSbeanarbitraryset.Prove:
(a)
@Sisclosed.
(b)
S0isopen.
(c)
Theexterior
ofSisopen.
(d)
ThelimitpointsofSformaclosedset.
(e)
S
D
S.
13.
Prove:
(a)
.S
1
\S
2
/DS0
1
\S0
2
(b)
S0
1
[S0
2
.S
1
[S
2
/0
14.
Prove:
(a)
@.S
1
[S
2
/@S
1
[@S
2
(b)
@.S
1
\S
2
/@S
1
[@S
2
(c)
@
S@S
(d)
@SD@S
c
(e)
@.ST/@S[@T
15.
Showthat
kXkDmaxfjx
1
j;jx
2
j;:::;jx
n
jg
isanormonR
n
.
16.
Supposethat.A
i
;
i
/,1i k,aremetricspaces.Let
ADA
1
A
2
A
k
D
˚
XD.x
1
;x
2
;:::;x
k
/
ˇ
ˇ
x
i
2A
i
;1ik
:
IfXandYareinA,let
.X;Y/D
Xk
iD1
.x
i
;y
i
/:
(a)
ShowthatisametriconA.
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534 Chapter8
MetricSpaces
(b)
LetfX
r
g
1
rD1
Df.x
1r
;x
2r
;:::;x
kr
/g
1
rD1
beasequenceinA.Showthat
lim
r!1
X
r
D
b
XD.bx
1
;bx
2
;:::;bx
k
/
ifandonlyif
lim
r!1
x
ir
Dbx
i
; 1ik:
(c)
ShowthatfX
r
g
1
rD1
isaCauchysequencein.A;/ifandonlyiffx
ir
g
1
rD1
isa
Cauchysequencein.A
i
;
i
/,1i k.
(d)
Showthat.A;/iscompleteifandonlyif.A
i
;
i
/iscomplete,1ik.
17.
Foreachpositiveintegeri,let.A
i
;
i
/beametricspace. LetAbethesetofall
objectsoftheformXD.x
1
;x
2
;:::;x
n
;:::/,wherex
i
2A
i
,i1.(Forexample,
ifA
i
DR,i 1,thenADR
1
.)Letf˛
i
g
1
iD1
beanysequenceofpositivenumbers
suchthat
P
1
iD1
˛
i
<1.
(a)
Showthat
.X;Y/D
X1
iD1
˛
i
i
.x
i
;y
i
/
1C
i
.x
i
;y
i
/
isametriconA.
(b)
LetfX
r
g
1
rD1
Df.x
1r
;x
2r
;:::;x
nr
;:::/g
1
rD1
beasequenceinA.Showthat
lim
r!1
X
r
D
b
XD.bx
1
;bx
2
;:::;bx
n
;:::/
ifandonlyif
lim
r!1
x
ir
Dbx
i
; i1:
(c)
ShowthatfX
r
g
1
rD1
isaCauchysequencein.A;/ifandonlyiffx
ir
g
1
rD1
isa
Cauchysequencein.A
i
;
i
/foralli 1.
(d)
Showthat.A;/iscompleteifandonlyif.A
i
;
i
/iscompleteforalli1.
18.
LetCŒ0;1/bethesetofallreal-valuedfunctionscontinuousonŒ0;1/. Foreach
nonnegativeintegern,let
kfk
n
Dmax
˚
jf.x/j
ˇ
ˇ
0xn
and
n
.f;g/D
kf gk
n
1Ckfgk
n
:
Define
.f;g/D
X1
nD1
1
2n1
n
.f;g/:
(a)
ShowthatisametriconCŒ0;1/.
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Section8.1
IntroductiontoMetricSpaces
535
(b)
Letff
k
g
1
kD1
beasequenceoffunctionsinCŒ0;1/.Showthat
lim
k!1
f
k
Df
inthesenseofDefinition8.1.14ifandonlyif
lim
k!1
f
k
.x/Df.x/
uniformlyoneveryfinitesubintervalofŒ0;1/.
(c)
Showthat.CŒ0;1/;/iscomplete.
19.
ShowthatMinkowski’sinequalityisfalseif0<p<1.
20.
Supposethat0<p<1.Showthatifuandvarenonnegative,then
.uCv/
p
u
p
Cv
p
:
UsethistoshowthatifX,Y2R
n
,
.X/D
n
X
iD1
jx
i
j
p
; and .Y/D
n
X
iD1
jy
i
j
p
;
then
.XCY/.X/C.Y/:
IsanormonR
n
?
21.
SupposethatXDfx
i
g
1
iD1
isin`
p
,wherep>1.Showthat
(a)
X2`
r
forallr>p;
(b)
Ifr>p,thenkXk
r
kXk
p
;
(c)
lim
r!1
kXk
r
DkXk
1
.
22.
Let.A;/beametricspace.
(a)
Supposethatfu
n
gandfv
n
garesequencesinA,lim
n!1
u
n
Du,andlim
n!1
v
n
D
v.Showthatlim
n!1
.u
n
;v
n
/D.u;v/.
(b)
Concludefrom
(b)
thatiflim
n!1
u
n
D uandv isarbitraryinA, then
lim
n!1
.u
n
;v/D.u;v/.
23.
Prove: Iffu
r
g
1
rD1
isaCauchysequenceinanormedvectorspace.A;kk/,then
fku
r
kg
1
rD1
isbounded.
24.
Let
AD
(
X2R
1
ˇ
ˇ
thepartialsums
X1
iD1
x
i
;n1;arebounded
)
:
(a)
Showthat
kXkDsup
n1
ˇ
ˇ
ˇ
ˇ
ˇ
Xn
iD1
x
i
ˇ
ˇ
ˇ
ˇ
ˇ
isanormonA.
(b)
Let.X;Y/DkXYk.Showthat.A;/iscomplete.
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536 Chapter8
MetricSpaces
25. (a)
Showthat
kfkD
Z
b
a
jf.x/jdx
isanormonCŒa;b,
(b)
Showthatthesequenceff
n
gdefinedby
f
n
.x/D
xa
ba
n
isaCauchysequencein.CŒa;b;kk/.
(c)
Showthat.CŒa;b;kk/isnotcomplete.
26. (a)
Verifythat`
1
isanormedvectorspace.
(b)
Showthat`
1
iscomplete.
27.
LetAbethesubsetofR
1
consistingofconvergentsequencesXDfx
i
g
1
iD1
.Define
kXkDsup
i1
jx
i
j.Showthat.A;kk/isacompletenormedvectorspace.
28.
LetAbethesubsetofR
1
consistingofsequencesXDfx
i
g
1
iD1
suchthatlim
i!1
x
i
D
0.DefinekXkDmax
˚
jx
i
j
ˇ
ˇ
i1
.Showthat.A;kk/isacompletenormedvector
space.
29. (a)
ShowthatR
n
p
iscompleteifp1.
(b)
Showthat`
p
iscompleteifp1.
30.
ShowthatifXDfx
i
g1
iD1
2 `
p
andYD fy
i
g1
iD1
2 `
q
,where1=pC1=qD1,
thenZDfx
i
y
i
g2`
1
.
8.2COMPACTSETSINAMETRICSPACE
Throughoutthissectionitistobeunderstoodthat.A;/isametricspaceandthatthesets
underconsiderationaresubsetsofA.
WesaythatacollectionH ofopensubsetsofAisanopencoveringofT T ifT T 
[
˚
H
ˇ
ˇ
H2H
. WesaythatT T hastheHeine–BorelpropertyifeveryopencoveringH
ofT containsafinitecollection
b
Hsuchthat
T [
n
H
ˇ
ˇ
H 2
b
H
o
:
FromTheorem1.3.7,everynonemptyclosedandboundedsubsetoftherealnumbers
hastheHeine–Borelproperty.Moreover,fromExercise1.3.21,anynonemptysetofreals
thathastheHeine–Borelpropertyisclosedandbounded.Giventheseresults,wedefined
acompactsetofrealstobeaclosedandboundedset,andwenowdrawthefollowing
conclusion:
AnonemptysetofrealnumbershastheHeine–Borelpropertyifandonlyifitiscompact.
Section8.2
CompactSetsinaMetricSpace
537
Thedefinitionofboundednessofasetofrealnumbersisbasedontheorderingofthe
realnumbers:ifaandbaredistinctrealnumberstheneithera<borb<a.Sincethere
isnosuchorderinginageneralmetricspace,weintroducethefollowingdefinition.
Definition8.2.1
ThediameterofanonemptysubsetSofAis
d.S/Dsup
˚
.u;v/
ˇ
ˇ
u;v2T
:
Ifd.S/<1thenSisbounded.
Aswewillseebelow,aclosedandboundedsubsetofageneralmetricspacemayfail
tohavetheHeine–Borelproperty. Sincewewant“compact"and“hastheHeine–Borel
property"tobesynonymousinconnectionwithageneralmetricspace,wesimplymake
thefollowingdefinition.
Definition8.2.2
AsetTiscompactifithastheHeine–Borelproperty.
Theorem8.2.3
AninfinitesubsetT ofAiscompactifandonlyifeveryinfinitesubset
ofThasalimitpointinT:
Proof
SupposethatT hasaninfinitesubsetEwithnolimitpointinT.Then,ift2T,
thereisanopensetH
t
suchthatt2H
t
andH
t
containsatmostonememberofE.Then
HD[
˚
H
t
ˇ
ˇ
t2T
isanopencoveringofT,butnofinitecollectionfH
t
1
;H
t
2
;:::;H
t
k
g
ofsetsfromHcancoverE,sinceEisinfinite.Therefore,nosuchcollectioncancoverT;
thatis,Tisnotcompact.
NowsupposethateveryinfinitesubsetofThasalimitpointinT,andletHbeanopen
coveringofT.WefirstshowthatthereisasequencefH
i
g
1
iD1
ofsetsfromHthatcovers
T.
If>0,thenT canbecoveredby-neighborhoodsoffinitelymanypointsofT. We
provethisbycontradiction.Lett
1
2T.IfN
.t
1
/doesnotcoverT,thereisat
2
2T such
that.t
1
;t
2
/ . . Nowsupposethatn  2andwehavechosent
1
,t
2
,..., t
n
suchthat
.t
i
;t
j
/,1i <j j n. . If[
n
iD1
N
.t
i
/doesnotcoverT,thereisat
nC1
2T such
that.t
i
;t
nC1
/,1i n. . Therefore,.t
i
;t
j
/,1i <j j nC1. . Hence,
byinduction,ifnofinitecollectionof-neighborhoodsofpointsinT coversT,thereisan
infinitesequenceft
n
g
1
nD1
inTsuchthat.t
i
;t
j
/,i ¤j. Suchasequencecouldnot
havealimitpoint,contrarytoourassumption.
Bytakingsuccessivelyequalto1,1=2,..., 1=n,...,wecannowconcludethat,for
eachn,therearepointst
1n
,t
2n
,...,t
k
n
;n
suchthat
T 
k
n
[
iD1
N
1=n
.t
in
/:
DenoteB
in
DN
1=n
.t
in
/,1in,n1,anddefine
fG
1
;G
2
;G
3
;:::gDfB
11
;:::;B
k
1
;1
;B
12
;:::;B
k
2
;2
;B
13
;:::;B
k
3
;3
;:::g:
538 Chapter8
MetricSpaces
Ift2T,thereisanHinHsuchthatt2H.SinceHisopen,thereisan>0such
thatN
.t/H.Sincet2G
j
forinfinitelymanyvaluesofj andlim
j!1
d.G
j
/D0,
G
j
N
.t/H
forsomej.Therefore,iffG
j
i
g
1
iD1
isthesubsequenceoffG
j
gsuchthatG
j
i
isasubsetof
someH
i
inH(thefH
i
garenotnecessarilydistinct),then
T 
[1
iD1
H
i
:
(8.2.1)
Wewillnowshowthat
T 
[N
iD1
H
i
:
(8.2.2)
forsomeintegerN.Ifthisisnotso,thereisaninfinitesequenceft
n
g
1
nD1
inTsuchthat
t
n
[n
iD1
H
i
; n1:
(8.2.3)
Fromourassumption,ft
n
g
1
nD1
hasalimit
tinT. From(8.2.1),
t 2 2 H
k
forsomek,so
N
.
t/H
k
forsome>0.Sincelim
n!1
t
n
D
t,thereisanintegerNsuchthat
t
n
2N
.
t/H
k
[n
iD1
H
i
; n>k;
whichcontradicts(8.2.3).Thisverifies(8.2.2),soTiscompact.
AnyfinitesubsetofametricspaceobviouslyhastheHeine–Borelpropertyandisthere-
forecompact.SinceTheorem8.2.3doesnotdealwithfinitesets,itisoftenmoreconvenient
toworkwiththefollowingcriterionforcompactness,whichisalsoapplicabletofinitesets.
Theorem8.2.4
AsubsetT ofametricAiscompactifandonlyifeveryinfinitese-
quenceft
n
gofmembersofThasasubsequencethatconvergestoamemberofT:
Proof
SupposethatT iscompactandft
n
gT.Ifft
n
ghasonlyfinitelymanydistinct
terms,thereisa
tinT suchthatt
n
D
t forinfinitelymanyvaluesofn;ifthisissofor
n
1
<n
2
<,thenlim
j!1
t
n
j
D
t. Ifft
n
ghasinfinitelymanydistinctterms,thenft
n
g
hasalimitpoint
tinT,sothereareintegersn
1
<n
2
< suchthat.t
n
j
;
t/ <1=j;
therefore,lim
j!1
t
n
j
D
t.
Conversely,supposethateverysequenceinThasasubsequencethatconvergestoalimit
inT. IfSisaninfinitesubsetofT,wecanchooseasequenceft
n
gofdistinctpointsin
S.Byassumption,ft
n
ghasasubsequencethatconvergestoamember
tofT.Since
tisa
limitpointofft
n
g,andthereforeofT,T iscompact.
Theorem8.2.5
IfTiscompact;theneveryCauchysequenceft
n
g
1
nD1
inTconverges
toalimitinT:
Section8.2
CompactSetsinaMetricSpace
539
Proof
ByTheorem8.2.4,ft
n
ghasasubsequenceft
n
j
gsuchthat
lim
j!1
t
n
j
D
t2T:
(8.2.4)
Wewillshowthatlim
n!1
t
n
D
t.
Supposethat> 0. . Sinceft
n
gisaCauchysequence,thereisanintegerN suchthat
.t
n
;t
m
/<,n>mN.From(8.2.4),thereisanmDn
j
Nsuchthat.t
m
;
t/<.
Therefore,
.t
n
;
t/.t
n
;t
m
/C.t
m
;
t/<2; nm:
Theorem8.2.6
IfTiscompact;thenTisclosedandbounded.
Proof
Supposethat
tisalimitpointofT. Foreachn,chooset
n
¤
t 2B
1=n
.
t/\T.
Thenlim
n!1
t
n
D
t. Sinceeverysubsequenceofft
n
galsoconvergesto
t,
t 2 2 T,by
Theorem8.2.3.Therefore,T isclosed.
ThefamilyofunitopenballsHD
˚
B
1
.t/
ˇ
ˇ
t2T
isanopencoveringofT.SinceTis
compact,therearefinitelymanymemberst
1
,t
2
,...,t
n
ofT suchthatS [
n
jD1
B
1
.t
j
/.
IfuandvarearbitrarymembersofT,thenu2B
1
.t
r
/andv2B
1
.t
s
/forsomerandsin
f1;2;:::;ng,so
.u;v/.u;t
r
/C.t
r
;t
s
/C.t
s
;v/
2C.t
r
;t
s
/2Cmax
˚
.t
i
;t
j
/
ˇ
ˇ
1i<j n
:
Therefore,Tisbounded.
TheconverseofTheorem8.2.6isfalse;forexample,ifAisanyinfinitesetequipped
withthediscretemetric(Example8.1.2.),theneverysubsetofAisboundedandclosed.
However,ifTisaninfinitesubsetofA,thenHD
˚
ftg
ˇ
ˇ
t2T
isanopencoveringofT,
butnofinitesubfamilyofHcoversT.
Definition8.2.7
AsetT istotallyboundedifforevery>0thereisafinitesetT
withthefollowingproperty:ift2T,thereisans2T
suchthat.s;t/<.Wesaythat
T
isafinite-netforT.
Weleaveittoyou(Exercise8.2.4)toshowthateverytotallyboundedsetisboundedand
thattheconverseisfalse.
540 Chapter8
MetricSpaces
Theorem8.2.8
IfTiscompact;thenTistotallybounded.
Proof
WewillprovethatifT isnottotallybounded,thenT isnotcompact.IfT isnot
totallybounded,thereisan>0suchthatthereisnofinite-netforT.Lett
1
2T.Then
theremustbeat
2
inT suchthat.t
1
;t
2
/>. (Ifnot,thesingletonsetft
1
gwouldbea
finite-netforT.) Nowsupposethatn2andwehavechosent
1
,t
2
,...,t
n
suchthat
.t
i
;t
j
/,1i <j j n. . Thentheremustbeat
nC1
2T suchthat.t
i
;t
nC1
/,
1i n.(Ifnot,ft
1
;t
2
;:::;t
n
gwouldbeafinite-netforT.)Therefore,.t
i
;t
j
/,
1i <j nC1. Hence,byinduction,thereisaninfinitesequenceft
n
g
1
nD1
inT such
that.t
i
;t
j
/,i ¤j.Sincesuchasequencehasnolimitpoint,T isnotcompact,by
Theorem8.2.4.
Section8.2
CompactSetsinaMetricSpace
541
Theorem8.2.9
If.A;/iscompleteandT isclosedandtotallybounded;thenT is
compact.
Proof
LetSbeaninfinitesubsetofT,andletfs
i
g
1
iD1
beasequenceofdistinctmembers
ofS.Wewillshowthatfs
i
g
1
iD1
hasaconvergentsubsequence.SinceTisclosed,thelimit
ofthissubsequenceisinT,whichimpliesthatTiscompact,byTheorem8.2.4.
Forn   1,letT
1=n
beafinite1=n-netforT. Letfs
i0
g
1
iD1
D fs
i
g
1
iD1
. SinceT
1
is
finiteandfs
i0
g
1
iD1
isinfinite,theremustbeamembert
1
ofT
1
suchthat.s
i0
;t
1
/  1
forinfinitelymanyvaluesofi. Letfs
i1
g
1
iD1
bethesubsequenceoffs
i0
g
1
iD1
suchthat
.s
i1
;t
1
/1.
Wecontinuebyinduction. Supposethatn>1andwehavechosenaninfinitesubse-
quencefs
i;n1
g1
iD1
offs
i;n2
g1
iD1
. SinceT
1=n
isfiniteandfs
i;n1
g1
iD1
isinfinite,there
mustbemembert
n
ofT
1=n
suchthat.s
i;n1
;t
n
/   1=nforinfinitelymanyvaluesof
i. Letfs
in
g1
iD1
bethesubsequenceoffs
i;n1
g1
iD1
suchthat.s
in
;t
n
/ 1=n. . Fromthe
triangleinequality,
.s
in
;s
jn
/2=n; i;j j 1; n1:
(8.2.5)
Nowletbs
i
D s
ii
,i   1. . Thenfbs
i
g
1
iD1
isaninfinitesequenceofmembersofT. Mo-
roever,ifi;j   n, thenbs
i
andbs
j
arebothincludedinfs
in
g
1
iD1
,so(8.2.5)impliesthat
.bs
i
;bs
j
/   2=n;thatis,fbs
i
g
1
iD1
isaCauchysequenceandthereforehasalimit,since
.A;/iscomplete.
Example8.2.1
LetTbethesubsetof`
1
suchthatjx
i
j
i
,i1,wherelim
i!1
i
D
0.ShowthatTiscompact.
Solution
WewillshowthatT istotallyboundedin`
1
. Since`
1
iscomplete(Exer-
cise8.1.26),Theorem8.2.9willthenimplythatT iscompact.
Let>0.ChooseNsothat
i
ifi >N.LetDmax
˚
i
ˇ
ˇ
1in
andletp
beanintegersuchthatp>.LetQ
D
˚
r
i
ˇ
ˇ
r
i
DintegerinŒp;p
.Thenthesubset
of`
1
suchthatx
i
2Q
,1iN,andx
i
D0,i>N,isafinite-netforT.
CompactSubsets of
CŒa;b
InExample8.1.7weshowedthatCŒa;bisacompletemetricspaceunderthemetric
.f;g/Dkf gkDmax
˚
jf.x/g.x/j
ˇ
ˇ
axb
:
WewillnowgivenecessaryandsufficientconditionsforasubsetofCŒa;btobecompact.
Definition8.2.10
AsubsetTofCŒa;bisuniformlyboundedifthereisaconstantM
suchthat
jf.x/jM
if axb and
f 2T:
(8.2.6)
AsubsetTofCŒa;bisequicontinuousifforeach>0thereisaı>0suchthat
jf.x
1
/f.x
2
/j if x
1
;x
2
2Œa;b; jx
1
x
2
j<ı; and f f 2T: (8.2.7)
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