c# pdf free : Add hyperlink pdf file SDK software service wpf winforms windows dnn TRENCH_REAL_ANALYSIS29-part247

282 Chapter5
Real-ValuedFunctionsof
n
Variables
MembersofRhavedualinterpretations:geometric,aspointsontherealline,andalge-
braic,asrealnumbers. Weassumethatyouarefamiliarwiththegeometricinterpretation
ofmembersofR
2
andR
3
astherectangularcoordinatesofpointsinaplaneandthree-
dimensionalspace,respectively.AlthoughR
n
cannotbevisualizedgeometricallyifn4,
geometricideasfromR, R
2
,andR
3
oftenhelpustointerpretthepropertiesofR
n
for
arbitraryn.
AswesaidinSection1.3, theideaofneighborhoodisalwaysassociatedwithsome
definitionof“closeness”ofpoints.Thefollowingdefinitionimposesanalgebraicstructure
onR
n
,intermsofwhichthedistancebetweentwopointscanbedefinedinanaturalway.
Inaddition,thisalgebraicstructurewillbeusefullaterforotherpurposes.
Definition5.1.1
Thevectorsumof
XD.x
1
;x
2
;:::;x
n
/ and YD.y
1
;y
2
;:::;y
n
/
is
XCYD.x
1
Cy
1
;x
2
Cy
2
;:::;x
n
Cy
n
/:
(5.1.1)
Ifaisarealnumber,thescalarmultipleofXbyais
aXD.ax
1
;ax
2
;:::;ax
n
/:
(5.1.2)
Notethat“C”hastwodistinctmeaningsin(5.1.1):ontheleft,“C”standsforthenewly
definedadditionofmembersofR
n
and,ontheright,foradditionofrealnumbers.However,
thiscanneverleadtoconfusion,sincethemeaningof“C”canalwaysbededucedfrom
thesymbolsoneithersideofit.Asimilarcommentappliestotheuseofjuxtapositionto
indicatescalarmultiplicationontheleftof(5.1.2)andmultiplicationofrealnumberson
theright.
Example5.1.1
InR
4
,let
XD.1;2;6;5/ and YD
3;5;4;
1
2
:
Then
XCYD
4;7;10;
11
2
and
6XD.6;12;36;30/:
Weleavetheproofofthefollowingtheoremtoyou(Exercise5.1.2).
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Section5.1
Structureof
R
n
283
Theorem5.1.2
IfX;Y;andZareinR
n
andaandbarerealnumbers;then
(a)
XCYDYCX.vectoradditioniscommutative/:
(b)
.XCY/CZDXC.YCZ/.vectoradditionisassociative/:
(c)
Thereisauniquevector0;calledthezerovector;suchthatXC0DXforallXin
Rn:
(d)
ForeachXinRn thereisauniquevectorXsuchthatXC.X/D0:
(e)
a.bX/D.ab/X:
(f)
.aCb/XDaXCbX:
(g)
a.XCY/DaXCaY:
(h)
1XDX:
Clearly,0D.0;0;:::;0/and,ifXD.x
1
;x
2
;:::;x
n
/,then
XD.x
1
;x
2
;:::;x
n
/:
WewriteXC.Y/asXY.Thepoint0iscalledtheorigin.
AnonemptysetV DfX;Y;Z;:::g,togetherwithrulessuchas(5.1.1),associatinga
uniquememberofV witheveryorderedpairofitsmembers, , and(5.1.2),associatinga
uniquememberofV witheveryrealnumberandmemberofV,issaidtobeavectorspace
ifithasthepropertieslistedinTheorem5.1.2. Themembersofavectorspacearecalled
vectors.WhenwewishtoemphasizethatweareregardingamemberofR
n
aspartofthis
algebraicstructure,wewillspeakofitasavector;otherwise,wewillspeakofitasapoint.
Length, Distance,andInnerProduct
Definition5.1.3
ThelengthofthevectorXD.x
1
;x
2
;:::;x
n
/is
jXjD.x
2
1
Cx
2
2
CCx
2
n
/
1=2
:
ThedistancebetweenpointsXandYisjXYj;inparticular,jXjisthedistancebetween
Xandtheorigin.IfjXjD1,thenXisaunitvector.
IfnD1,thisdefinitionoflengthreducestothefamiliarabsolutevalue,andthedistance
betweentwopointsisthelengthoftheintervalhavingthemasendpoints;fornD2and
nD3,thelengthanddistanceofDefinition5.1.3reducetothefamiliardefinitionsforthe
planeandthree-dimensionalspace.
Example5.1.2
Thelengthsofthevectors
XD.1;2;6;5/ and YD
3;5;4;
1
2
are
jXjD.1
2
C.2/
2
C6
2
C5
2
/
1=2
D
p
66
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284 Chapter5
Real-ValuedFunctionsof
n
Variables
and
jYjD.3
2
C.5/
2
C4
2
C.
1
2
/
2
/
1=2
D
p
201
2
:
ThedistancebetweenXandYis
jXYjD..13/
2
C.2C5/
2
C.64/
2
C.5
1
2
/
2
/
1=2
D
p
149
2
:
Definition5.1.4
TheinnerproductXYofXD.x
1
;x
2
;:::;x
n
/andYD.y
1
;y
2
;:::;y
n
/
is
XYDx
1
y
1
Cx
2
y
2
CCx
n
y
n
:
Lemma5.1.5(Schwarz’sInequality)
IfXandYareanytwovectorsinR
n
;
then
jXYjjXjjYj;
(5.1.3)
withequalityifandonlyifoneofthevectorsisascalarmultipleoftheother:
Proof
IfYD0,thenbothsidesof(5.1.3)are0,so(5.1.3)holds,withequality.Inthis
case,YD0X.NowsupposethatY¤0andtisanyrealnumber.Then
0
Xn
iD1
.x
i
ty
i
/
2
D
Xn
iD1
x
2
i
2t
Xn
iD1
x
i
y
i
Ct
2
Xn
iD1
y
2
i
DjXj
2
2.XY/tCt
2
jYj
2
:
(5.1.4)
Thelastexpressionisasecond-degreepolynomialpint.Fromthequadraticformula,the
zerosofpare
tD
.XY/˙
p
.XY/2 jXj2jYj2
jYj2
:
Hence,
.XY/
2
jXj
2
jYj
2
;
(5.1.5)
becauseifnot,thenpwouldhavetwodistinctrealzerosandthereforebenegativebetween
them(Figure5.1.1),contradictingtheinequality(5.1.4). Takingsquarerootsin(5.1.5)
yields(5.1.3)ifY¤0.
IfX D D tY, , thenjXYj D jXjjYj D jtjjYj2 (verify), , soequalityholdsin(5.1.3).
Conversely,ifequalityholdsin(5.1.3),thenphastherealzerot
0
D.XY/=jYk2,and
Xn
iD1
.x
i
t
0
y
i
/
2
D0
from(5.1.4);therefore,XDt
0
Y.
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Section5.1
Structureof
R
n
285
y
t
y = p(t)
r
1
r
2
Figure5.1.1
Theorem5.1.6(TriangleInequality)
IfXandYareinR
n
;then
jXCYjjXjCjYj;
(5.1.6)
withequalityifandonlyifoneofthevectorsisanonnegativemultipleoftheother:
Proof
Bydefinition,
jXCYj
2
D
Xn
iD1
.x
i
Cy
i
/
2
D
Xn
iD1
x
2
i
C2
Xn
iD1
x
i
y
i
C
Xn
iD1
y
2
i
DjXj
2
C2.XY/CjYj
2
jXj
2
C2jXjjYjCjYj
2
(bySchwarz’sinequality)
D.jXjCjYj/
2
:
(5.1.7)
Hence,
jXCYj
2
.jXjCjYj/
2
:
Takingsquarerootsyields(5.1.6).
Fromthethirdlineof(5.1.7),equalityholdsin(5.1.6)ifandonlyifXYDjXjjYj,
whichistrueifandonlyifoneofthevectorsXandYisanonnegativescalarmultipleof
theother(Lemma5.1.5).
Corollary5.1.7
IfX;Y;andZareinR
n
;then
jXZjjXYjCjYZj:
Proof
Write
XZD.XY/C.YZ/;
andapplyTheorem5.1.6withXandYreplacedbyXYandYZ.
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286 Chapter5
Real-ValuedFunctionsof
n
Variables
Corollary5.1.8
IfXandYareinR
n
;then
jXYjjjXjjYjj:
Proof
Since
XDYC.XY/;
Theorem5.1.6impliesthat
jXjjYjCjXYj;
whichisequivalentto
jXjjYjjXYj:
InterchangingXandYyields
jYjjXjjYXj:
SincejXYjDjYXj,thelasttwoinequalitiesimplythestatedconclusion.
Example5.1.3
TheanglebetweentwononzerovectorsXD .x
1
;x
2
;x
3
/andY D
.y
1
;y
2
;y
3
/inR3 istheanglebetweenthedirectedlinesegmentsfromtheorigintothe
pointsXandY(Figure5.1.2).
X
0
Y
Y
X
X−Y
θ
Figure5.1.2
ApplyingthelawofcosinestothetriangleinFigure5.1.2yields
jXYj
2
DjXj
2
CjYj
2
2jXjjYjcos:
(5.1.8)
However,
jXYj
2
D.x
1
y
1
/
2
C.x
2
y
2
/
2
C.x
3
y
3
/
2
D.x
2
1
Cx
2
2
Cx
2
3
/C.y
2
1
Cy
2
2
Cy
2
3
/2.x
1
y
1
Cx
2
y
2
Cx
3
y
3
/
DjXj
2
CjYj
2
2XY:
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Section5.1
Structureof
R
n
287
Comparingthiswith(5.1.8)yields
XYDjXjjYjcos:
Sincejcosj1,thisverifiesSchwarz’sinequalityinR
3
.
Example5.1.4
Connectingthepoints0,X,Y,andXCYinR
2
orR
3
(Figure5.1.3)
producesaparallelogramwithsidesoflengthjXjandjYjandadiagonaloflengthjXCYj.
0
X
Y
Y
Y
X
X
X+Y
X+Y
Figure5.1.3
Thus,thereisatrianglewithsidesjXj,jYj,andjXCYj.Fromthis,weseegeometrically
that
jXCYjjXjCjYj
inR
2
orR
3
,sincethelengthofonesideofatrianglecannotexceedthesumofthelengths
oftheothertwo.Thisverifies(5.1.6)forR
2
andR
3
andindicateswhy(5.1.6)iscalledthe
triangleinequality.
Thenexttheoremlistspropertiesoflength,distance,andinnerproductthatfollowdi-
rectlyfromDefinitions5.1.3and5.1.4.Weleavetheprooftoyou(Exercise5.1.6).
Theorem5.1.9
IfX;Y;andZaremembersofRn andaisascalar,then
(a)
jaXjDjajjXj:
(b)
jXj0;withequalityifandonlyifXD0:
(c)
jXYj0;withequalityifandonlyifXDY:
(d)
XYDYX:
(e)
X.YCZ/DXYCXZ:
(f)
.cX/YDX.cY/Dc.XY/:
288 Chapter5
Real-ValuedFunctionsof
n
Variables
LineSegmentsin
R
R
R
n
TheequationofalinethroughapointX
0
D.x
0
;y
0
;´
0
/inR3 canbewrittenparametri-
callyas
xDx
0
Cu
1
t; yDy
0
Cu
2
t; ´D´
0
Cu
3
t; 1<t<1;
whereu
1
,u
2
,andu
3
arenotallzero.Wewritethisinvectorformas
XDX
0
CtU; 1<t<1;
(5.1.9)
withUD.u
1
;u
2
;u
3
/,andwesaythatthelineisthroughX
0
inthedirectionofU.
Therearemanywaystorepresentagivenlineparametrically.Forexample,
XDX
0
CsV; 1<s<1;
(5.1.10)
representsthesamelineas(5.1.9)ifandonlyifVDaUforsomenonzerorealnumbera.
Thenthelineistraversedinthesamedirectionassandtvaryfrom1to1ifa>0,or
inoppositedirectionsifa<0.
TowritetheparametricequationofalinethroughtwopointsX
0
andX
1
inR3,wetake
UDX
1
0in(5.1.9),whichyields
XDX
0
Ct.X
1
X
0
/DtX
1
C.1t/X
0
; 1<t<1:
ThelinesegmentfromX
0
toX
1
consistsofthosepointsforwhich0t1.
Example5.1.5
ThelineLdefinedby
xD1C2t; yD34t; ´D1; 1<t<1;
whichcanberewrittenas
XD.1;3;1/Ct.2;4;0/; 1<t<1;
(5.1.11)
isthroughX
0
D .1;3;1/inthedirectionofUD .2;4;0/. Thesamelinecanbe
representedby
XD.1;3;1/Cs.1;2;0/; 1<s<1;
(5.1.12)
orby
XD.1;3;1/C.4;8;0/; 1<<1:
(5.1.13)
Since
.1;2;0/D
1
2
.2;4;0/;
Listraversedinthesamedirectionastandsvaryfrom1to1in(5.1.11)and(5.1.12).
However,since
.4;8;0/D2.2;4;0/;
Section5.1
Structureof
R
n
289
Listraversedinoppositedirectionsastandvaryfrom1to1in(5.1.11)and(5.1.13).
Settingt D1in(5.1.11),weseethatX
1
D.1;1;1/isalsoonL.Thelinesegment
fromX
0
toX
1
consistsofallpointsoftheform
XDt.1;1;1/C.1t/.1;3;1/; 0t1:
ThesefamiliarnotionscanbegeneralizedtoR
n
,asfollows:
Definition5.1.10
SupposethatX
0
andUareinR
n
andU¤0.Thenthelinethrough
X
0
inthedirectionofUisthesetofallpointsinR
n
oftheform
XDX
0
CtU; 1<t<1:
Asetofpointsoftheform
XDX
0
CtU; t
1
tt
2
;
iscalledalinesegment.Inparticular,thelinesegmentfromX
0
toX
1
isthesetofpointsof
theform
XDX
0
Ct.X
1
X
0
/DtX
1
C.1t/X
0
; 0t1:
NeighborhoodsandOpenSets in
R
R
R
n
HavingdefineddistanceinR
n
,wearenowabletosaywhatwemeanbyaneighborhood
ofapointinR
n
.
Definition5.1.11
If>0,the-neighborhoodofapointX
0
inR
n
istheset
N
.X
0
/jD
˚
X
ˇ
ˇ
jXX
0
j<
:
An-neighborhoodofapointX
0
inR2 istheinside,butnotthecircumference,ofthe
circleofradiusaboutX
0
.InR3itistheinside,butnotthesurface,ofthesphereofradius
aboutX
0
.
InSection1.3westatedseveralotherdefinitionsintermsof-neighborhoods:neigh-
borhood,interiorpoint,interiorofaset,openset,closedset,limitpoint,boundarypoint,
boundaryofaset,closureofaset,isolatedpoint,exteriorpoint,andexteriorofaset.Since
thesedefinitionsarethesameforR
n
asforR,wewillnotrepeatthem.Weadviseyouto
readthemagaininSection1.3,substitutingR
n
forRandX
0
forx
0
.
Example5.1.6
LetS bethesetofpointsinR2 inthesquareboundedbythelines
x D ˙1,y D D ˙1, , exceptfortheoriginandthepointsontheverticallinesx D D ˙1
(Figure5.1.4,page290);thus,
SD
˚
.x;y/
ˇ
ˇ
.x;y/¤.0;0/;1<x<1;1y1
:
290 Chapter5
Real-ValuedFunctionsof
n
Variables
EverypointofSnotonthelinesyD˙1isaninteriorpoint,so
S
0
D
˚
.x;y/
ˇ
ˇ
.x;y/¤.0;0/;1<x;y<1
:
Sisadeletedneighborhoodof.0;0/andisneitheropennorclosed.TheclosureofSis
SD
˚
.x;y/
ˇ
ˇ
1x;y1
;
andeverypointof
SisalimitpointofS. TheoriginandtheperimeterofSform@S,the
boundaryofS.TheexteriorofSconsistsofallpoints.x;y/suchthatjxj>1orjyj>1.
TheoriginisanisolatedpointofS
c
.
y
x
(1, 1)
(−1, 1)
(1, −1)
(−1, −1)
x
Figure5.1.4
Example5.1.7
IfX
0
isapointinR
n
andrisapositivenumber,theopenn-ballof
radiusraboutX
0
isthesetB
r
.X
0
/ D
˚
X
ˇ
ˇ
jXX
0
j<r
. (Thus,-neighborhoodsare
openn-balls.)IfX
1
isinS
r
.X
0
/and
jXX
1
j<DrjXX
0
j;
thenXisinS
r
.X
0
/.(ThesituationisdepictedinFigure5.1.5fornD2.)
Thus,S
r
.X
0
/containsan-neighborhoodofeachofitspoints,andisthereforeopen.
Weleaveittoyou(Exercise5.1.13)toshowthattheclosureofB
r
.X
0
/istheclosedn-ball
ofradiusraboutX
0
,definedby
Section5.1
Structureof
R
n
291
S
r
.X
0
/D
˚
X
ˇ
ˇ
jXX
0
jr
:
X
0
X
1
X
r 
r− X
1
−X
0
Figure5.1.5
Openandclosedn-ballsaregeneralizationstoR
n
ofopenandclosedintervals.
Thefollowinglemmawillbeusefullaterinthissection,whenweconsiderconnected
sets.
Lemma5.1.12
IfX
1
andX
2
areinS
r
.X
0
/forsomer>0,thensoiseverypointon
thelinesegmentfromX
1
toX
2
:
Proof
Thelinesegmentisgivenby
XDtX
2
C.1t/X
1
; 0<t<1:
Supposethatr>0.If
jX
1
X
0
j<r; jX
2
X
0
j<r;
and0<t<1,then
jXX
0
jDjtX
2
C.1t/X
1
tX
0
.1t/X
0
j
Djt.X
2
X
0
/C.1t/X
1
X
0
/j
tjX
2
X
0
jC.1t/jX
1
X
0
j
<trC.1t/rDr:
TheproofsinSection1.3ofTheorem1.3.3(theunionofopensetsisopen,theintersec-
tionofclosedsetsisclosed)andTheorem1.3.5anditsCorollary1.3.6(asetisclosedif
andonlyifitcontainsallitslimitpoints)arealsovalidinRn.Youshouldrereadthemnow.
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