Orscyete metric space: Lat
if
olt)
Re Suts:
i In
Evermetric
cx
sel +anc
be a
y+
x
7
R Sueh
if-
cliscvete topolojn ever
sel
is
open
space is a topolojicalSpace buL the m47
ma nol be 2
tnu
anc
9
ol
x-R
i
)= 1x-lisusual metric space
Discrele lopoloi: Px) is also a topoloj on X anc is Calleo a
oliscrete topelo
.
Tncliscrete
Calleo
anc
topolojn
x
on
Lopaleu, is also a
is
inoliscrele topolo
*j3
T
Open Ball: BLdo, Y)=zey
ollz,Zo)LYj,wheve
o
eTPCK
Closecl Ball
exol na)Y
B)
sphere: So,
)
ey
=
dz,
Theorem: Ever open ball in
Open
set;
i
Because
e0t,
16
A=
s
2
)cA
m: Ever
Result: 9n
Remayk:
Ever
ii
=
(I-o.S,
2
is
an open intenal
that e BlY)CA
12 usual melic space)
Such
1-Ss)
metric
?
A
.
space ,becaus
2]CA
open is
cliscrete metic
open ball is an open sei
oliscrefe metric
Ever
metric space
tt o.J) = (o:S,
open in dliscrete
Eyample: B-3)s
heoYe
Y7o
ZotY)
BMo)=2-Y,
not open in
t,2s
o.5)
vj
=
usua
neA, 3 a real number
Example:
H-{t,
7)
open ball
space
are
in
BoY)
space
metric space ( o))
Y
x
Singleton sets in oliscvete
Subset in oliscvete metric space
is
Y2
metie
space
both open ancl closecl
Resulf: Sinltons sets in uual metnc space are close
= 1R
x
=
because Let
A-, x)UNi )
A
open
A i
A1s
closed
Scanned with CamScanner
6uestion:iiy+
ol M,7)=
olX)
LD Y4
iv yt9,
=
max
ola)
+13
b|
lx| lql3
l)
l)i
-)
dl)=Jx-
space
not melric space
is
is
metnc
nol
is nolmelnc
cl)
V I olis
ametic
metmc space
isnot me tric space
l)
then
space
space
olx)- min (1, olLu)) s
So
ol)is an
space anol cl,)bn
14
metnc
oltx)).then
(,
space
Xol) is nol metic
A seuence in y is a
space
(Xl).
metnc
Consiclen
a
Seyuence:
oleno te b
is usuall
Sevuence
ils vane
functionw
* nisseyuence
x
in X
set.e
X-Remark: I
n is Finite number then nis
inite othenis
X.
x,
n
inhnile
Convexjent Sevuence:
sevuencen
A
Convev{ent in X if
va
val numbe
natu
Such
nalu
oltn, x)
Callec
n
Remavk:)U
Lm
n
n2
n
Example:x- 1R,
Lim
Xn =
ne
olu,)
tov
that
a
space
in a meic
(X,ol)
iven o
there exist
n
is Converjen
is
a
vnzn.
E
B(X, €)then
=
lx-z|
ol(An,X) =
Take
Xna
E= too
nz 96
htS
Remavk:
Limit ot Seyuences alwa unvu
is nol
Converse
but
Seyuence
Ever
isbounoleo
i)
Ever Convejent
not
Converjent
bounolecd
is
but
n
n
,neN
i
7
E
fov
=-
tnue
space (X, ol) is
metric
Sevuence {n
Caucha Seyuence:
a nalural number
o
exist
there
Called Cauch f for a fiven
o Such that for all m,n2 no
is a
A
n
Exampl
ov
-) olln,
Ol2n,Zm) E
Xm)
x
min
Xm) =
b
1R
as
nm
ollxr)= 1x-j
n is a cauchy
Kn=CoSnA
n
sevuence
Scanned with CamScanner
n
oll 7) |*-al
no
6
npmon.
where
ol(Xni Xm)n is a cauch sequnCe
Re Sult: Even Convevfent Sevuenca is a Cauch1 Sevuen ce lonverse
ma ov ma not be tvuL
=
a Cauch SeyunnCe
Xn
in k
l-j,
(o,1]c
ola)
12,
=is
Xo (o, 1].
but
0
but nol Coniverfent in x, Nn
Comp lete spacg: Amelric space Ko)
is known aJ Complele metic
ever cauch seuene is Converjenž in X
Space
Examplex: j 2, ct) ,lCid), (2.l),
Ccel), (o,i),cd), U, el) are all
Complete metnc space
yaople
=
X
X=2
a
t
iBut
Bur
(o,1), ort), la, b) ete are nol Complele
wis not Comple te 10- set vational numberi) ol= 1X-]
Continuotus function
={x: [4. b]
cla,b)
R,ir
olcg)= max)-jco
te Carb
ContYacl1on; Let (x,el) be a
bLalI
.
metric space A self mafp :X
X
Contraction mapfpin. if there exist a veal numbe
a
such
to
be
saiol
is
S Complete.
that
ollfex, fe))
L
ol
)
g
t)
6YoHa),
cl1)
Fixed poinz: Consiole a self map T: X
X on d space
Df X is Callec ixec poinl of TfT=X
heorem Everq Contraction map is unifsmn Conbiuous
X
A
poinZ
Continuous metric space: Let (x,cl),ly, ') be metric space and
a
Continuouus
function.Then
an
fov
a
y
anc/
aEX
be
t:K
there is a rea numbev -SCaE)
Yeal number Eo,
Such thati
that
xEX
d'f),
Continuous metnc space: A
olXa)
Unilorm
unifoml Continuos on
number
if
ven
olepenols on
Such thal
xEX
CE),shich
DavticulaY poinl
eneralieol
X
a
triangular Inesualit z: Foy
space
Then
metic
oltxZ)L oltX2a)++ol(%0 )+-_-
fea))
e
&
anj
y
saicl to be
these is a rea
nol on the
fx
function
,
o
eonlzanc
o
is
a) 28
X
ntX,where
Xa(n
,
n)
Scanned with CamScanner
X 1
Banach Fixeol point theorem: Let LXd) be
anc T be a Contya ttion map on x, lhen
n x.
poin
4
e Complete
T
has a
Eyereise: Let x= clo,
x cleine
and TX
i2 is exist
Finel the xeol point of T
Theotem: Let (X,ol) be a Complete metnic space
self map If
uniayue fineol
i
T
x
point
is a Conlraction
metic
unnu
space
fixeol
as Tt)- {(xlt)
4))
anc :X-x
map,
be a
Then T hay a
ADPLicotion
of Bannach Contracion theorem: et F:Rxa. b]
2 be
Conbinuous Funçhion. such
that |EO,t)- FAt)1EL|x-a1
anc tela, bJ, then
X
anc
e2
the diHerential euation
cpt)
subject
to
la)=has
uniayu
a
solutiom where
L
an posibue rea
number
Note The inidial valeue problem has a univue Soltion i the
a
folloaing
interal eyualion
t)
has
univue Solution
FCu,u) cleu
BuPATion: Solve.
oiHerential evuation by xeg point
methoo
(X+t)t
Subjecl to tonolition
lD) =o anc oLt42
t3
i clt 7+t)esubject to conolition xu)=4 ano 14tL3.
R be Continueus anh
Imptict Function theorem: Lt F:Rx(a,b]
12
G,t)= Fat)+
Fxa, b)
be olehineoa
Assume that there euisis a number kl
such that o
KL
Then the euation Ft)=o
lehine a uniuue Coninuols function
o 1
tem
uestion: Fl,t) t+ sinttx)that
*shos
nt)
is
Continuous
and
=
à
Satishe
Suestion:
FXit)=
o
F:2x1
thal Xlt) is
RR
d
X-3+
d
that
Fx, t)- x(t-) +
Shots
Contintuotls anc satis fes
Fx,t)=o
Fx,t)=o
BueSon:D Mappinp in elemenlarj jeome
ty
which hauve a single
fined puoint
such
Dwhich hare in hnitel man1 tixeo Po ints
Scanned with CamScanner
bluestion: Let x={x tM 1x21jC 2 ancl let the
mapping Txx
a contrachon anc
T
be clehnec bj T = 4
then
the
b
=
point
ixeol
Bonach
Remark;
smallest
In
ano can nol
7:X
If
be
omitleol
X
satis hes
is essenbial
Completeness
theorem
ol TT)
dl7)
when
+1
ano) T ha
is uninue
point
hnecl
then
Fineol point,
pornt
hmeol
a
hag
Continuous funetion
Eve
functionrom
a
space v(a
normet
fbe
Lineay Function : et
F:NF
space).nen
veclor
Flhelol
of
helo
to
space)
and
an
4
tor
1éw
functio
a
Linear
to
nai
be
is saic
bfCJ)=af)4
+bd)
a.bEF
space W
M
a
norme0
fom
operator:
uneay
T:w
et
Extension of
Subspace
"A"
a
Consicler
rotov.
ope
lineay
a
space
M
be
to a normeol
H
olehineo
A function
T:AM
b
of N.
T1A
olenoteal
A
b
7 to
restncton
ancis
of
Calleol
Subsefž
is
1 TA
a
for
if
of
exlensiDn
-Thu
an
1
S
The operaly aineay
operatoT:AM,a
anc
funcion
anol
A of
an
edenSin
then
Ti's
TR=T
eh.
that
M is such
a
a
ect
ax
a
TT
T:w
of T
ope
rator is alwa binea
Linear
of
Remayk: Therestnicton vestncl10n oF a Linea opermtoy
a uniyue
vatoy
ope
Lineay
is
alsa
exzension ef a
There
one
than
more
be
iij There Can
A function P:U
space
a
uinea
be
v
het
xeu
Finite Function:
tov
fov
all
xeU
hnite
is
Convex
to be Finile it
a
p
is said
function
Funclion: A
p:UPis
Saio to be
xeU
Convex
pu20
property)
ous
propet
noTmf
semi
homejeneous
o
homeene
function
(+ve
0
a2
XEU
=
ap)b
pLa)=
+P)
i)pLr)P)
Rpd
i
where E
Eyampl: The
space
(suba olotihve propert
l-1:V
funion
Dom
saCohve
where Vis
hunction
Scanned with CamScanner
a
hormec
Hahn-Banac h theoremfo real space et p be a inite Convex
unction odehined on a real space (Lineav space) V anc let Ube
a Subspace of U. Lat ,:U
R be a Linear funcion S.
)Plc)
xetU-O
Then
Can be extended to a lineay function f olehned on
t
U
Such
that
kx)pl)
preper
the
set
ilh
oYn's emma: het y be a Daytially odereol
that eve Chain in X has an upper bounol. Then x Contashg
maxima element
pavtial ordlered set: A Set M is Callecl partiall avclereo set
velation
pavlial
is
binay
on
M, that is a
cleine
oder
ifa
anol satishes the conolitions
which is wntlen
)
Reflexivilj
a4a
9f aLb
(Antisymmetnic)
bea then a=b
a tb be C then a L CCtransibive
maximal element, of sep, where P, s) is 4
Nofe: B
Partiall oderecl set,we mean an element me Ssuch thal
anq SE Si mS
spaces
A
topologica
function
be
X,Y
Let
HomeomaYphism
y is saio to be a homeomphism if
hx
Continuotus
Continuous
is
F
i
f
is
isbijecbire
aeM
u
a
t
is
mapping
the
of
properti
mophism
Homeo
aiven b
T(0)
=
odEF
X
Transtation Mappinj:
12:NT(0)
Continuws Functhon in
sContimwy
on
X
it
F
8en
Zo
X
Da
X,éd
oixed
Topolojical Spaces: A funchim fy
v
euey
at
point
ol K
is ConTihuos
Scanned with CamScanner
emm:
ball
inNThen
Remavk
space ond
N be a nomed
el
B(o
Y) =
B(o,)
be an open
Zot YGloj t)
Bo;v)v)-yBlo; /))=B(
0; 1)
hemma: Le 7 be a surjecbive Continuous uineay operalr rom a
Benach space
to a Canach space M (i:e T:NM)
to n
hen
1
open bal BCo;I), the image
TB(o; 1)) Contains an en ball in
Mwith Center at the ori8ion
The Open Mapping Theorem:
het n and M be lanach spaces anol
7:N M be a Surjecbive Continuous Lineay operaloY 1 hen 7
is an open mapping
CovolLan et
N enod M be
Bannach spaces and T:
5 M bea
biiechve Continuous LineaY openalor.Then T is a homeomophisnm
anc (M, -.) be notmeo spaces and
Deknibions Let A,0)
p-{)
xeEM}
on
muliplicoD
scala
uestion:
F
a
oblibim
d(7)= loy
CX1) EP
space
a o1mec
Pal
= Il-+- }|llo
lCll,
Garaph o T
rom
mapping
da)
qpll= (lIiz1P/l
1pt
Ile
P
2
llxql,=max(Iklllljllo)
Anpther norm on P is
n
(J)+O7 )=+x"7)-
ana
spa ces N and M
two
normed
an
space
M
Danach
(ATN-3M)
to
Banach Space w
For
vephafT
callecl
he
dehne the set CaTO T)eNis
ano
spaces
tnic
So Gre
me
are
M
Remark: i since N and
inelucecl
metnc
the
under
P=
prodUCL
wxM
spaces,
their
HausclorH
bb
In
space
metic
on
nom
Pis
the
M
spcces
n
topuloical
two
1eneral tr
GT
of
aLgo
4
mapping T:N-3M
G
the
the
ph
ph
Subspact
of
closeo
moy nol be
M
Scanned with CamScanner
case
a
ot
norme
spaces
ef
the
eph
conbinu
D the
connibn
exba
some
closeol
ith
is alwo
ppnT:NM
9n
we hae a maYe Complete resul
spaces
and
M
Banach
be
and
Closeo raph Theorem: het N
Continucus
T
operatoy
is
Then
i£
Lineay
M
a
be
T:N
The vaph of T is a closeod subspace3of Ny
th
and
nameo
spaes
M be
N
and
Opevatoy:
at
loseo Lineay
operalor
space
ineav
A
nome
a
Subspace
of
X be a
fo all Seruences
.
closeolit
be
to
Sasol
M is
T:x
X
in hich
to
Convege
to
eN xe
that
Such
jeM
Conve3
M be
and
nw
Theorem2: et
Lineay operator T: X
in XxM
NA
Theorem:
het
the seuence | Tnot
x
name
M
ano
=T
spaces ancl
X
be
n
image
in M
sub ace
aSp
o4
is Close
GT
iH
closeo
is
anc
spaceg
and
normeal
ancd M be
closec
is
T
Then
openalsY
Lineay
n
T:W-5M
T:N-SM
bounlec
opeator
Linear
closecl
Surjecbve
a
M
be
Theorem: het TA
bounclecl
is
it
Texist,
inverselT) of
the
vanant
in
is
it
i
If
if
omeln
Calleo
is
mapping
then
A
ano
4.ol),
ISomel
map b(X,ol)
a
be
S
V
olislance.
isanisomel
is a
If
oltfoo
X
feg))= ol)
M of a metnic space X
Subset
RareloRNoLhere olensej:A
closure
X
its
olense) in
noshere
Raraloy
be
to
is Sa no intenor poinlsS
Mhas
Mea jer (or
1SL
be
to
said
Cateior):
A
meaerin
Subsel M
x
to
l
to
Sal
non-meajer
be
space
is
the union of Counlabl
ifMis
is are in x
shich
sets
each
ef
many
A
Subset
Me4qerl2CaLeag)
Non
in y
ofa metic
F
M
M
of
is
space x
metic
not meajer
a
Scanned with CamScanner
în
is
y
spaces)
Lamplote
metie
Baire 's Catejon Theoem
it
il
is non- meage
then
at
in
X
deast
If ametic
itself Herce
U Ak
is Complete,
x d
space
if x+ i's Comple te an0
(Ak Close d)
one A Conlalns a non-epty qpen Subet
Unitovm Boundlecinesc Theolem Let (7,) be a seuence of
Y fom a lanach Space K mo
bounoled Lineay operalor n: K
a nomec space y Such that T,is
ever
xe
for
bounolel
where
C
nomsllUnll
,llC
a
is
is
n= v23
he
seanuence
ot
real number. Then the
C
hal
Such
a
is
there
is
bounoleolthat
n= \,2,3,
IThll C
Buestion: Shous that T: 2-2
TSopen
Ls
themapping R
--
lehinecl ba G»&)
given
b
Gr)
R
G)
1o)
an oper mappi
map
closec
nol
open
mappin need
6uestion: Shos thal an
Sets onto close sets
a nam on XxY
dehines
+llqll
Buestion lCll= xll
oehnec!
are
anol
X
Y
XxY of nomeo space
prooluct
on the
Question
pl&)=
buestion: show
Ty
1lUag)ll
max 1l, lgi
Y
= (li«e/
linear
ofa
that theraph GT)
iS a
vecto
Subspace
o
la2
operat
XxY.
XxY with
U=XxY
V=
ith
Shos
thal
thal
Shol
spaces.
Banach
are
Buestion: Tf x anolY
space.
Gannach
a
= llxll +lgll is
nom lon)ll
Eigenvalueg: An
ejenvalue f
SUUare
malin
t=Cogk) is a
NX has Solution xao0
Ax=
numbe Such thot
thi's
then
Solution
x
ti's
x#o,
has
Tf Ax=
ELjenvec
N
eisenvelue
to
CoTesponalins
otH
egenvecloy
an
called
is
Eijen SpaceThe eijenvectors Covesponoling to ejenialuen
ancd the tero vector tom a vector Subspace of y which is
callecl the ejenspace et A Coresponoling to ejenvalue ^
x
Scanned with CamScanner
of
Specturm The set ef
specturm of A. It is notedb TA)
Resolvent Sel: The Complement of spectum
A
all egenvalues
Complex plane
CA)
eoYem
ode
ancd
is
the
Caled the
the
in,
ofA
A-lenoteol b
set
of
vesolvenl
lalled the
is as
inePA)=-TUAJ
Ejenvalues ofa matni
1 he
ejen
values
t
an
soltiong
the
ot
bj
A=Ciy)
iven
are
n-Yosed Svuare malrix
the charactensbc evuation det(A- I)
A. Hence A has
Gt least one eitenValue (oncl al mest n-numemcally olHerent
f
senualues)
Lheu1emEisenvatues of an Operator): All matrices represenb
a iven lineaY operator
T: y
on a hinite limensiona
x
noY eo
space
Yelotive to Various bases for
x
x have the
Sarne ejhenvalues
Similay motricey An nyn matnx 12 is
saol tu be Similay
an
nyn
matrix 11, if there exisls
to
non- singular
matrin C Such that
T hen
hulols.
CTC
T
and
T
12 are Calleol Similar matrice8.
Remark i Two matriceg Yepresenting the Same Linea
inear
opesato 7 on hnile dimenSieinal nameo space
Yelative
ty
ave
X
to an bas
Similay
a
io Similay malice8 have Che
Seme eijenuakies3
Basic Concepl
T-
Resdvent operator rs
2-T)=
TAIJ
Scanned with CamScanner
X# of be a Complex homeol space and
a lineay openalor with olomajin D CT)C
A
vejulay
X.
of Tis & Complex number Such that
Regular value: Let
T:DT)
velue
x
R,UT) exists 2T)
Gset shich
is bouncleol
uy
RT)
is olehineod on
olense inx
Resolvent set: The resolvent set T) of Tis
the set of
allYejulaY value8 of T
Spectum: The complemend of resolvent set in the complex
is Calle
the spectrum of T ancd eC(T)
is Calleo
plane
a speciyal value efT.
PUT)
TT)=
spectrum
cliscrete
spectum
Poin speclvum:The point
A
not
exist.
does
RT)
Such
set
that
the
pT) is
of 7.
eigenvalue
an
callec
is
EGp(T)
is the sel
Continuous SpecLrum: The Continuous spectum T)
such that R,T) eists anol sabivhes R(T) iolehineo ona
set shich is dense in x but nol satisfes RT) is bounoleo!.
Resiolual Spectum: The resrolual Spectrum
G(T) is the Set
Such that Pa1)eisIS(anol may or mon not be bounoleol) bul
not saiShe RT) is olehneol on a set which is olense
iny
Note The spectrum of a lineay operator on a hnite olimensiona
space is Dure pvint spectum
OCT)=
TAT) = Q),where
T
ineay
operato
So everspecta
value LS an eijenvalu
an inhnite climensional, then Linear operatoy
9F is
Can have speclral Values hich are nol eigenualues
T X X
boundeod OpenazoA Lineat openalor
>M
is Calleo
T:N
is
-
s
boundlec
if
3c7o
Such
thal
l11Tl
cllxl
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Remarks 9t T y
Lineay and Kis
Xis bounoleo anc Lineay
Completer enol iF fov Some
vesolvent RT) exist anol
the
isolehneol on the wholl Space Xthen tov thal
thi
Yesolvent
is
bounaled..
emmg: het
y be a Complex Banach space
1:X
operatoy ano
PT)» Asume that
is
bTis
e
bounledThen RCT)
space y ano
aT
is
Xa
eay
Linea
closecl
on the shole
olehned
is bounolecl
Theorem (Lnverse): et
Te BCXx), wheve Xis a Canach spaxe.
then (L-T)exists
TIlL,
as a bounoleol Lineay operatay
on the hole
Space x ano
=
(1-T)
T +TT
If
TT
here
he
sene on he izhl
is Converzent in
te nom on U%)
Spectrum close Theorem: The
esolvent set PT) of a boundlol
linea operator 1 on a Complex Banach space X is
open.
hence the speclroum T)
is closeol
De0ye sentation 1 heorem Resalueni: For X and
T as in pectum
closedd theorem ancd eve N€ fT) the resolvent RT) has the
represenlation
Jo
the seies being absbleilel Converjenl tr eve
olisk
iven bg
I-ol
in
Thi's disk is a subset
the
of fuT)
Spectum1heotem;
The spectum T) ef
operato T:x
X on a Complex
Ganngch
und Lies in
Hence
the
the olisk
Yesolvenl
iuen
b
I
set gT)
Tis
in Ühe open
Complex Plane
a bounoleol
space
TI
Kis Compacl
not empt
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Linear
Spectval vacius: The spectral
vadius YgT) of an perator
Te BlX, x) on a Complex anach space x is he vacius
1
T)=Sup
eGT)
the
Smallest closeel
ef
cdisk centereod al the origin
Complex
plane and tontainint CT)
Remavks
T)IlTI
Guestion: Let
where
VEX
Anstwe:
eestion
X= Cc[o, ')
S
Hxed.
specl rum
Hinol 0C1)
=
vlo,ij
Finc a Linear
isa
3iven
Answe
anol olefine
ef be
(T)=mT
n
T:x
(wole
(hat
(a, b]
openalov T:
inlenal la, b]
bb Tx= Ux
Vx
TT)
is Closeal)
XK
c[o,1]-
clo,1] ,shoje
T: CCo)
Clo) S.t T= V.
buestion: f Y ij the eigenspace Corvesoncling to an eenvaluk
an operato T> wha is the spectrum
of
Ty?
Anter:
C(T)=
Hermitian matrix: The enenvalues of a Hermibian
matnx Azlj)
are rea
skes-Heamtian matrix: The eigenvalues ot a Skes- Hemban
malri
A=jik) ae pune imajinar or tero
Unidar
matriy:
The eijenvalues
of a unitaj malrin haue
hane
absolute value
spectal Mappin 7heorem to polnomias Let x be a Cemplex
Banach space, Te B(x,x)
ano
+
Then
PT))
of the openalov
pT)= T+
PlaT))
that
is
(n t0)
the spectum T(PT))
the
n-)
n-
+t
thove
d,L
3L5 pveu'sel
Consi sls
consi
of all
values which the ponomial p
assume on the spectvum aT)
of T
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Theovem (Linea
Convesponoling
inoleperolune)
Eigenvect
rs
to aliHerent eigenvalues N
Lineay operatoY
on a
T
anolepenolent set.
vecto
Space
X
--
X:, V2j
--
"
- n
hn
of a
Constbtule a Linearly
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