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Chapter 4 Hilbert Space
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4.1 Inner product space
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Inner product
CEE :),(
0,0),()( xholdxxi
Exforlinearisxii ),()(
E : complex vector space
is called an inner product on E if
Eyxxyyxiii ,),(),()(
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Inner product space
),(
xxx ,
E : complex vector space
is an inner product on E
With such inner product E is called
inner product space. If we write
,then is a norm on E and hence
E is a normed vector space.
Show in next some pages
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Schwarz Inequality
Eyxyxyx ,,
E is an inner product space
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yxyxTherefore
yxyx
havewebyThenyxyxayax
andathenyx
yxaTaking
CaEyxyaxyax
EyxyxyxHence
xy
yx
y
yxx
y
yx
y
yxx
havewey
yxtTaking
ytyxtxtyxtyx
Rtanyfor
andyxthenyxIfCase
yxyxthenyxIfCase
),(
),(
(*).),(),(),(
1,),(
),(
(*),,),(Re
,),(Re
),(Re
),(Re),(Re),(Re20
,),Re(
),Re(2),(0
0,0,0),(:2
),(,0),(:1
222
222
222
2
2
2
2
22
2
2
2
22
2
222
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Triangular Inequality for ∥ .∥
Eyxyxyx ,
E is an inner product space
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yxyx
yx
InequalitySchwarzbyyyxx
yyxx
yyxx
yyxx
yxyxyx
EyxanyFor
2
22
22
22
22
2
2
,2
,Re2
,Re2
,
,
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Example 1 for Inner product space
nCE
nnn Cinzzzandzzz ,,,, 11
n
iiizzzz
1,
Let
For
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Example 2 for Inner product space
221
2 ;,,)(Ni
izzzNE
)(,, 22121 Nzzzandzzz
Ni
iizzzz,
Let
For
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Example 3 for Inner product space
),,(2 LE
),,(2 Lgandf
gdfgf ,
Let
For
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Exercise 1.1 (i)
)(, 2 Nzz
summableiszzNjjj
zz ,
For Show that
and hence
is absolutely convergent
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Njj
Njj
Njjj
jNj
j
Njjj
zz
zz
zz
zz
22
22
2
1
2
1
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Exercise 1.1 (ii)
)(2 NShow that is complete
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Nixz
sayCinsequenceconvergentaiszthen
CinsequenceCauchyaisz
Niforthen
Ninkmforzz
nkmforzz
nkmforzz
tsNnanyFor
zzz
zzz
withNinsequenceCauchyabezLet
inin
ni
ni
kimi
Nikimi
km
n
lim
,
,,
,
,
..,0
,,
,,
)(
22
22212
12111
2
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.)(
)(lim
2
,,
max
2..
0
2
2
2
0
21
10
completeisNTherefore
NinxzHence
xzxz
nnforthen
xxxLet
nnTake
nnxztsNn
andNiFor
nn
Nii
Niinin
ii
iiinii
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Hilbert space
),( E
.)(2 casespecialareCandN n
),,(2 L
An inner product space E is called
is complete Hilbert space if
is a Hilbert space of which
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Exercise 1.2
Define real inner product space and
real Hilbert space.
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4.2 Geometry for Hilbert space
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Theorem 2.1 p.1
Ex
E: inner product space
M: complete convex subset of E
Let
then the following are equivalent
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Theorem 2.1 p.2
zxyxsatisfiesMyMz
min
My
MzzyxysatisfiesMy 0),Re(
(1)
satisfing (1) and (2).
(2)
Furthermore there is a unique
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0),Re(
,0
),Re(2
)0()(0
)0(),Re(2
)()(
)1()(
10
)2()1(
2
222
2
2
zyxy
havewelettingBy
zyzyyx
ff
fzyzyyxyx
zyyx
zyxflet
andMzanyFor
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zxyxHence
zxyx
zxyxzxyxyx
xzyxyx
yxxzyx
yzyx
MzFor
Mz
min
),Re(
),Re(
),Re(
),Re(0
,
)1()2(
2
2
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21
122211
122211
21212
21
21
0
,Re,Re
,,
,0
),2()1(
:
yythen
yyxyyyxy
yyxyyyxy
yyyyyy
thenandsatisfyyandyIf
yofUniqueness
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nn
n
nm
nmnmnm
nmnmnm
nmnm
nmnm
n
nn
Mz
zxyxthen
nasyztsMycompleteisMSince
nmasnmnm
zz
xzz
zxzxzz
xzxzzxzxxzz
xzxzzxzx
zxxzzz
sequenceCauchyaiszClaimn
zxtsMzConsider
zxLet
yofExistence
lim
0..,
],011
241
21
2
2422
,Re22
4
,Re2
[
:
1..
.inf
:
2222
2222
222
22
22
222
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Projection from E onto M
MEt :
Mt
The map
of Thm 1 is called the projection from E onto M.
y is the unique element in M which satisfies (1)
defined by tx=y, where
and is denoted by
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Corollary 2.1
)()( 2 idempotentisttti
Mtt
)()( econtractivistyxtytxii
Let M be a closed convex subset of a Hilbert
has the following
properties:
space E, then
)(0),Re()( monotoneistyxtytxiii
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0),Re((*),)(
),Re(
(*)),Re(
)),(Re(0
0),Re(
,0),Re()(
.)(
2
2
2
tytxtytxyxByiii
yxtytx
tytxyx
tytxyxtytx
tytxtytxyx
tytxtytxyx
txtyyty
tytxxtxii
obviousisi
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Convex Cone
0, MxMx
A convex set M in a vector space is called
a convex cone if
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Exercise 2.2 (i)
tIs
MxxyEyN 0),Re(;
NM tsandtt
Let M be a closed convex cone in a Hilbert
Put
Show that
space E and let
I being the identity map of E.
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tIs
Xxtxxsx
txytxxtx
txyxtx
ytxxtx
ytxxxtxx
Mtxcetxy
Mcetxxtxtxxtx
NyandExanyFor
0),Re(),Re(
),Re(
),Re(
),Re(
sin,0),Re(
0sin,0)0,Re(),Re(
,
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Exercise 2.2 (ii)
0)( iftxxt
( t is positive homogeneous)
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txxtthen
Mzce
ztxxtx
ztxxtxztxxtx
thenIfCase
Mcexttxt
thenIfCase
MzandXxanyFor
)(
1sin,0
)1
,Re(
),Re(),Re(
,0:2
0sin),(0)0()(
,0:1
,
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Exercise 2.2 (iii)
Exsxtxx ,222
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222
22
22
22
),Re(2
,
]0),Re(
0)0,Re(
sin,),Re(
sin,),Re(0[
0),Re(:
,
txsxx
txsx
txsxtxsx
txsxxtxsxx
tsItIsSince
sxtxthen
txxtx
tIscetxxtx
NsxandMtxcesxtx
sxtxClaim
ExanyFor
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Exercise 2.2 (iv)
0; txExN
0; sxExM
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0;
0
)2(
0;
0
)1(
sxExMHence
sx
txsxtx
xtxMx
txExNHence
tx
txsxsx
xsxNx
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Exercise 2.2 (v)
sxztxy ,
;0),Re( sxtxxandsxtx
,0),Re(,, zyandNzMyzyx
then
conversely if
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sxzSimilarly
txythen
zw
zwywyzwyxy
Mwanyforthen
zyandNzMyzyxifConversly
sxtx
sxtxsxtxsxtxsxtx
sxtxxandsxtxxSince
sxtxxtxxsx
ExanyfortIsSince
,
0),Re(
),Re(),Re(),Re(
,
0),Re(,,,,
0),Re(
),Re(2
,
22222
222
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Exercise 2.2 (vi)
MxxyEyMN 0),(::
M is a closed vector subspace of E. Show that
In the remaining exercise, suppose that
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Mz
Myyzhence
Myyz
yz
My
yz
yz
yz
yz
subspacevectorclosedaisMce
My
iyz
iyz
yz
yz
MyyzNz
obviousisIt
0),(
0),Im(
0),Re(
0),Im(
0),Im(
0),Re(
0),Re(
sin
,
0),Re(
0),Re(
0),Re(
0),Re(
0),Re(""
.""
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Exercise 2.2 (vii)
both t and s are continuous and linear
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.
,
.
)(
)(
)()(
)()(
,,
222
22112211
22112211
22112211
221122112211
222111
2121
continuousaresandtthen
xsxandxtxsxtxxSince
lineararesandt
sxxsxxs
andtxxtxxt
NsxxsandMtxxtwhere
sxxstxxtxx
sxtxxandsxtxx
CandExxanyFor
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Exercise 2.2 (viii)
tsENstEM ker;ker
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tsENhaveweSimilarly
KersysyMy
sMthatshowTo
MytEy
tEyytyMy
tEMthatshowTo
ker,
0
ker)2(
""
"''
)1(
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Exercise 2.2 (ix)
Eyxtyxytx ,),(),(
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),(),(
),(),(),(
),(),(),(
tyxytx
tytxtysxtxtyx
tytxsytytxytx
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Exercise 2.2 (x)
MzandMy
such that x=y+z
tx and sx are the unique elements
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zsxSimilarly
ytx
tNMzcetytx
linearistcetztytx
MzandMywherezyx
,
kersin,
sin,
,
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4.3 Linear transformation
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We consider a linear transformation from
vector space Y over the same field R or C.
a normed vector space X into a normed
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Exercise 1.1
T is continuous on X if and only if
T is continuous at one point.
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.
)(
,
)(
..0,0
.""
.""
0
00
0
XoncontinuosisTHence
TsTx
sxTsx
Xsanyforhence
xxT
TxTxxx
tsanyforthen
xatcontinuousisTthatAssume
obviousisIt
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Theorem 3.1
XxxcTx
0c
T is continuous if and only if there is a
such that
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XxxcTx
thencchoosewe
xTx
xx
Txx
XxanyFor
Txxts
xatcontinuousisT
continuousisT
XoncontinuousisTExerciseBy
xatcontinuousisTthatobviousisIt
,1
1
1
0
1..0
0
""
.,1.3
.0""
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Theorem 3.3Riesz Representation Theorem
Xy 0
X
Xxxyx ,)( 0
Let X be a Hilbert space and
Furthermore the map
such that then there is
0y
is conjugate linear and 0y
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)(),(
,)(
)()(),)(
(
),(),(
)()()(
.
,
,
.dim,ker
0
0
020
00
0020
0
20000
00
0
0
xxy
havewethenxx
xyletweifHence
xxxxx
x
xxvxxx
xxvx
scalaraand
MinelementnonzerofixedaisxMvwhere
xvxwritecanweXxanyFor
ensionaloneisMthenMLet
thatassumemayWe
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4.4 Lebesgue Nikondym Theorem
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Indefinite integral of f
fd
,,
AfdAA
)(
Let
Suppose that
and f a Σ –measurable function on Ω
be a measurable space
has a meaning;
then the set function defined by
is called
the indefinite integral of f.
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Property of Indefinite integral of f
nA0)(
0)(0)( AwheneverA
ν is σ- additive i.e. if
nn
nn AA )(
is a disjoint sequence, then
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Absolute Continuous
,,,,, spacesmeasurebeandLet
0)(0)( AwheneverA
ν is said to be absolute continuous
w.r.t μ if
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Theorem 4.1Lebesgue Nikodym Theorem
and
spacesmeasurebeandLet ,,,,
AhdAA
,)(
with
),,(1 Lh
Suppose that νis absolute continuous w.r.t.
μ, then there is a unque
such that
Furthermore ..0 eah
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)1.4()1(
)(
..
Re,
)(
1)(
,)(
),,(
,
)(2
1
212
12
2
2
Xffgddgf
fgdfgdfgdfdf
tsXg
ThmonpresentatiRieszbyX
f
ddfdffSince
Xffdfbydefined
Xonfunctionallineartheconsiderand
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Let
L
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0)(
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onxeaxgClaim
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nn
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4.5 Lax-Milgram Theorem
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Sesquilinear p.1
),(),(),( 22112211 xxBxxBxxxB
CXxxxfor 2121 ,,,,
CXXB :),(
Let X be a complex Hilbert space.
),(),(),( 22112211 xxBxxBxxxB
is called sesquilinear if
![Page 69: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/69.jpg)
Sesquilinear p.2
EyxyxryxB ,),(
B is called bounded if there is r>0 such that
XxxxxB 2),(
B is called positive definite if there is ρ>0 s.t.
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Theorem 5.1The Lax-Milgram Theorem p.1
1S
XyxySxByx ,),(),(
Let X be a complex Hilbert space and B a
a bounded, positive definite sesquilinear
functional on X x X , then there is a unique
bounded linear operator S:X →X such that
and
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Theorem 5.1 The Lax-Milgram Theorem p.2
1S
rS 1
Furthermore
exists and is bounded with
![Page 72: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/72.jpg)
1
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xxxx
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atederuniquelyisxandDDthen
XyyxByxtsXxXxDLet
![Page 73: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/73.jpg)
*
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closedisDthatshowtoFirst
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mnmnmn
nn
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ioncontradictay
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000
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rSxrxS
xSxr
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![Page 76: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/76.jpg)
4.7 Bessel Inequality and parseval Relation
![Page 77: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/77.jpg)
Propositions p.1
kEk tt
ne
neLet
Ut
be an orthogonal system in a
Hilbert space X, and let U be the closed vector
subspace generated by
Let be the orthogonal projection onto U
and where kk eeE ,,1
![Page 78: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/78.jpg)
Proposition (1)
Xxexextk
jjjk
1),(
![Page 79: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/79.jpg)
k
jjjk
ikikkii
i
k
jjjiki
k
jjjk
exextHence
xtextxtexeBut
eexte
kiFor
thenextLet
1
1
1
,
),(1,,
),(),(
,,1
,
![Page 80: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/80.jpg)
Proposition (2)
xtxtXxFor Ukk
lim,
![Page 81: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/81.jpg)
kUkUk
k
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andUyanyFor
sin,lim
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1
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Proposition (3)
).,(),(),(1
yexeytxt j
k
jjkk
For each k and x,y in X
![Page 83: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/83.jpg)
),(),(
),(,),(
),(,),(),(
1
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Proposition (4)
).,(),(),(1
yexeytxt jj
jUU
For any x,y in X
![Page 85: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/85.jpg)
),(),(
),(),(lim
),(lim),(
0
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),(),(
),(),(),(
),(),(
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Proposition (5)
Xxxxei
i
,),(
2
1
2
Bessel inequality
![Page 87: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/87.jpg)
22
1
2),( xxtxe U
jj
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Proposition (6)
XUXxxxei
i
,),(
2
1
2
An orthonormal system ne
is called complete if U=X
( Parseval relation)
![Page 89: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/89.jpg)
XUHence
oncontraditiaxext
xtxtx
xxttsXx
thenXUthatSuppose
xxtxe
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iiU
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222
22
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2
![Page 90: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/90.jpg)
Separable
A Hilbert space is called separable
if it contains a countable dense subset
![Page 91: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/91.jpg)
Theorem 7.1
2
A saparable Hilbert space is isometrically
nCisomorphic either to for some n
or to
![Page 92: Chapter 4 Hilbert Space. 4.1 Inner product space](https://reader031.vdocuments.us/reader031/viewer/2022012312/56649f535503460f94c7842b/html5/thumbnails/92.jpg)
.inf.inf
.,
,
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kk
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