Chaper 3 Weak Topologies. Reflexive Space.Separabe Space. Uniform Convex Spaces.
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Transcript of Chaper 3 Weak Topologies. Reflexive Space.Separabe Space. Uniform Convex Spaces.
The weakest Topology
Recall on the weakest topology which
renders a family of mapping
continuous
IiYX ii :
topological space
arbitary set
To define the weakest topology on X
such that
i is continuous from X to iY
for each Ii
Let ),( ii YFw )(1iw
must be open in X
For any finite set IF (*) ,)(1
Fiiw
iw : open in iY
The family of the sets of the form (*)
form a base of a topology F of X
The topology is the weakest topology
that renders all i continuous
)()(
)(
)(
..
)()(
)(
""
1
11
xxHence
wx
wxNn
tsN
Xinopeniswandwx
Yinopeniswandwx
IieachFor
ini
ni
in
ii
ii
Fiiin
ini
i
inii
i
ii
ii
Fiii
wx
FiwxNn
thenFiNNLet
wxNn
tsN
wx
ieachFor
Fiwxwhere
wUconsidertosufficientisIt
)(
)(
,max
)(
..
)(
,
,)(
)(""
1
1
1
Proposition III.2
IiYXZ ii
i
Let Z be a topological space and
Then is continuous
is continuous from Z to IiYi
Zinopenisw
w
wU
wULet
continuousisHence
continuousisceZinopenisw
Xinopenisw
thenYinopeniswIf
IieachFor
Fiii
Fiii
Fiii
Fiii
i
i
i
i
)()(
)(
)(
)(""
sin))((
,)(
,
""
1
11
111
1
11
1
Definition : The weak topology
is the weakest topology on E such that
),( EE
REf :
is continuous for each Ef
.),(log
,,
,),(),(
,),(),(
,,:
,,:
,,
..
sec
2121
12
11
2
1
HausderffisEonEEytopotheHence
OOandOyOx
EinopenEEisO
EinopenEEisOthen
zfEzO
andzfEzOLet
yfandxf
tsRandEf
ThmBanachHahnofformgeometricondBy
EyxLet
f
f
Proposition III.4
Ex 0
0x
FixxfExV i ,,: 0
Let ; we obtain a base of
neighborhood of by consider
sets of the form
where
Efi ,0 , and F is finite
Proposition III.5
nx
Efxfxfweaklyxx nn ,,
xxn
Let be a sequence in E. Then
(i)
(ii) if strongly, then
xxn weakly.
nn
nn
n
n
n
n
nnn
n
xxand
xIICorollaryBy
Efxfxf
ibythenweaklyxxIfiii
weaklyxxIIIopositionBy
xfxf
xxfxxfxfxfSince
stronglyxxAssumeii
IIIopositionByi
inflim
sup,2.
,,
)(,)(
,1.Pr
,,
,,,
.)(
1.Pr)(
xfxfei
xfxfHence
ffffSince
boundisxiiiBy
xfxfiBy
xfxfxff
xfxfxff
xfxfxff
xfxfxfxf
xfxfiv
nn
nn
nn
n
n
nnn
nnn
nnn
nnnn
nn
,,..
0,,
0,
)(
0,,)(
,,
,,,
,,,
,,,,
,,)(
Exercise
),( EE ),( FF
Let E , F be real normed vector space
consider on E and F the topologies
and
Then the product topology on E X F is
),( FEFE
respectively.
niEfexfxExFor
niets
EofeeebasisaChoose
UrxBSuppose
UVtsofVnhdopenEEaisthere
xofUnhdopenanfor
andExforthatshowtosufficientisIt
openEEissetopenstrongathatshowTo
ii
n
ii
i
n
,,1,)(,
,,11.
,,,
.)(
...),(
,.
.),(
1
21
,0
0
UrxBV
thenrntsChoose
nxBVthen
nixxfExV
letweifHence
nixxfifn
xxf
exxf
exxfxx
i
i
n
ii
n
iii
n
iii
),(
,..
),(
,,1,;
,,1,
,
,
,
0
0
0
0
10
10
100
]dim
)(dim
11)(,),(
:
,[
0,;dim
,,10,
0..
,,0
,,1,;
)1,0(:
)1,0(:
.
1;
1
0
1
0
00
1
0
0
),(
nE
nRthen
isxfxfx
bydefinedRE
mapthethenysuchnoisthereIf
xfExcobecause
niyf
andytsEyisThere
Effandwhere
nixxfExVlet
andBxLetpf
BSClaim
closedstronglyisS
xExSLet
n
n
n
ii
i
n
i
EE
),(0
000
000
0000
000
000
0
0
00
,,10,
.
1
.0,)(lim
1)0(,.
)(
EE
i
t
Sx
SVytx
Vytxthen
nixytxfBut
Sytxei
ytx
tstistherethentgand
xgxoffuncontinuousisg
tyxtgfunctiontheConsider
closedweaklyisC
openweaklyisC
CVthen
CVandxofnhdaisV
thenxfExVLet
Cxxfxfts
RandEfisthereCxLet
closedweaklyisC
thenclosedstronglyisCIfshowTo
c
c
c
0
0
0
.
,,;
,,..
0,
.
,:
Remark
CxfExH f ,,
The proof actually show that every
every strongly closed convex set
is an intersection of closed
half spaces
Corollary III.8],(: E
),( EE
If
is convex l.s.c. w.r.t. strongly topology
then
In particular, if
is l.s.c. w.r.t.
)),(( EEweaklyxxn
then )(inflim)( nn
xx
nn
n
xx
thenEEweaklyxxifHence
cslEEisthen
convexandcslstronglyis
convexandcontinuousstronglyis
thenxxLet
nObservatio
inflim
)),,((
..),(
..
,)(
:
Theorem III.9
FET :
),( EE
Let E and F be Banach spaces and let
be linear continuous (strongly) , then
T is linear continuous on E with
to F with ),( FF
And conversely.
continuousEEisTxfx
csuEEisTxfx
cslEEisTxfx
cslEEisTxfx
continuousstronglyisTxfx
EEEfromcontinuousisTxfx
thatshowtosufficientisit
IIIopositionBy
),(,
..),(,
..),(,
..),(,
,
)),(,(,
2.Pr
),(),(
log..)(
)(
),(...)(..
),(
),(),(log
..)(
)),(,()),(,(
sup,
FEFtoEfromcontinuouisT
TheoremGraphClosed
ytopostrongtrwFEinclosedisTGthen
FEinconvexisTG
FEFEtrwclosedisTGei
FEFEiswhich
FFEEytopoproduct
trwFEinclosedisTGthen
FFFtoEEEfrom
continuouslinearisTthatposeConversely
Proposition III.11
Einxxx n,,,,0 21
Ef 0
One obtains a base of a nhds for a
by considering sets of the form
nixffEfV i ,,1,,; 0
Lemma III.2
n
iiin tsR
11 ..,,
n ,,, 1 Let X be a v.s. and
are linear functionals´on X such that
0)(,,10)( vnivi
)()(
0
0)()(
)()(
..,,,
,)()0,,0,1(
)()(
)(,),(),()(
:
1
1
1
1
1
1
1
uu
Xuuu
Xuuu
tsRandzeroallnotR
XFSince
RinsetconvexclosedisFRXFThen
XuuuuuF
bydefinedbeRXFLet
i
n
i
i
n
iii
n
iii
n
n
n
n
Efxffthen
xxTake
Efxf
xffIIILemma
fthen
nixfifparticularIn
nixfEfV
followsasVtakemay
fts
EEforofVnhdaisThere
i
n
ii
i
n
ii
i
n
ii
i
i
,)(
,
,)(2.
0)(
,,10,,
,,1,;
1)(..
),(0.
1
1
1
Corollary III.14
RExxfEfH ,0,,;
If H is a hyperplane in E´ closed w.r.t
Then H is of the form
),( EE
Vffor
VffEither
convexisV
Exxx
nixffEfVwhere
HVistherethenHfSuppose
REwherefEfH
formtheofisH
n
i
c
)()13(
)()13(
0,,,,
,,1,;
,
,0,)(;
21
0
0
*
xfEfHthen
EfxfftsEx
IIITheoremBy
continuousEEisthen
ofnhdEEaisW
fVWgfgFrom
fVWgfgFrom
,;
,)(..
13.
),(
0.),(
)()()13(
)()()13(
00
00
Lemma 1 (Helly) p.1
Efff n ,,, 21
Rn ,,, 21
Let E be a Banach space,
are fixed.
and
Then following statements are
equivalent
n
iii
n
iii
n
iii
n
iii
n
iii
n
i
n
iiiii
n
iin
f
havewelettingBy
sf
sxf
sxfi
sandRLetiii
11
1
11
1 1
121
,0
,
,)(
,,,)()(
)(
.')(
)()(
,,,,,,)(
:
,,,)()(
21
21
E
E
n
n
nn
Bthen
holdtdoesnithatSuppose
BiThen
xfxfxfx
byREmapthedefine
andRLetiii
)(
,
)(
..,,,
,)(
11
11
21
iiscontradictwhichf
Bxxf
Bxx
tsRaisthere
principleseparationstrictlyby
RinsetconvexclosedaisBSince
n
iii
n
iii
E
n
iii
n
iii
E
nn
nE
Lemma 2 (Goldstine)
EBJ
EB
Let E be a Banach space. Then
is dense in
w.r.t the weak* topology
EonEE ),(
VJx
nifJx
niffJx
nixf
tsBxLemmaHelly
ff
haveweR
thatnoteandnifLet
nifEV
formtheinofnhdabeVandBLet
i
ii
ii
E
i
n
iii
n
ii
n
iii
n
ii
i
E
,,1,
,,1,,
,,1,
..,0
,
,,,
,,1,
,,1,;
111
21
),(..)(
,),(..
,(,,(,
,2.Pr
,(,
,)(,))((
,(,,(,::
sin,
.""
1
1
1
11
1
EEtrwcompactisBJB
ClaimbyandEEtrwcompactisBSince
EEEtoEEEfrom
continuousisJ
IIIopositionBy
RtoEEEfrom
continuousisJfthen
EfJfJf
EfanyFor
EEEEEEJClaim
isometryisJceBJBThen
reflexiveisEthatAssume
EE
E
EE
.
,
),(..
),(ker),(
)),,(,()),(,(
)),(,()),(,(
,9.
),(..""
reflexiveisEHence
EJEthen
BJBLemmaGoldstineBy
EEtrwcompactisJB
EEthanweaisEEbecause
EEEtoEEE
fromcontinuousisJ
EEEtoEEE
fromcontinuousisJIIIThmBy
normwithcontinuousisJ
EEtrwcompactisBthatSuppose
EE
E
E
Exercise
Suppose that E is a reflexive
Banach space . Show that
evere closed vector subspace M
of E is reflexive.
].),(
,,1,,ˆ;
ˆ,
)(
)()(ˆ[
.),(:
,,1
,,1,,;
.),(
),(...
),(...
),(...,
0
0
0
0
0
00
xofnhbEEais
nixxfExMV
EfclosedstronglyisMSince
Mxifxf
MxifxfxfLet
xofnhbEEaisMVClaim
niMfwhere
nixxfMxVformthein
xofVnhbMMaandMxanyFor
EEtrwcompactisMBB
EEtrwclosedisMthen
closedstronglyisM
EEtrwcompactisBreflexiveisESince
c
i
ii
c
i
i
EM
E
reflexiveisMTherefore
MMtrwcompactisBHence
BV
BMVtsI
EEtrwcompactisBSince
setsopenEEoffamilyaisMV
ClaimbyandBMVthen
BVthatsuch
setsopenMMoffamilyaisVIf
M
M
n
i
M
n
i
cn
M
Ic
MI
c
MI
I
i
i
),(..
..,,
),,(...
),(
),(
1
11
.
,
,
,
.),1(
.)2(
.
),(..
),(..
),(),(
.)1(
reflexiveisE
JEandEbetweenisometryisJSince
reflexiveisJEExerciseby
EofsubspaceclosedisJESince
reflexiveisEbythen
reflexiveisEthatSuppose
reflexiveisEHence
EEtrwcompactisB
EEtrwcompactisB
ThmBornbakiAlaogluBy
EEEEThen
reflexiveisEthatSuppose
E
E
Corollary 2
),( EE
Let E be a reflexive Banach space.
Suppose that if K is closed convex
and bounded subset of E . Then
K is compact w.r.t
).,(..
),(..,
0,
),(..7.
,
EEtrwcompactisK
EEtrwcompactismBreflexiveisESince
msomeformBKboundedisKSince
EEtrwclosedisKIIIThmby
convexandclosedstronglyisKSince
E
E
Uniformly Convex
yxwithByx E,
A Banach space is called
uniform convex if for all ε>0 ,
there is δ>0 such that if
12
yxthen
Counter Examplefor Uniformly Convex
),( 2 RConsider
is not uniform convex.
2121 ),( xxxx
see next page
12
,4
11
4111
41
2
41
2
4
2
)log(2
)1,0(,
0
2
22
22
2
2222
2222
yx
haveweTake
yx
yx
yxyxyx
ThmramParalleyxyxyx
yxandByxanyFor
anyFor
VJxtsBx
EEtrwBindenseisJBSince
EEtrwofnhbaisV
fEVLet
ftsfwithEf
convexityuniform
ofdefinitiontheinasbeletGiven
JxtsBx
forthatshowtosufficientisit
stronglyEinclosedisJBSince
JBthatshowTo
ELet
E
EE
E
E
E
..
,,..
,...
,2
,;
21,..1
0,0
..
,0
,
1,
.
ˆsin,ˆ)ˆ(ˆ
ˆ
12
)2
1(2
,2
ˆ
ˆˆ,,2
2,ˆ,
2,,
)ˆ,(sin
ˆ..ˆ
,..
,..
,..
,,..
.[
:
ioncontradicta
WxJcexJJxxxJxxBut
xx
fxx
xxxxff
fxfandfxf
VxJJxcethenhaveWe
WVxJtsBx
EEtrwopenisW
EEtrwclosedisBJx
EEtrwclosedisB
EEtrwcompactisBSince
BJxWThennotSuppose
BJxClaim
E
E
E
E
cE
E