Chaper 3 Weak Topologies. Reflexive Space.Separabe Space. Uniform Convex Spaces.

Chaper 3

Weak Topologies. Reflexive Space .Separabe Space. Uniform Con

vex Spaces.

III.1

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

Proposition III.1

nx

Iixxxx inin )()(

Let be a sequence in X, then

F F( ) iY

)()(

)(

)(

..

)()(

)(

""

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

III.2

Definition and properties ofthe weak topology σ(E,E´)

Definition σ(E,E´)

E: Banach space

E´: topological dual of E

EfExxfxf ,,)(

see next page

Definition : The weak topology

is the weakest topology on E such that

),( EE

REf :

is continuous for each Ef

Proposition III.3

The topology ),( EE on E

is Hausdorff

.),(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.

(iii) if xxn weakly, then

nx is bounded and

nnxx inflim

(iv) if xxn weakly and

ffn strongly in E´, then

xfxf nn ,,

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.

Proposition III.6

Edim

),( EE

If ,then

is strong topology on E.

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

Remark

Edim ),( EE If ,then

is strictly weaker then the

strong topology.

]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

III.3

Weak topology, convex setand linear operators

Theorem III.7

EC Let be convex, then

C is weakly closed

if and only if

C is strongly closed.

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

cslEEisthen

closedEEisAthen

convexandclosedstronglyisAthen

RxExALet

..),(

),(

.

,)(;

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

Remark),(, EEE

EEj :

Efxffxj ,),(

On is weak topology

by

xxj )(

EEj )(

In genernal j is not surjective

E is called reflexive

If

III.4

The weak* topologyσ(E′,E)

The weak* topology),( EE

xff ,

is the weakest topology on E´

such that

is continuous for all Ex

Proposition III.10

The weak* topology on E´

is Hausdorff

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

Proposition III.12

Exxfxfn ,,

nfLet

(i)

be a sequence in E´, then

),(* EEforffn

ffn (ii) If strongly, then

),( EEforffn

),( EEforffn

(iii) If

then

),( EEforffn

ffn *

nf

(iv) If then

is bounded and

nn

ff inflim

ffn * xxn (v) If and

strongly, then

xfxf nn ,,

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

Proposition III.13

EfxfftsEx ,)(..

RE :If

then there is

is linear continuous´w.r.t ),( EE

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

III.5 Reflexive spaces

Remark),(, EEE

EEj :

Efxffxj ,),(

On is weak topology

by

xxj )(

E

j is isometry

j(E) is closed vector subspace of

EEj )(

In genernal j is not surjective

E is called reflexive

If

Lemma 1 (Helly) p.1

Efff n ,,, 21

Rn ,,, 21

Let E be a Banach space,

are fixed.

and

Then following statements are

equivalent

Lemma 1 (Helly) p.2

1; xExBE

..,0 tsBx E

nixf ii ,,2,1,

(i)

(ii)

where

Rf n

n

iii

n

iii

,,, 21

11

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

Theorem(Banach Alaoglu-Bornbaki)

1; fEfBE

is compact w.r.t. ),( EE

Theorem

EB

A Banach space E is reflexive

if and only if

is compact w.r.t weak topology

),( EE

),(..)(

,),(..

,(,,(,

,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

Corollary 1

E

Let E be a Banach space. Then

E is reflexive if and only if

is reflexive

.

,

,

,

.),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

x y

2

yx

Counter Examplefor Uniformly Convex

),( 2 RConsider

is not uniform convex.

2121 ),( xxxx

see next page

x

y

2

yx (0,1)

(1,0)

(0,-1)

(-1,0)

Examplefor Uniformly Convex

),( nRConsider

is uniform convex.

222

211 ),,( nn xxxxx

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

Theorem

A uniformly convex Banach space E

is reflexive.

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