Applications of Hahn Banach Theorem

E: normed vector space, assumed

to be real for definitions

Known:2

,

..,

xxyandxy

tsEyExeachFor

Takingx

yx

We have

xxxandx

tsExExeachFor

,1

..,

Corollary 1

yxxx

tsExthenEyxIf

,,

..,

yxxx

yxxx

yxx

yxyxx

tsEx

,,

0,,

0,

0,

,.

Proof:

EofsposeparatesE int

Corollary 2

.EinllyisometricaembededbecanE

E

Proof in next page

This corollary implies that

We may consider E as embedded in

as normed space, then is a

complete space which is the

completion of E.

E

xxjHence

xjxxjxxjxxx

xxxandxtsEx

handothertheOn

xxxxxxxjxj

ExForPf

ExxxjeiisometryanisjClaim

onetooneisjHence

xjxj

xxjxxj

xxxxtsEx

CorollarybyxxIf

ExjandlinearisEEj

Exforxxxxj

byEinxjdefineExFor

xxx

)(

)()(),(,

,1..

,

sup,sup),(sup)(

:

)(,..;:

.

)()(

),(),(

,,..

1,

.)(:

,),(

)(,

111

21

21

21

21

A dual variational principleletandEofssvabeFLet ,..

FxxxExF 0,

yxFxdist

haveweExanyforThen

Fy

inf),(

,

xxxxxFxxFx

,max,max1,1,

),()(

,

,

,supinf

,

,

,

,1

1,

Fxdistyxffunctionallineara

MFyRyxFxondefine

handothertheOn

xxyxHence

xxyx

yxxyx

yxxyxyxx

Fyfor

havewexwithFxanyFor

xFxFy

.,max),(

),(,

1..

.,

1),(

sup

),(sup

1,

,

,

xxFxdisHence

FxdisxfandFfBut

fandEfei

fbydenotedstillnormsamethewith

Eondefinedbetoextendedbecanf

thatimpliesTheoremBanachHahn

yx

Fxdis

yx

Fxdisf

xFx

E

FyR

FyRM

Example 1,0CELet

Eonfunctionscontinuousallofspacethe

)(max,

1,0txxExFor

t

byEfDefine

dttxdttxxf 1

02

1 )()(, 2

1

xdttxxf 1

0)(,

1 Ef

1, EfActually

Why?See next page

.,0 functionfollowingthebexLetFor

2

1 2

11

1

-1

1

1)(sup

1)(lim1)(

1

1

0

E

xE

fHence

xff

xfxf

andx

Claim

1,

..1

xff

tsxwithExnoisThere

E

In this example

Exercise

yxFxdis

tsFynoisthere

FxExanyforthen

fxfExF

letweifthatShow

xffts

xwithExthatpropertythe

havenotdoesEfthatSuppose

),(

.

,,

ker0,

,.

1

FyFxEx ,,

yxFxdis ),(

Suppose that such that

, then

yxf

fFyceyxfxf

handothertheOn

yxfxf

xff

f

fFfcex

f

f

xx

Fxdis

yx

xFx

kersin,,,

,

)1(,

,1

1,sin,,

,max

),(

1,

impossibleiswhich

f

byyxfyx

fFycexfyx

yxfyx

yx

yxfzf

andzEz

thenyx

yxzLet

yxfxfHence

,

(*),1

kersin,,1

,1

,,

,1,

,

(*),

Applications of Mazm- Orlich Theorem

)(SPf

is the space of the probability

measure of S

k

kssp

,,

,,

1

1

kisp ii ,,1,)(

11 ),,( k

kwhere

svES .:

)()(1

k

iiissdpp

)(,,

,,

1

1 SPss

p fk

k

Mazm-Orlich Theorem2,1: isublinearREp ii

2.1: iES ii

))(())((.

2,1)(

2211

*

slslts

iplwithElI iiii

))(())(((

)()(

2211 pppp

SPpForII f

Mazur-Orlicz (1953)

)()()()( SPppqpII f

ES :REq :

S : arbitary set E : real vector space

RS :sublinear

(I)

thatsuchqwithE *

Ssss )()(

ExxxHence

Exxx

Exxx

Exxx

Exxx

pf

xxxxqxandEClaim

and

qqttqEERETake

)(

)(

)(

)(

)(

:

)()()(:

,,,)(,,

1

1

1

1

1

111*

1

2

12121

Corollary 1

))((inf))((infsup pqsfPpSsq

Let

,,qE be as in Mazur-Orlicz Theorem

then

))((infsup))((inf

))(())((inf..

)(

))((inf

)())(()(

))((

..)((*)

)(

)(

)(

*

)(

*

spqthen

SsspqtsqwithE

impliesandholdsIIthen

pqTake

SPppqII

Sss

tsqwithEI

TheoremOrlichMazurBy

SssdefineandRLet

SsqSPp

SPp

SPp

f

f

f

f

))((inf))((infsup

))((inf))((infsup

,0

))((inf))((infsup

)())(())((infsup(*)

))(())((infsup

..0

)(

)(

)(

*

pqsHence

pqs

havewetakingBy

pqs

SPppqsBy

Ssss

tsqwithEGiven

SPpSsq

SPpSsq

SPpSsq

fSsq

Ssq

f

f

f

Corollary 2

))((inf))((infsup sqsSsSsq

0),( SPp f

If in Corollary 1

Ss

sastisfies the condition:

For each

qwithE * ))(())(( ps

there is such that

then

))((inf))((inf..

))((inf,0

2))((inf

2))(())((ˆ))((ˆ))((

))(())((ˆ..ˆˆ

,

))(())((

..,

))((..)(,0

))((inf

))((inf))((inf

))((inf))((inf

,))((inf))((inf))((infsup

)(

*

*

)(

)()(

)(

)(

pqsqei

sqhavewetakingBy

sqthen

pqpssqthen

sqstsqwithE

thatimpliesThmBanachHahnsthisFor

qwithEps

tsSspandthisFor

pqtsSPpGiven

pqifarbitary

finiteispqifpqLet

pqsqthatshowtosufficentisit

sqpqsSince

SPpSs

Ss

Ss

f

SPp

SPpSPp

SPpSs

SsSPpSsq

f

f

ff

f

f

Example p.1

Sssfsfs n ))(,),(()( 1

RSff n :,,1 S: arbitary set

),,(,max)(, 11

nini

n xxxxxqRE

defined by

)(maxinf)(infsup1)(11

pfsf iniSPp

n

iii

Ss fn

ES :Then

)(maxinf)(infsup

))((inf))((infsup

1

)(max))((

,)())((

),,()(

..,

1)(1

)(

1

1

11

1

1

pfsf

pqs

CorollaryBy

pfpq

andSssfsthen

Rxxxxx

tsEanyforRESince

iniSPp

n

iii

Ss

SPpSsq

ini

n

iii

nn

n

iii

nn

fn

f

Example p.2

1

11)()(

n

n

iii

n

iii pfsf

..),( tsSsSPp f

is q-convex

nipfsf ii ,,1)()(

qEps ,))(())((

Example p.3

niEsssfsf nr

r

jjij

r

jjji ,,1,,,,)( 1

111

n

iii

Ss

n

iii

Sssfsf

nn 11

)(maxinf)(infsup1

1

Then

RSfi :In particular, S is a convex set in a linear map

is convex i.e.

This implies von Neumann Minimax Theorem

n

iii

Ssi

niSs

n

iii

Ss

SsSsq

ii

r

jjj

r

jjji

r

jjiji

fr

r

sfsfsf

sqs

haveweCorollarybythen

convexqisthen

sfpfthenssLet

sfsfpf

niSPss

panyFor

nn 111

1

11

1

1

)(maxinf)(maxinf)(infsup

))((inf))((infsup

,2

)()(,

)()(

,,1),(,,

,,

11

n

i

m

jjiij

n

i

m

jjiij

n

i

m

jjiji

n

i

m

jjiji

n

iii

n

iii

i

mm

jjiji

i

nm

mnij

aa

aa

ff

haveweresultpreviousBy

convexisfThen

af

byRSfDefine

niforandESLet

aGiven

TheoremMiniNoumannvonthatshowTo

nmmn

nmmn

nmmn

1 11 1

1 11 1

11

1

1

11

1111

1111

1111

maxminminmax

minminminmax

)(minmin)(minmax

,

)(

:

,,1,,

,

max

Duality map p.1

ExLet E be a real reflexive Banach space. For

xxJ )(

2,,:)( xxyxyEyxJ

J is a Duality map. If E is a Hilbert space, then

J(x) is w-compact (see next page)

)(

),(

),(),(),(

),()(,

.)(

)(,

,)(

2211

2211

221122112

222

2122112211

2122112211

2121

xJyyHence

yyx

xyyxyyx

xxxxyxyxyy

xxxyyyy

andxJyyFor

convexisxJthatshowTo

compactwisxJclosedand

convexboundedisxJandreflexiveisESince

Lemma p.1

qSC ,)(

FfyfftsSy )()(.. 00

Let S be a compact convex subset of a

)(sup)()(: sffqbyRSCqSs

topological linear space. Define

If

and F is the space of all affine functions then

linear function +constant

FffsubsetfiniteeveryforA

thatshowtosufficientisIt

AthatshowtoNow

AHence

fsfyftsSy

sffsf

sfsfff

sffqf

AClaim

yffSyAletFfanyFor

if

Fff

f

Ss

SsSs

SsSs

Ss

f

f

i

,,

)()(min)(..

)(max)()(min

)(min)(max)()(

)(max)()(

:

)()(;,

11

10

01

0

1 1

1

1)(

11)(

1

0)()(max..

0)()(minmax

)()(minmax

)()(maxmin

)(minmax)(maxmin

1

,,1)(

,,

1

1

1

if

iii

j jijij

Ss

jiij

Ss

iiiSPp

jjj

Ssi

iSPp

iii

i

f

f

AsHence

fsftsSsthen

fsf

fsf

fpf

shph

CorollaryBy

iforffhlet

FofffsubsetfiniteanyGiven

Theorem p.1

1,,),(min..0)(

xExyxatsxJy

REEa :Let E be a real reflexive Banach space

EyExyxcyxa ,),(

is bilinear such that

(i) There is c>0 such that

(ii)

Theorem p.2

Exyxax ),()( 0

EThen for each

0yEy 0

there is a unique

with such that

GgggbySCGLet

SCfsffqbyRSCqDefine

EofsubsetcompactwisSthen

whereKSLetGin

stillisGinelementsofncombinatiolinearfinite

everyandcontinuouswisGinfunctionEvery

ExxxaGLet

reflexiveisEcecompactwisKthen

EinballunitclosedthebeKLet

thatassumeMay

Ss

)()(:

)()(max)()(:

,.

);(),(

sin,

,

0

0

),(max

),(max

)(),(max

)(),(max

)(max))((

)(,,

,,

)(0))((:

002

00

00)(0

000

00

000

11

1

1

0

0

xxxx

xyxax

xyxax

xyxa

xsomeforxyxa

sggqpq

GPgg

panyFor

GPppqClaim

xJy

xy

Ky

KSy

r

iii

Ss

r

iii

fr

r

f

Exxyxa

Ggyg

Ggyggandythen

functionsaffineallofspacetheisFwhere

FfyfftsSyLemmaBy

Ggg

Ggg

Ggg

tsqwithSC

GPppq

linearisG

ThmOliczMazurby

f

)(),(

)(0

)()(0

)()(..

0)(

0)(

0))((

..)(

)(0))((

0

0

00

00

0),(

)(

,1,

,

..1,

0),(

)(),(),(

:

01

01

001

01

001

0101

00101

00

10

10

yy

yyxathen

xJyy

yythen

xyy

yythenyySuppose

xyyyy

tsxwithExreflexiveisESince

Exyyxathen

ExxyxayxaIf

Uniqueness

Variational Inequality(Stampachia-Hartmam) p.1

E: reflexive Banach space

C: closed bounded convex set in E

ECf :

(i) f is monotone i.e.

segment in C.

satisfies

Cyxyxyfxf ,0),()(

(ii) f is weakly continuous on each line

Variational Inequality(Stampachia-Hartmam) p.2

Cy 0Then there is such that

Cxxyyf 0),( 00

Ssytakingby

ysyf

monotoneisfceysyf

yssf

syhsqpq

SPss

panyFor

shhqbyRSCqLet

xhxbySCSCSLet

CyxyxxfxyhLet

ii

i

ii

iSy

iii

Sy

iiii

Sy

iii

Syiii

f

Ss

1

1

1

1

11

1

1

,,0

),(sup

sin,),(sup

),(sup

),(sup)())((

)(,,

,,

)(sup)()(:

),()()(:,

,),(),(

Sxxyxf

Sxyxxf

SxxyhxhTherefore

yxfxfHence

yxfxffromBut

yxfxf

yxfxxfxfxxf

yxfxxfxfxxf

yxxfxxf

xyhxh

SxeachforThen

FhyhhtsyLemmaBy

Sxxh

Sxx

tsqwithSC

SPppq

ThmOliczMazruby

f

0),(

0),(

),()),((0

),()),((

),()),(((*),

),()),((

),(),()),((),(

),(),()),(),((

),()),((

),()),((

(*))()(..

0)),((

0))((

..)(

)(0))((

0

0

0

0

0

0

0

0

0

0

00

0)),(

,0

0)),(

0)(),(

0),(

)1(

,10

0),(

00

0

0

0

0

00

xyyf

havewettakingBy

xyzf

xytzf

zyzf

yttxzlet

CxtFor

CxxyyfthatshowtoNow

t

t

tt

t

Applications of Mazm- Orlich Theorem

Inequality after mixing of functions

Theorem

RSff n :,,1

Let S be an arbitary set.

RSgg m :,,1

The following two statements are equivalent:

Sssgsf

Im

jjj

n

iii

mn

11

11

)()(

,)(

)(max)(min

),()(

11pgpf

haveweSPpanyForII

jmj

ini

f

)(max

)(,)(

)()(min

)(

)()(

)(,),()(

)(,),()(

,,max)(

,,max)(

,

1

1

11

12

11

11

2

11

1

21

pg

Ibypg

pfpf

SPpanyFor

III

Sssgsgs

Sssfsfs

yyyyxp

xxxxxp

RERELet

TheoremOrliczMazmAppling

jmj

m

kjj

n

iiii

ni

f

m

n

mjnj

nini

mn

))(())((..

))((

))(,),((

)(max

)(min

)(max

))(,),(())((

),(

)()(

2211

22

12

1

1

1

1111

ppppei

pp

pgpgp

pg

pf

pf

pfpfppp

SPpanyFor

III

m

jmj

ini

ini

n

f

Sssgsfand

yyyforyy

xxxforxx

tsRandRei

Ssss

tsipwithE

holdsTheoremOrliczMazmofIstatementthe

m

jjj

n

iii

mjmj

m

jjj

nini

n

iii

mn

n

n

iiii

11

111

111

11

2211

*

)()(

,,max

,,max)(

.),,(),,(..

))(())((

.2,1

)(

1

1

1

1

1

1

11

11

,

.1

1

1

),1,,1(

1

),1,,1(

,,10

0

,0

0,

,,max,)(

m

nn

ii

n

ii

n

ii

n

ii

i

i

j

i

n

n

iii

mn

Similarly

eiHence

thenxtakeFinally

thenxtakeThen

ni

k

ijifx

kkxtakeFirst

xxxBy

andthatshowtoremainsIt

00

00,

.1

1,1

1,1[

max"["

."["

,,,,,:

)(,),()(

)(,),()(

,,max)(

,,max)(

,

1

1

1

111

111

*1

12

11

11

2

11

1

21

ii

ji

n

ii

n

iii

n

iii

ini

i

n

ii

nnn

m

n

mjnj

nini

mn

kthen

ijifxandkxtakeFinally

Hence

haveweixtakeThen

haveweixtakeFirst

xxp

clearisIt

pthenEIfClaim

Sssgsgs

Sssfsfs

yyyyxp

xxxxxp

RERELet

TheoremOrliczMazmAppling

)(max)(min

)(

)(max)(max

)(

)))((()))(((

)(

))(())((

..,,,,

)()(..,

11

11

2211

2211

2211

1211

11

11

pgpf

SPpanyfor

pgpf

SPpanyfor

pppp

SPpanyfor

Ssss

pandp

tsand

functionallinear

Sssgsfts

jmj

ini

f

jmj

ini

f

f

mn

n

iii

n

iii

mn

Minimax Theorem of Von Neumann

11,,

mnmnij

andaAFor

),(),(

1111minmaxmaxmin

jijiij

jijiij aa

nmmn

),(),(

1

),(),(

11

),(),(

11

),(),(

1111

111

11

1

minmaxmaxmin

minmaxmin

,minmax

,max

)(

jijiij

jijiij

m

jijiij

jijiij

mn

jijiij

jijiij

mn

jijiij

jijiij

aa

aa

aa

aa

nmmn

nmn

nm

m

)ˆ,ˆ(

ˆˆ

),(

)(,,

),(,),,(

),(),(

),(

,

,Pr

)(

1)(

1

1)(

1

)(

1

)(

1

1

)()()1()1(

1

1

11

pthen

andLet

Spthen

SPpLet

Sforag

af

SLet

TheoremeviousApply

mk

kk

nk

kk

k

kk

k

kk

fl

i

n

iiji

j

m

jiji

mn

TheorempreviousofIstatement

TheorempreviousofIIstatementei

pgpf

pg

a

a

a

pfpf

apgandapf

jmj

ini

jmj

n

iiij

mj

n

iiij

m

jj

j

m

jij

n

ii

i

n

iii

ni

n

iiijj

m

jiji

)(

)(..

)(max)(min

)(max

ˆmax

ˆˆ

ˆˆ

)(ˆ)(min

ˆ)(ˆ)(

11

1

11

11

11

11

11

),(),(

11

),(

0

),(

0

11

),(

0

),(

0

11

1 1

0

1 1

0

1

0

1

0

1100

1111

11

minmaxmaxmin

,minmax

,

,

)()(

..,

jijiij

jijiij

mn

jijiij

jijiij

mn

jijiij

jijiij

mnm

ji

n

iijj

n

ij

m

jiji

m

jjj

n

iii

mn

aa

aa

aa

aa

Sssgsf

ts

nmmn

nm

Lemma VI.1 (Riesz-Lemma)

boundeddomainCCu ,:, 12 B\

Let

For any

),( xv

fixed , apply Green’s second

identity to u and in the domain

and then let 0 we have

dsn

xu

n

uxudxxu

xx

),(),(),(