Mazm-Orlicz Theorem

Linearly open

ifopenlinearlycalledisEC

.spacevectorrealabeELet

CofpoerioranisCofpoeach intintint

EyCxanyforei ,.

smallsufficientistifCtyx

Lemma 1

IntCwithEinsetconvexthebeCLet

.spacevectorrealabeELet

ThenIntCxExlet .,00

tsffunctionlinearnonzeroisthere .

Cxxfxf )()(0

.0 IntCassumeMay

00ˆ, yCCletIntCyIf

000ˆ yxx

Cyyfxf ˆˆ)ˆ()ˆ(0

Cyyyfyxf )()(000

Cyyfxf )()(0

Proof of Lemma 1 p.1

)()(1)(

,0)1(

1)(,

:

)()(:

)(,

min

000

00

00

*

00

xPxPxg

For

xPIntCxSince

pf

RxPxgClaim

GgThen

RxgRxGLet

CoffunctiongaugekowskithebePLet

CC

C

C

C

Proof of Lemma 1 p.2

CxxfxfHence

gextendsfcexf

xgxPxf

CxforThen

ExxPxf

thatsuchgextendingEf

TheoremBanachHahnApply

ExxPcexPxg

For

C

C

CC

)()(

sin,)(

)(1)()(

)()(

0)(sin,)(0)(

,0)2(

0

0

0

*

00

Remark

EofsubsetconvexopenanbeCLet

.spacevectorrealnormedabeELet

ThenCxExlet .,00

tsEf .

Cxxfxf )()(0

Proof of Remark p.1

Ef

CxxrxPxf

CxxrxP

xifrxP

xifCxrtsr

CxxfxfHence

byxf

openisCcexPxf

CxforThen

CxxPxfand

xftsEf

LemmaofprooftheFrom

CassumeMay

C

C

C

C

C

10

10

10

00

0

0

0*

)()(

)(

1)(

1.0

)()(

(*))(

sin,)(1)(

(*))()(

)(1.

1

0

Corollary To Lemma 1

EinsetconvexopenlinearlyabeCLet

.spacevectorrealabeELet

ThenCwith .0

tsEf .*

Cxxf 0)(

Proof of Corollary

CxxfHence

impossibleiswhich

tifytftyxf

smalllysufficientistifCtyx

openlinearlyisCSince

yfwithCyChoose

CxsomeforxfSuppose

Cxxff

tsEf

LemmaofprooftheFrom

0)(

0,0)()(0

,

0)(

0)(

)()0(0

.0

1*

Convex cone .spacevectorrealabeELet

ifconeconvexaisP

Pxxanyfork,,

1

zeroallnotbutPxi

k

iii

,01

Define P

.: sublinearbeREpLet

.spacevectorrealabeELet

thenxpExPlet ,0)(

coneconvexopenlinearlyaisP

?

Proof of ? P.1

.

0

00

0)(

)()(

,,10)(

,

.)1(

1

1

11

1

coneconvexaisPHence

zeroallnotbutPx

zeroallnotbut

xp

xpxp

kixp

PxxanyFor

coneconvexaisPthatshowTo

ii

k

ii

i

ii

k

ii

i

k

iii

k

ii

i

k

Proof of ? P.2

.

,

,0)(

,0

0,)()(

0,)()(

)()()(

,

.)1(

openlinearlyisPHence

smalllysufficientistifPtyx

smalllysufficientistiftyxp

smalllysufficientistif

typt

tytpxp

typxptyxp

EyPxanyFor

openlinearlyisPthatshowTo

Lemma 2( 証明很重要 )

.: sublinearbeREpLet .spacevectorrealabeELet

0)( xpExPLet

thenlandElPIf ,0,, * :equivalentarestatementsfollowingThe

PxxlI 0)()(EonpltsII .0)(

Proof of Lemma 2 P.1

QRClaim

rrrrRLet

tEinspoofnscombinatioconvex

finiteallofsettheisQei

tEofhullconvexthebeQLet

xlxpx

byREtDefine

III

obviousIII

22121

2

2

:

,0,),(

.int

.

))(),((

:

)()(

)()(

Proof of Lemma 2 P.2

QRHence

Rv

xl

Ibyxl

Px

thenxpxp

thenxpIf

xlxp

xlxp

xlxp

Exwheretxv

QvFor

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

i

n

iii

i

n

iii

n

iii

22

1

1

1

11

1

11

11

1

11

0)(

)(0)(

,0)()(

,0)(

))(),((

))(),((

))(),((

,1,0

Proof of Lemma 2 P.3

Exandrrxlxprr

tswithRei

QvRuvfufei

QvRuvufts

Ronffunctionallinearnonzeroaisthere

LemmatoCorollaryBy

convexandopenlinearlyisC

vRQRCConsiderQv

,0,0)()((*)

.0),(.

,)()(.

,0)(.

,1

.

)(

21212211

22

21

221

2

2

2

22

Proof of Lemma 2 P.4

0

,0

0)(

0)(

0,0

0,0:

)()(

),0,0(),(

0,0

,0,0

1

2

21

21

12

21

21

212211

Hence

impossibleiswhichl

Exxl

Exxl

thenIf

Claim

Exxpxl

havewerrLet

Exrrallforabovefrombundedisrr

Proof of Lemma 2 P.5

Exxpxlthen

Hence

impossibleiswhichP

Exxp

Exxp

thenIf

)()(

0

,

,0)(

,0)(

0,0

2

1

1

2

1

12

Main Lemma p. 1

.: sublinearbeREpLet .spacevectorrealabeELet

.setnonemptyarbitaryanbeSLet

.: mapabeRSLet

:var entequiarestatementsfollowingThe

Main Lemma p. 2

Sssl 0))((

tsplwithElI .)( *

k

iiispII

1

0))(()(

Sofsssubsetfiniteallforn,,

1

0,,01

n

alland

Proof of main Lemma P.1

QPII

SbygeneratedconeconvexthebeQLet

PSuppose

ltake

ExxpthenPIf

III

Iby

sl

IbyslsP

III

n

iii

n

iii

n

iii

)(

.

0

0)(,

)()(

)(0

))((

)())(())((

)()(

1

11

Proof of main Lemma P.2

(**)0)(

(*)0)(0

0)()(),()(

.

0)(

.0,

,1

.0

,

*

Qxxl

Pyylr

rrylxlylrxl

PyandQxallforei

Czzl

tslEl

LemmatoCorollaryBy

CwithconeconvexopenlinearlyisC

PyQxyxPQCConsider

Proof of main Lemma P.3

SsslandEonpl

havewelbylname

Sssl

LemmabyEonplts

0))((

,Re

0))((

2,.0

Theorem p. 1

onfunctionalsublinearabepLeti

2,1iFor

andEspacevectorrealai

.ii

EtoSfrommapabe

:var entequiarestatementsfollowingThe

.setarbitaryanbeSLet

Theorem p. 2

Ssslsl ))(())((2211

tsplplwithElElI .,,)(2211

*

22

*

11

n

iii

n

iii

spspII1

221

11))(())(()(

Sofsssubsetfiniteallforn,,

1

0,,01

n

alland

Proof of Theorem P.1

)()(),(

,

,

)1(

)(),(

:

)()(,

:

21

*

22

*

11

21

21

21

21

vlulvul

throughElEl

llpairabydrepresentebe

canEonlfunctionallinearEvery

thatobserveandLemmamainapplyThen

sss

byESLet

vpupvu

byREpLet

EEELet

Proof of Theorem P.2

))(())((

0))(())((

0)(,)((

0))(()3(

))(())((

0))(())((

0)(),((0))(()3(

,,)2(

122

111

122

111

12

11

1

2211

2211

21

221121

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

spsp

spsp

ssp

sp

slsl

slsl

sslsl

plplpll

Corollary (Mazm-Orlicz) p. 1

.: sublinearbeREpLet .spacevectorrealabeELet

.setnonemptyarbitaryanbeSLet

.: mapabeRSLet

:var entequiarestatementsfollowingThe

.: mapabeRSLet

Theorem p. 2

Sssls ))(()(

tsplwithElI .)( *

n

iii

n

iii

spsII11

))(()()(

Sofsssubsetfiniteallforn,,

1

0,,01

n

alland

Proof of Corollary

TheoremapplyThen

andLet

ppandttpLet

EEandRELet

21

21

21

)(

S

1E

2E

R

1 1

p

2

2p

S

RE 1

1

idp 1

Rpp

2

2 EE

2

1l

idl 1

2l

ll 2

Mazm-Orlicz Thm implies Hahn-Banach Thm p.1

0,,0,0

,,,

)(..

int

,,

,

.

..

.:

.

21

21

*

k

k

and

GofssssubsetfiniteanyFor

Gsssei

EoGofmaptionidentificais

gGStake

ThmOrliczMazmIn

GonpgwithGgLet

ssvbeEGLet

sublinearbeREpLet

spacevectorrealabeELet

Mazm-Orlicz Thm implies Hahn-Banach Thm p.2

Gsslsg

Gsslsg

Gsslsg

Gsslslsg

tsEonplwithElei

I

holdThmOrliczMazminII

spsgsgk

iii

k

iii

k

iii

)()(

)()(

)()(

)())(()(

...

)(

)(

)()()(

*

111