applications of hahn banach theorem
DESCRIPTION
Applications of Hahn Banach Theorem. E: normed vector space, assumed to be real for definitions. Known:. Taking. We have. Corollary 1. Proof:. Corollary 2. Proof in next page. This corollary implies that. We may consider E as embedded in as normed space, then is a - PowerPoint PPT PresentationTRANSCRIPT
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
),(),(),(