Muroran-IT Academic Resources Archive
Title
Author(s)
Citation
Issue Date
URL
Totally Ordered Linear Space Structures and Hahn-Banach
Type Extension Theorem
Iwata, Kazuo
室蘭工業大学研究報告.理工編 Vol.8 No.2, pp.429-434, 1974
1974-10-15
http://hdl.handle.net/10258/3600
Rights
Type
Journal Article
See also Muroran-IT Academic Resources Archive Copyright Policy
Muroran Institute of Technology
TotaUy Ordered L
i
near Space Structures and
Hahn-Banach Type Extension Theorem
KazuoIwata
A
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.[
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r
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s
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s
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i
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i
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r
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e
sf
o
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，we now reach thefollowing.
. LetM b
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i
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e
a
rs
u
b
学a
c
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i
n
e
a
rformon
THEOREM 1
M.LetK b
eap
o
i
n
t
e
di;onvexc
o
n
ei
nE，andqagaugef
u
n
c
t
i
o
nonK. A
i
n
e
a
rform F OJl E
n
e
c
e
s
s
a
r
yands
u
f
f
i
c
i
e
n
tc
o
n
d
i
t
i
o
nt
h
a
tt
h
e
r
ee
x
i
s
t
sα l
e
x
t
e
n
d
i
n
gf ands
a
t
i
s
f
y
i
n
gF
(
γ)
<
_
q
(
γ
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l
l yEK i
st
h
a
tt
h
e
r
ee
x
i
s
t
sa
，必)ofL witht
h
efollowingpr
ψe
r
t
i
白:
t
.
o
.
l
.
s
. (L
(
i
) BfUCqc(L，
必)+;
(
i
i
) (
L，
~)+ i
sa
b
s
o
r
b
i
n
ga
t(
0，
1
)forL;
whereL i
st
h
eproductl
i
n
e
a
r学 αceExR，仰 dBf=
Cq {
(
y，
可
)
:q
(γ
)く守，yεK
}i
J
l L
.
{(x，~):
f(
劫<己 xEM}，
口
PROOF. Under t
h
eh
y
p
o
t
h
e
s
i
s，i
n L，Cq proves t
o bea convexcone
withoutv
e
r
t
e
xz
e
r
o
. And t
ot
h
i
s end，L canbeendowedwithap
a
r
t
i
a
l
l
y
，
0
3
'
'
)withp
o
s
i
t
i
v
econeCq • (
N
e
c
e
s
s
i
t
y
)By
orderedl
i
n
e
a
rs
p
a
c
es
t
r
u
c
t
u
r
e(L
h
y
p
o
t
h
e
s
i
s，de
五nmgφ (
x
，
~)= F(x)+
己 φisap
o
s
i
t
i
v
el
i
n
e
a
rformon(
L
，
0
3
"
)
with φ(0
，1)=1
. Hence (
t
o be p
r
e
c
i
s
e
)，t
a
k
eat
.
o.
1
ふ (L，見 1
) such t
h
a
t
BfUC
c
(
L
，
3
e
)
+
b
y
[
1
8
)
T
h
.
4
(
1
)
a
n
d
L
e
m
m
a
1
]
，
t
h
e
n
b
y
[
1
8
)
，
L
e
m
m
a
s
2
，
q
1
3(
1
) and 4
]，(L， φ(~1)) must become a t
.
o
..
1s
.a
sr
e
q
u
i
r
e
d
. (
S
u
白c
i
e
n
c
y
)
五ning~(x，~) =-f(x)+~， ~ i
sa n
o
n
i
d
e
n
t
i
c
a
l
l
y
z
e
r
ol
i
n
e
a
rformonM x
De
R'
3(
0，
1
)
. Thereforebyh
y
p
o
t
h
e
s
i
s，nowwitht
h
ea
i
do
f[
1
8
)
，Th.4(
2
)
](
c
f
.
[
1
9
)，p
. 46，f
o
o
t
n
o
t
e
]
)
，weget ap
o
s
i
t
i
v
el
i
n
e
a
rformφon(L
，
0
3
"
)extending
~. Hence t
h
e
r
ee
x
i
s
t
sal
i
n
e
a
r form F onE extendingfand s
a
t
i
s
f
y
i
n
g
x
，
~) = - F
(
x
)+~. Andhenceq
(ν
)<ザ impliesF(y)<_ηfora
l
lε
ν K，which
φ(
e
n
s
u
r
e
st
h
ea
s
s
e
r
t
i
o
n
.
Asf
o
rsome s
i
m
p
l
eexamples
ヲ
. Let E beR 2• Take a pointed convex cone K ={
(
α，
s
)
:
EXAMPLE 1
ホ
料
S
u
c
hb
e
i
n
gt
h
ec
a
s
e
，s
p
e
c
i
f
i
c
a
l
l
y
，o
u
rグ b
e
l
o
ww
i
l
lb
eo
fasymmetry.
Theya
r
ea
sq
u
o
t
e
db
e
f
o
r
e
;s
e
ef
o
o
t
n
o
t
e'
t
'
.
(
2
1
4
)
H
a
h
n
B
a
n
a
c
hT
y
p
eE
x
t
e
n
s
i
o
nTheorem
4
3
1
α>0
，orα=0 and s
;
?
O
}i
nE
. De
五neqonK t
o mean q
(
α，
s)=αifα>0
and q
(
O
，s)=si
fs
;?O，andqi
s agaugef
u
n
c
t
i
o
nonK
.With t
h
i
s
(
1
) l
e
tM be t
h
e砕 a
x
i
sandd
e
f
i
n
ef onM byf(a
，
0)=α;
(
2
) l
e
tM be {
(
O
，
O
)
} andd
e
f
i
n
ef(O
，
0)=0;
(
3
) l
e
tM be t
h
es
a
x
i
s andd
e
f
i
n
ef onM byf
(O
，
s)=s.
Then i
nc
a
s
eo
f(
1
)(
r
e
s
p
.(
2
)
)，notwithstanding BPCq i
s not absorbing
a
tb= (
(
0，
0
)，
1
)(
r
e
s
p
.a
tanyp
o
i
n
to
fM xR)f
o
rL，t
h
esu
自c
i
e
n
tc
o
n
d
i
t
i
o
n
smetenough. Whilei
nc
a
s
eo
f(
3
)，althoughfi
smajorized
o
fTheorem1i
a
i
l
st
o haved
e
s
i
r
e
de
x
t
e
n
s
i
o
n
. That i
s why
，choosing
byqonM nK，ff
t
h
ef
o
l
l
o
w
i
n
gf
o
u
rv
e
c
t
o
r
sb
，
c
=
(
(
1
，
0
)
，
2
)
，の=
(
(
1
，
ρ+
1
)
，
2
)
i
n
C
nda=
1
q a
(
一(
0，
ρ+1
)，-p) i
nB，
j w
herep being a
r
b
i
t
r
a
r
y
，there holds the equality
ρ
(b-c
)+α十 c
0. Namelyuponappealingt
o Theorem1
，noneof(L，必)+
1
2=
with B
fUC
L
ラ見)十 c
anbeabsorbinga
tbf
o
rL
.
qC (
REMARK 1
. InTheorem1，l
e
ti
np
a
r
t
i
c
u
l
a
rK =E (
w
i
t
hgaugep
)and
f
(
x
)
<
_
p
(
x
)f
o
ra
l
lxEM. Theni
tf
o
l
l
o
w
s(
r
e
s
p
.
)t
h
a
tCp i
s，byi
t
s
e
l
f
，absorb，1
)f
o
rLandt
h
a
tBpC
s，a
sabove
，p
o
s
i
t
i
v
e
l
yindependenti
nL
.
inga
t(
0
p i
Henceby [
1
8
)
，Lemma 1
]，t
h
e su
伍c
i
e
n
tc
o
n
d
i
t
i
o
nt
h
e
r
e
o
fi
s met enough.
Thiscorrespondst
ot
h
eu
s
u
a
le
x
t
e
n
s
i
o
ntheoremf
o
rl
i
n
e
a
rs
p
a
c
e
s
. Moreover
，
t
h
e“
i
f
"p
a
r
to
fTheorem1e
s
s
e
n
t
i
a
l
l
y(
a
n
daf
o
r
t
i
o
r
i
)c
o
v
e
r
s[
9
)，P
r
o
b
.3E
]
.
Meanwhile
，l
e
t P =(E
，必)十 bea maximal positive cone i
n E，which
o
E
E
. Letus t
a
k
et
h
i
so
p
p
o
r
t
u
n
i
t
yt
o make mention [
1
8
)，
i
s absorbinga
tu
Lemma3(
2
)
](
t
h
i
sp
l
a
y
sr
a
t
h
e
rw
e
l
li
nc
o
n
j
u
n
c
t
i
o
nwithLemma1i
b
i
d
.
)i
n
，xεP-αu
o
}o
f
connection with t
h
e Minkowski gaugep(x)=inf{α:α >0
P-uo・
SUPPLEMENT TO [
1
8
)，LEMMA 3(
2
)
]
. At五r
s
t
，needless t
os
a
y
(
1
) Asu
s
u
a
l
，usingp
(
x
)(
r
e
s
p
.i
nview o
ft
h
eorderedl
i
n
e
a
rs
p
a
c
e (E
，
.
3
e
)
)
， one candeduce t
h
i
s lemmaa
l
s
ov
i
at
h
e Hahn-Banach (
r
e
s
p
. Krein
、
)
e
x
t
e
n
s
i
o
ntheorem. Buta
sf
o
rt
h
i
slemma
，i
t
sproofgiveni
n1
8
)i
snotonly
s
e
l
f
c
o
n
t
a
i
n
e
dbuta
l
s
os
i
m
p
l
e
r thant
h
ea
b
o
v
e
.
Secondly t
h
i
s proof i
n terms o
ft
h
en
e
g
a
t
i
v
epartf- o
ffEE* i
.
e
.，
f-(
x
)=max{
-f
(
x
)O
}(
x
εE)，nowanewv
e
r
i
f
i
e
s
(
2
) AnfEE* r
e
q
u
i
r
e
dt
h
e
r
ewithf(u
)=l i
sg
i
v
e
nbyρi
nt
h
es
e
n
s
e
o
i
c
ev
e
r
s
a
. Thati
s
，f
(
x
)mustb
ee
q
u
a
lt
op(
-x)ρ(
x
)with
off-=p，andv
f一
(
幼 =p(
劫 fora
l
lxEE
.
This i
s knownbyp
(
x
)= 0 (XEP)，p(O)=O andρ(x)=inf{α: -x<αU
o
(
必)}=sup{戸:。
宅
三 suo<-x(先
)
}
口 -f(x)(
x
ε-P).
Concerning(
2
)，i
nf
a
c
tt
h
ef
o
l
l
o
w
i
n
gw
i
l
lbev
e
r
i
五e
d
.
(
3
) LetK b
eaconvexc
o
n
ei
nE whichi
sn
o
ti
d
e
n
t
i
c
a
l日 i
t
hE andi
s
a
b
s
o
r
b
i
n
ga
tbEE
. Then g ofgEE* i
st
h
e Minko
切 s
k
igauge ofK -b
ε
E:σ
(X)>O}CKC{XEE:σ
(
x
)
;
?
O
}
.
i
f
fg
(
b
)
=1and{x
Returningt
ot
h
es
u
b
j
e
c
t
，nextthereholds thefollowing，at
o
p
o
l
o
g
i
c
a
l
ラ
ラ
(
2
1
5
)
K.Iwata
4
3
2
v
e
r
s
i
o
no
fTheorem1
. Int
h
i
stheorem we l
e
t R be e
q
u
i
p
p
e
d with t
h
e
u
s
u
a
lt
o
p
o
l
o
g
y
.
THEOREM 2
. L
etE b
ea l
i
n
e
a
rt
o
p
o
l
o
g
i
c
a
l学a
c
e
，andl
e
tM，
，
f K，q
b
ea
si
nt
h
es
t
a
t
e
m
e
n
tofTheorem 1
. An
e
c
e
s
s
a
r
yands
u
f
f
i
c
i
e
n
tc
o
n
d
i
t
i
o
n
x
t
e
n
d
i
n
gfands
a
t
i
.
めI
Z
n
g
t
h
a
tt
h
e
r
ee
x
i
s
t
sac
o
n
t
i
n
u
o
u
sl
i
n
e
a
rformF onE e
F(
ν
)<
，
_q
(ν
)forαI
IYEK i
st
h
a
tt
h
e
r
ee
x
i
s
t
sa t
.
o
.
lふ (L
，
3
e
)with t
h
ef
o
l
lowingp
r
o
p
e
r
t
i
e
s
:
(
i
) BfUCqc(L，
必)+;
(
i
i
) (
ム3
e
)
+i
sa convexn
eighbourhooda
t(
0
，1
)forL;
st
h
et
o
p
o
l
o
g
i
c
a
l
l
うr
o
d
u
c
tExR andBf>Cqa
r
esamea
si
nTheorem1
.
whereL i
PROOF. Proceeda
si
nt
h
ep
r
o
o
fo
fTheorem1
，andcheckt
h
a
tφ(
x
，
~)
i
sc
o
n
t
i
n
u
o
u
sonL i
fando
n
l
yi
fs
oi
sF
(
x
)onE
. And L now being a
l
i
n
e
a
rt
o
p
o
l
o
g
i
c
a
ls
p
a
c
e
，t
ot
h
i
send
，wemayconsulttheproofof[
1
9
)
，Th.
3
]
. Thiscompletest
h
ep
r
o
o
fo
ft
h
et
h
e
o
r
e
m
.
N
o
t
i
c
et
h
a
t，s
i
m
i
l
a
r
l
ya
sp
o
i
n
t
e
do
u
ti
n1
9
)
，ourconditiono
f(
i
)p
l
u
s(
i
i
)
abovei
se
q
u
i
v
a
l
e
n
tt
ot
h
a
tt
h
e
r
ee
x
i
s
t
s a convexopen s
u
b
s
e
t Q:
3(
0，1
)i
n
L sucht
h
a
tBfUCqUQi
sp
o
s
i
t
i
v
e
l
yi
n
d
e
p
e
n
d
e
n
t
. Moreover
，t
h
i
st
i
m
es
i
m
p
l
e
computationg
i
v
e
st
h
ef
o
l
l
o
w
i
n
g
. Theses
i
m
p
l
i
f
yourc
o
n
d
i
t
i
o
no
fTheorem2
.
REMARK2
. Let U beaconvexO-neighbourhoodi
nE andp
u
t(
h
e
n
c
e
forth)D=Ux{
1
}，
B=(1/2U)x1where1={ρER:Iρ-11<1/2}. I
fBfUCqUD
i
sp
o
s
i
t
i
v
e
l
yindependentt
h
esamei
st
r
u
ef
o
rBfUCquB.
Let u
s nowo
b
s
e
r
v
esomec
o
r
o
l
l
a
r
i
e
sa
b
o
u
tTheorem2
. C
o
r
o
l
l
a
r
i
e
s2
and3mentionedbelowa
r
et
h
eu
s
u
a
le
x
t
e
n
s
i
o
ntheoremsi
nt
h
ec
o
n
t
e
x
to
f
l
i
n
e
a
rt
o
p
o
l
o
g
i
c
a
ls
p
a
c
e
s
.
COROLLARY 1
. Let E，M andf b
e as i
n Theorem2
. L
et K b
ea
l
i
n
e
a
rs
u
b
学a
c
eofE withM cK，qagaugef
u
n
c
t
i
o
nonK ωi
t
hf(
ェ)<q(x)
fora
l
lxEM. 1ft
h
ec
o
n
d
i
t
i
o
n
)
(
P
t
h
e
r
e
i
s
a
c
o
n
v
e
x
0
・n
eighbourhoodU i
nE n
o
tm
e
e
t
i
n
g{
yEK:
1
q
(
ν
)
=1
}
i
se
n
j
o
y
e
d
，t
h
es
u
f
f
i
c
i
e
n
tc
o
n
d
i
t
i
o
nofT
h
e
o
r
e
n
t2 i
ss
a
t
i
s
f
i
e
d
.
PROOF. LetL，BfandCq bea
si
nq
u
e
s
t
i
o
n
. Takingt
h
es
u
b
s
e
tD =Ux
{
1
}o
fL，supposet
h
a
tBfUCqUDwerenowp
o
s
i
t
i
v
e
l
ydependenti
nL
. Then
r
e
f
e
r
r
i
n
gt
oRem. 1，t
h
e
r
ewoulde
x
i
s
tboth五n
i
t
emanyr
e
s
p
e
c
t
i
v
ev
e
c
t
o
r
s，
s
a
y
，(
x
，
~r)
ε
Bf>
(ゐ払
ε
)
C
，
(
u
"
1
)
E
D
a
n
d
c
o
r
r
e
s
p
o
n
d
i
n
g
s
c
a
l
a
r
s仏
>
0
，
ß8~
0
，
r
q
7
o
rι = 0，
ん >0，7t>0)sucht
h
a
t
t> 0 (
(
*
)
q(~
7tut)~ q
(-~
arxr)-q(~
み -f(~ αrXr)-q(~
s
8
Y
8
)
s
s
Y
s
)
>-~α，'~r- ~ s
s
r
;
.= ~ 7
，
t= 1
(
2
1
6
)
H
a
h
n
B
a
n
a
c
hT
y
p
eE
x
t
e
n
s
i
o
nT
h
e
o
r
e
m
433
whichc
o
n
t
r
a
d
i
c
t
st
h
eh
y
p
o
t
h
e
s
i
ss
i
n
c
eL
:ItutEKnU. HencebyRem.2and
by [
1
8
)
，Lemma 1
]，t
h
ep
r
o
o
fi
scomplet
疋d
.
REMARK3
. Thec
o
n
v
e
r
s
eo
ft
h
i
sr
e
s
u
l
ti
sn
o
talwaysv
a
l
i
d
. Thati
s，
(
P
s，
undert
h
eremainingh
y
p
o
t
h
e
s
e
s，notalwaysn
e
c
e
s
s
a
r
yf
o
rc
o
n
c
l
u
s
i
o
n
.
1) i
Counterexamplesa
r
ee
a
s
i
l
yo
b
s
e
r
v
e
d(
c
f
.e
.g
.，E玄. 2 b
e
l
o
w
)
. Ont
h
eo
t
h
e
r
h
ec
o
n
d
i
t
i
o
n
hand，t
(
F
) t
h
e
r
ei
sa convexO
n
e
i
g
h
b
o
u
r
h
o
o
dU i
nE n
o
tm
e
e
t
i
n
g {XEM:
f(x)=l}
i
sr
a
t
h
e
rn
e
c
e
s
s
a
r
yf
o
rt
h
i
si
m
p
l
i
c
a
t
i
o
n(
f
o
rt
h
ep
r
o
o
f
，c
f
. Cor
.3 b
e
l
o
w
)，
butt
h
i
snowf
a
i
l
st
obes
u
伍c
i
e
n
tf
o
ri
t
. Thesef
a
c
t
sseemt
oi
l
l
u
s
t
r
a
t
et
h
e
s
i
g
n
i
f
i
c
a
n
c
eo
fourc
r
i
t
e
r
i
o
n
.
E
a
s
i
l
y(
r
e
s
p
. As a matter o
fc
o
u
r
s
e
)C
o
r
o
l
l
a
r
y 1y
i
e
l
d
st
h
ef
o
l
l
o
w
i
n
g
伍c
i
e
n
c
yp
a
r
to
fC
o
r
o
l
l
a
r
y3
)
.
C
o
r
o
l
l
a
r
y2 (
r
e
s
p
.t
h
es
u
Buto
fc
o
u
r
s
e，t
obes
h
o
r
t
，thesecorollariesaref
u
l
l
ydonebyTheorem
2i
t
s
e
l
f
. Forr
e
f
e
r
e
n
c
e
，d
e
t
a
i
l
sa
r
eg
i
v
e
na
su
n
d
e
r
.
COROLLARY2
*
. L
etE，
1
¥
4
.andfb
ea
si
nTheorem2
. L
etpb
eag
.
αuge
onE withf(x)~p(劫 for a
l
lxEM. 1ft
h
ec
o
n
d
i
t
i
o
n
(
P
pi
sc
o
n
t
i
n
u
o
u
sa
tt
h
eo
r
i
.
五
;
t
n
2)
t
ゐ
se
可
n
1
)0)
fm叫~on
PROOF. With t
}
)
e ∞nvex O-neighbourhood U= {
γ
ε E: ρ
(
γ
)<1
}，a
p
r
i
o
r
i，D =Ux{
1
}cCp f
o
l
l
o
w
s
. (
A
l
t
e
r
n
a
t
i
v
e
l
y
，Cp3
(
O
，1
)i
sr
e
a
d
i
l
yopeni
n
L
.
) Hence，af
o
r
t
i
o
r
i，t
h
ea
s
s
e
r
t
i
o
nf
o
l
l
o
w
sfromTheorem2
.
COROI
.LARY 3
Let E，
M andf b
ea
si
n Th
ω rem2
. A n紅 白s
a
r
y
ands
u
f
f
i
c
i
e
n
tc
o
n
d
i
t
i
o
nt
h
a
tf canb
ee
x
t
e
n
d
e
dt
oac
o
n
t
i
n
u
o
u
sl
i
n
e
a
rfoれm
F onE i
st
h
a
t(
F
)ofRem. 3 i
ss
a
t
i
s
f
i
e
d
. 1ft
h
es
u
f
f
i
c
i
e
n
りI oft
h
ec
o
作
品i
t
i
o
ni
sme
ムt
h
e
r
eαu
t
sa
tl
e
a
s
to
n
eF s
u
c
ht
h
a
tF
(
x
)=
1
=1fora
l
lxEU
.
料
.
PROOF. This i
s viewed a
sas
p
e
c
i
a
lc
a
s
eo
f Theorem2(
f
(
エ)=q(x)
(
x
εMnK)p
l
u
sM =K
)
. (N
田 e
s
s
i
t
y
) By Theorem 2，t
h
e
r
ea
r
e both t
.
o
..
1s
.
(L
，
3
e
)andconvexO-neighbourhoodU sucht
h
a
tB，
j Ux{
1/2}C(L
，3
e
)
+h
o
l
d
.
I
ff
(
u
o
)= 1f
o
rsome Uo
EMnU，t
h
e
r
e would f
o
l
l
o
w (-uo
，-1/2) (
u
o
，1
/
2
)
ε(L
，
3
e
)
七 anobvious contradiction. (Su伍 ciency)Takingtheconvexsubset
D = U x{
1
}o
fL，supposet
h
a
tBfUDwerenowp
o
s
i
t
i
v
e
l
yd
e
p
e
n
d
e
n
t
. Then
i
twouldf
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. T o extend an identically-zero linear form on M in the
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sense of Corollary 3
，w e have at least U= E
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，we have at least D=Ex{
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9
7
4
)
References
1
) E
. STIEMKE: むberp
o
s
i
t
i
v
eLusungenhomogenerl
i
n
e
a
r
e
rGleichungen. Math.Ann.
4
0
3
4
2
.
7
6(
1
9
1
5
)，3
2
) W.B
.CARVER: Systemo
fl
i
n
e
a
ri
n
e
q
u
a
l
i
t
i
e
s
. Annalso
fMath. (
2
)2
3(
1
9
2
2
)，
2
1
2
2
2
0
.
3
) L
.L
.DIENES: De五n
i
t
el
i
n
e
a
rdependence. Annals o
fMath.2
7(
1
9
2
5
)，
5
7
6
4
1
. S
t
u
d
i
aMath. 1(
1
9
2
9
)
，223-239.
4
) S
. BANACH: Surl
e
sf
o
n
c
t
i
o
n
n
e
l
l
e
sl
i
n
e
a
i
r
e
s1
5
) S
. MAZUR: uberkonvexeMengeni
nl
i
n
e
a
r
e
nnormiertenRaumen. S
t
u
d
i
aMath.4
(
1
9
3
3
)
，70-84.
6
) V
.L
.KLEE，
J
r
.
: Convexs
e
t
si
nl
i
n
e
a
rs
p
a
c
e
s
. DukeMath.]
.
， 1
8(
1
9
5
1
)，
4
4
3
4
6
6
.
a
r
t1
，chaps. I
IandV
.Wiley
7
) N.DUNFORDandJ
.T
.SCHWARTZ: Linearo
p
e
r
a
t
o
r
s，P
s
)
，1nc，NewYork，1
9
5
8
.
(
1
n
t
e
r
s
c
i
e
n
c
eP
u
b
l
i
s
h巴r
h
a
p
.1
1
. Moscow
，1
9
6
2
.(
R
u
s
s
i
a
n
)
.(
J
a
p
a
n
e
s
et
r
a
n
sl
.by
8
) D.A.RAIKOV: Vectors
p
a
c
e
s，c
Y
.YOSHIZAKI: TokyoToshoCo.，1
9
6
6
.
)
.KELLEY，1
.NAMIOKAandCOAUTHORS: L
ineart
o
p
o
l
o
g
i
c
a
ls
p
a
c
e
s，c
h
a
p
s
.1and
9
) ]
.L
9
6
3
.
4
.D.VanNostrandCo. I
n
c
.，Princeton，1
1
0
) A.WILANSKY: Functionala
n
a
l
y
s
i
s，c
h
a
p
s
.3and 1
2
.B
l
a
i
s
d
el
!publishingCo.，New
9
6
4
.
York，1
噂
.E
. EDWARDS: Functional a
n
a
l
y
s
i
s，c
h
a
p
.2
. Holt，Rinehart and Winston，1
n
c
.，
1
1
) R
NewYork，1
9
6
5
.
1
2
) D. M. TOPPING: Some homological pathologyi
nv
e
c
t
o
rl
a
t
t
i
c
e
s
. Can.J
.Math.1
7
1
1
4
2
8
.
(
1
9
6
5
)，4
e
c
t
o
r
i
e
l
st
o
p
o
l
o
g
i
q
u
e
s，c
h
a
p
. 1e
tI
I
. Elements de math
1
3
) BOURBAKI，N.: Espacesv
品
目l
a
t
i
q
u
e，l
i
v
r
eV.Hermann，P
a
r
i
s，1
9
6
6
.
，
i
!16，i
!1
7
. Springer-Verlag，B
e
r
l
i
n，1
9
6
6
.
1
4
) G
.KoTHE: Topologischel
i
n
e
a
r
eRaume，1
IandV. TheMacmillanCo.，
1
5
) H. H.SCHAEFER: Topological v
e
c
t
o
rs
p
a
c
e
s，chapsI
NewYork，1
9
6
6
.
1
6
) B
.Z
.VULIKH: I
n
t
r
o
d
u
c
t
i
o
nt
ot
h
e theory o
fp
a
r
t
i
a
l
l
y ord巴red s
p
a
c
e
s，c
h
a
p
. XII1
.
Wolters-Noordho
，
妊 Ltd.，Groningen，TheNetherlands，1
9
6
7
.
1
7
) MATH.SOC.OFJAPAN: Sugaku-Jiten (
D
i
c
t
i
o
n
a
r
yo
f Mathematics)，2nd e
d
.，p
.5
9
8
.
u
b
l
i
s
h
e
r
s，Tokyo，1
9
6
8
.
1wanamiShoten，P
!
yorderedl
i
n
e
a
rspaces
t
r
u
c
t
u
r
e
sands
e
p
a
r
a
t
i
o
ntheorem. Hokkaido
1
8
) K.1wATA: Total
.1
，No.2(
1
9
7
2
)，2
1
1
2
1
7
.
Math.J
o
u
r
.(
S
a
p
p
o
r
o
)，Vol
o
t
a
l
l
y ordered l
i
n
e
a
r space s
t
r
u
c
t
u
r
e
s and s
e
p
a
r
a
t
i
o
ntheoremi
nr
e
a
l
1
9
) 一一一一一一: T
l
i
n
e
a
rt
o
p
o
l
o
g
i
c
a
ls
p
a
c
e
s
. Mem.Muroran1
n
s
t，Tech.(
l
¥
1uroran，
J
a
p
a
n
)，
Vol
.8，No
4
3
4
8
.
1(
1
9
7
3
)，
圃
(
2
1
8
)
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