Publicacions
Ma emü iques,
Vol
34
(1990),
341-347
.
A
bs ac
CHEBYSHEV
COEFFICIENTS
FOR
L
1
-PREDUALS
AND
FOR
SPACES
WITH
THE
EXTENSION
PROPERTY
JOSÉ
M
.
BAYOD
AND
M
.
CONCEPCIÓN
MASA
We
apply
he
Chebyshe
coe icien s
A
and
Ab,
ecen ly
in oduced
by
he
au ho s,
o
ob ain
some
esul s
ela ed
o
ce ain
geome ic
p ope ies
o
Banach
spaces
.
We
p o e
ha
a
eal
no med
space
E
is
an
Ll-p edual
i
and
only
i
(E)
=
1/2, and
ha
i
a
( eal
o
complex)
no med
space
E
is
a
'P,
space,
hen
Ab(E)
equals
Ab(K),
whe e
oá
is
he
g ound
ield
o
E
.
In
his
no e,
IK
will
be
he
eal
o
complex
ield,
and
E
a
no med
space
o e
IK
;
when we
wan
o
s a e
a
esul
only
o
he
eal
case
o he
complex
case,
we
will
indíca e
i
speci icaly
.
We
will
use he
no a ions
o
[2],
o
which
we
e e
o
all
concep s
o
he
heo y
o
no med
spaces
which
may
appea
wi hou
de ining
hem
he e
.
I
S
is
a
non
emp y
subse
o
E,
he
numbe
(S)
=
in
sup
lix
-
y¡¡
YEE-ES
is
called
he
Chebyshe
adius
o S,
and
b(S) deno es he diame e
o
S
.
De lni ion
.
We
will
call
he ini e
Chebyshe
coe
icien
o
E
he
eal
numbe
A
(E)
=
sup{ (S)/b(S)
:
SC
E,
S
ini e,
b(S)
>
0},
and
he
bounded
Chebyshe
coe icien
o
E
he
eal
numbe
Ab(E)
=
sup{ (S)/b(S)
:
SC
E,
0
<
b(S)
<
oo}
.
I is
easy o
p o e
ha ,
in gene al,
1/2
<
A
(E)
<
Ab(E)
<
1
.
34
2

J
.M
.
BAYOD,
M
.C
.
MASA
Mo eo e ,
when
E
is
ini e
dimensional,
we
ha e
A
(E)
=
ab(E)
.
Speci ically,
he
Chebyshe
coe icien s
associa ed
o
he
scala
ields
a e
A (R)
=
Ab(R)
=
1/2
, (C)
=
Ab(C)
=
1/4
.
Le
us
ecall
ha
a
P,,(aá)
space,
whe e a
is
a
eal
numbe
g ea e
han
o
equal
o
1,
is
a
Banach
space
E
o
which
any
o
ollowing equi alen condi ion
holds
:
(i)
Gi en
wo
Banach
spaces,
F
and
G,
a
linea
isome y
in o,

:
F
-->
G,
and
a
bounded
linea
ope a o ,
L
:
F
->
E,
he e
exis s
a bounded
linea
ope a o
L
:
G
-+
E,
which
ex ends
L,
in
he
sense
o
L
o
0
=
L,
and
such
ha
JILII
<
aJILI1
(a-ex ension p ope y)
.
(ii)
Gi en
a Banach
space
F,
and a
linea
isome y,

:
E
-+
F,
he e
exis s
a
p ojec ion,
P
:
F
-+
O(E),
such
ha
JIPI¡
<
a
(a-p ojec ion
p ope y)
.
I
is
said ha
a
Banach
space
E
is
a
N
a space,
whe e
a
is
a
eal
numbe
g ea e
han
o
equal
o
1,
when
he e
exis s
a
collec ion
(E
.
y )
.,
E
o
ini e
dimensional
subspaces
o
E,
which
is
upwa ds
di ec ed,
hei
union
is
dense
in
E
and
e e y
one
o
hem
is
a
Pa
.(K)
space
.
No e
ha
a
.
Banach
.
.space
is
.a
n
L
l
-p edual
space
i
and
only
i i is
aN,,
space
o
e e y
a
>
1
([2,
heo em
2,
pg
.
232])
.
Theo em
1
.
I
¡he
Banach
space
E
is
an
L
1
-p edual,
hen
a
(E)
=
a
(K)
.
P oo -
We
ix
an a >
1
.
Gi en
ha
E
is
an
L
l
-p edual,
i is
a
N
a
space,
and,
so,
he e
exis s
a
collec ion
(Ej-,E
o
subspaces
o
E
acco ding
o
he
de ini ion
abo e
.
We
pu
F
=
U
E,,
which,
because
i
is
dense
in
E,
sa is ies
y
E
a (E)
=
a (F)
.
Le
S be
a
ini e
subse
o
F
wi h
mo e
han
one
poin
.
The e
exis s
yE P
such ha
S
C
E
.y
,
and,
i
we
indíca e
wi h
subíndices he
Chebyshe
adii
in
subspaces
o
E,
we
ha e
(S)
=
,
(S)
:5
e,
(S)
:5
b(S)
.A (E-,)
:5
b(S)
.a
.a
(K),
whe e
he
las
inequali y
is
due
o
E,
E
P,,(IK)
.
The e o e,
A
(E)
=
a
(F)
<
a
.A
(K),
o
e e y
a
>
1,
so
A
(E)
<
A
(K)
.
On
he
o he
hand,
by
he
Hanh-Banach
heo em,
he e
exis s
a
p ojec ion
o
no m
1
om
E
o
K,
so
a (K)
<
a (E)
.
I
*E
is
a
non-s anda d
enla gemen
o
E,
hen,
o e
he
se
in*E
=
{x
E
*E
:
3y E
E,
lix
-
y¡¡
is
a
ini e
hype eal
numbe }
o he
ini e
elemen s
o *E,
conside
he
equi alence
ela ion
"x
is
in ini ely
close
o y",
deno ed
by
x
-
y,
and
de ined
by
"l[x
-
y¡¡ is
in ini esimal"
.
In
he
quo ien
se ,
deno ed
E,
he
no m
¡IxII
=
s lixil,
í
E
E,
is
de ined,
and
he
esul ing
no med
space
is
called
an
in ini esimal
hull
o
E
.
CHEBYSHEV
COEFFICIENTS
FOR
L'-PREDUALS

34
3
Lemma
2
.
A (É)
=
A (E)
.
P oo
.
.
Le
S
be
a
ini e
subse
o
E
wi h
mo e
han
one
poin
.
Then,
S
is
a
ini e
subse
o
in*E
wi hou
in ini ely
Glose
poin s
.
I is
ob ious
ha
6(S)
=
6(S)
and (S)
<
(S)
.
We
suppose
ha
(S)
<
(S),
and
ake
a
eal
numbe
such
ha
(S)
<
<
(S)
.
Then,
he e
exis s
c
E
in*E
such
ha
S
C
B[c,
],
and
so,
lix
-
cil
<
(S)
o
e e y
x
E
S
.
Since
S
is
ini e,
*S
=
S,
and
we
ha e
a
c
E
*E
such
ha
lix
-
cil
<
(S)
o
e e y
xE
*S
.
Applying
he
T ans e
P inciple,
he e
exis s
a
s anda d
elemen
c
E
E
such
ha
lix
-
cil
<
(S)
o
e e y
x
E S,
and
again
because
S
is
ini e,
his
would
imply
SC
B[c,
p],
wi h
p
=
m
S
lix
-
cil
<
(S)
.

The e o e,
i is
ue
(S)
=
(S) and we
conclude
A
(E)
<
A
(E)
.
Le
S
be
now
a
ini e
subse
o
in*E
wi h
some
poin s no
in ini ely
Glose
.
Then,
S
is
a
*- ini e
subse
o
*E
wi h
some
poin s
no
in ini ely
Glose
and S
is
a
ini e
subse
o
E
such
ha 6(S)
-*6(S)
.
Since
he
ela ion
(T)
<
6(T) .A (E)
is
ue
o
e e y
ini e
subse
T
o
E,
by
he
T ans e
P inciple,
we
ha e * (T)
<_*6(T)
.A (E)
o
e e y
*- ini e
T,
and,
in
pa icula ,
* (S)
<*6(S)
.A (E)
-
6(S)
.A
(E)
.
Le
be
a
hipe eal
numbe
such
ha
>* (S),
-
6(S),A (E)
.
The e
exis s
a
c
E
*E
such
ha
SC
B
[c, ],
and
hen,
Iix
-
¿Ii
=
s jjx
-
cil
-
lIx
-
cil
<
'-
6(S)
.X
(E),
lx
E
S
.
Since
he
i s
and
las
membe s
a e
s anda d,
we
ha e
11
i-c11
<
6(S)
.A (E),
o
e e y
xE
S, so
ha
(S)
<
6(S)
.A (E),
and we
conclude
A (E)
<
A (E)
.
Theo em
3
.
I
E
is
a
eal
Banach
&pace
such
ha
A
(E)
=
1/2,
hen
E
is
an
L
l
-p edual
.
P oo
:
We
conside
an
in ini esimal
hull
E
o
E
.
Then,
E
has
he
adial
in e sec ion
p ope y
(2,4),
ha
is,
gi en
ou
closed
balls
in
E
wi h
he
&ame
adius,
which
in e sec
in
pai s,
he
o al
in e sec ion
is
non
emp y
.
Indeed,
le
p
be
a
posi i e
eal
numbe ,
and
le
xl,
í2, i3,
24
E
E,
such
ha
II
.ii
-
xi
il
<
2p,
o
i,
j
=
1, 2,
3,
4
.
We
ake
S
=
{xi,
i2,
,
i3,
¡J,
a
ini e
subse
o
E
wi h 6(S)
_<
2p,
and,
so,
(S)
_<
p
.
Fo
e e y
na u al
numbe
p,
he e
exis s
c
p
E
E
such
ha
Iixi
-Q¡
< p
+
(1/2p),
o
i
=
1, 2, 3,
4,
and,
consequen ly,
he e
exis s
c
p
E
in*E
such
ha
lixi
-
CP11
< p
+
(1/p),
o
i
=
1, 2,
3,
4
.
We
conside
now
he
sequence
(cp)PEN
in
*E,
which
can
be
enla ged
o
an
in e nal
sequence
(cp)pE* N
in
*E
.
The
se
o
index
p
E
*h1
such
ha
lixi
-c
p
jj
<
p+(1/p),
o
i
=
1,
2,
3,4
is
an
in e nal
subse
o
*N
con aining
all
s anda d
na u al
numbe s,
and,
so,
i
we
wo k
in
a
sui ably
sa u a ed
model
(c
.
[3]),
i
con ains
an
in ini e
index,
wE
*NI
.
Then,
he
elemen
c,,
E
*E
is
34
4

J
.M
.
BAYOD,
M
.C
.
MASA
ini e,
because
Ilxi
-
c
w
ll
<_
p,
so
ha
we
can
ake
c
,
E
E
hus
e i ying
ha
j1x
i
-
c
u ,
11<
p,
o
i
=
1,
2, 3,
4
.
The e o e,
E
is
an
L
1
-p edual
([2,
heo em
6,
pg
.
212]),
ha
is,
E'
is
an
L
l
space
.
P ojec ing
E'
o e
E' by
means
o
he
unc io i
T
E
É'
->
TSE
E
E',
we
ha e
ha
E'
is
also
an
L
i
space
([2,
heo em
3,
pg
.
1620,
and
hen
E
is
an
L
l
-p edual
.
Lemma
4
.
ab(E)
is
he
in mum
o
he
posi i e
eal
numbe s
such
ha
o
e e y
-y
>
0,
whene e
(xcJaEI
C
E
is
a
y
-Cauchy
ne
( ha
is,
gi en
e
>
0,
he e
exis s
ceo
E
I
such
ha
lixa
-
xpll
<
-y
+
e,
o
e e y
pai
o
subíndices
a,
,p
E
I
g ea e
¡han o
equal
o
ao), (xcjc EI
has
some
y-limi
x in
E
( ha
is,
gi en
e
>
0,
he e
exis s
ao
E
I
such
ha
Iix
a
-
xjj <_
y
-I-
e,
o
e e y
subindex
a
E
I
g ea e
han o
equal
o
ao)
.
P oo
. .
Le
S
be a
bounded
subse
o
E, wi h
mo e
han
one
poin
.
Fo
e e y
na u al
númbe
n,
we
conside
S(n)
=
S
x
{n} and
he
bijec ion
x
-
x(n)
=
(x,
n)
.
We
ake
now
I
=
U
S(n)
wi h
he
o de
ela ion
nEN
a
<p

a
=
/
0
V
(a
=
x(n),
,Q
=
y(-),
n
<
m),
which
makes
I a
di ec ed
se
.
O e
i ,
we
build
he ne
(xa)aC-I
de ined
by
x,
=
x
i
a
E
I
is
such
ha
a
=
x(n),
o
some
nE
N
.
Thus,
(xa)aEI
ls
a
6(S)-Cauchy
ne
in
E,
wi h
ange
S,
and
such
ha o
e e y
y
E
S
and
e e y
a
E
I,
he e
exis s
a
/3
E
I, /i >_
a,
which
sa is ies
xp
=
y,
ha
is,
o
e e y
y
E
S
he e
exis s
and
in ini e
index
0
which
sa is ies
xp
=
y
.
We
call
A'(E)
he
in imum
o
he
posi i e
eal
numbe s
such
ha
o
e e y
y>
0,
e e y
y-Cauchy
ne has
an
y-limi in
E,
and
le
be
g ea e
han
a'(E)
.
The
p e iously
builded
ne
(xa)aEI
has
a
6(S)-limi x
E
E,
and,
so,
Ilx
a
-
xil ;5
ó(S)
o
e e y
in ini e
índex
a
.
Hence,
we
ha e
¡¡y
-
xjj
S
6(S)
o
e e y
y
E
S,
and,
because
bo h
membe s
a e s anda d,
lix
-
y¡¡
<
6(S)
o
e e y
y
E
S,
ha
is,
SC
E[s, b(S)j
.
Thus,
(S)
<_
6(S)
o
e e y
>
A'
(E),
and
ab(E)
<
A'(E)
.
Con e sely,
i
(xa)a j
is
a
,.
-y-Cauchy ne
in
E
o
some
y
>
0,
we
pu
S
«
_
{xp
:
3
E
I,
l >_
a},
o
e e y
a
E
I
.
Thus,
e e y
se
S
I
is
bounded and we
can
suppose
ha
i
has
mo e
han
one
poin
(o he wise he
p oo
is
i ial)
;
he e o e,
gi en
e
>
0,
he e
exis s
a
E
I
such
ha
ó(S
a
)
<
y+e/Ab(E)
.
Then,
(Sa)
:5
ó(Sa)
.
Aj(E)
<
(y
+
e/Ab(E))-ME)
=
- Y-
,
b(E)
+
e,
and,
so,
he e
exis s,
c
E
E
E
such
ha
Iixp-ce¡¡
<_
y,Ab(E)+e
o
e e y
,Q
>
a
.
Hence,
c
E is
a
(y
.Ab(E)
-I-
e)-limi
o
(xa)aEI,
and,
since
his
is
ue o
e e y
e
>
0
and
o
e e y
y-Cauchy
ne
in
E,
i
ollows
ha
A'(E)
<
Ab(E)
.
CHEBYSHEV
COEFFICIENTS
FOR
L'-PREDUALS

34
5
Theo em
5
.
I
E
is
a
P1(K)
space,
hen
Ab(E)
=
Ab(K)
.
P oo
.
I
we
embed
K
in o
E
by
means
o
a
linea
isome y,
iden i ying
i
wi h
a
onedimensional subspace
o
E,
he
Hahn-Banach
heo em
assu es
he
exis ence
o
a
p ojec ion
o
no m
1,
P
:
E
--->
K,
whence
we
deduce
Ab(K)
<
Ab(E)
.
We
will
p o e
he
ecip oca¡ inequali y
in se e al
s ages
:
(I)
In
he
i s
place,
we
obse e
ha
i
E
E
Pl(K)
and
E
is
an
in ini esi-
mal
hull o
E,
hen
Ab(E)
<_
Ab(E),
because, conside ing
he canonical
linea
isome y
E
--+
E,
he e
exis s
a
con ac i e
p ojec ion
E
-+
E
.
(II)
Le
F
be
a
non
emp y
se
.
We
deno e by
P0(I',
K)
he
se
o
all
bounded
unc ions
om F
o
K, wi h
he
uni o m
no m
.
Gi ing
o
he
disc e e
o-
pology,
we
know
ha
1°°( ,
K)
is
linea ly
isome ic
o
he
space
C(
I',
K)
o
con inuous
unc ions
wi h
alues
in
K
de ined
o e
he
S one-Cech
compac i i-
ca ion
o
;
so,
l
00
(17,K)
E
PI(K)
([2]),
and A
b
(100(
,
K))
<
ab(¡-( ,K)),
by
(I)
.
(III)
We
suppose
ha
F
is
a
ini e se
.
We
will
p o e
ha ,
in
his
case,
Ab(1'(17,
K))
<
Ab(K)
.
Because
100(17,
K)
is
a
ini e
dimensional,
we
know
ha
i s
bounded
and
ini e
Chebyshe
coe icien s
a e equal, as a e
hose
o
K
.
Le
p
be
a
eal
numbe ,
p
>
A (K),
and
le
S
=
{xl
.
. .
. .
.
xx,,}
be
a
ini e
subse
o
1
0°
(17,
K)
.
Fixed
^y
E
F,
we
conside
he
ini e
subse
o
K S
7
=
{xl
(y),
. .
.
,
x
.(-y)}
;
hen,
(Sy)/S(S7)
<
A
(K)
<
p,
and
(S
.
y )
< p
.ó(S
.,)
=
p
.

max

I
xi(y)
-
xi(y)j
<p
.

max

11xi
-
xj1l
=
p
.b(S)
.
1<i,
j<n

1
:5i,
,1<n
Then,
he e
exis s
a
cen e
c,
E
K
such
ha
S
7
C
B[c
7
,
p
.b(S)]
CK
.
We
de ine
hus
a
unc ion
om
F
in
K
which
associa es
c
.,
o
e e y
-y,
and
which
sa is ies
Ilxi
-
cil
=
sup
I
xi(
-
y)
-
cl
C
p
.b(S),
i
=
1,
. .
.
,
n,
YE
so ha
S
C
B[c,
p
.b(S)]
C
1
00
(17,
K)
.
The e o e,
(S)
<
p
.b(S),
and
we
conclude
A
(l'(
K))
<
A
(K)
.
(IV)
The
inequali y
A
b
(100(F
,
K))
<Ab(K)
is
alid also
when
F
is
an
in ini e
se
.
Indeed,
le
(xa)aEI
be
a
y-Cauchy
ne
in
1
00
(F,K),
o
y
>
0
.
We
ake
X
o
=K
U
F
U
I
in
o de
o
build
a
supe s uc u e
X
wi h
base
X
o
and
o e
i
a
polysa u a ed
nons anda d
model
sa is ying
he
Ro-isomo phism
p ope y
(c
.
[3,
sec
.
0
.4
.])
.
In
his
case,
we
can
iden i y
¡
00
(17,
K)
o
¡
00
(w,
K),
o
e e y
in ini e
na u al
numbe
w, since
¡'(w,
K)
is
isome ically
isomo phic
o
¡
00
(INI,K),
and
his
is
so o
¡
00
(I',K)
([4,
heo em
2
.11])
.
Le
p be
a
na u al
numbe
.
The e
exis s
an
index
a
p
E
I
such
ha
Ilxa
-
xp11
<
y
+
1/(2p),
when
a
0
E
`I,
a,/l
>
a
p
.
We
conside
he
se

34
6

J
.M
.
BAYOD,
M
.C
.
MASA
S =
{x,,
:
a
E
*I,a
>_
a
p
},
an
in e nal
*-bounded
subse
o
1°°
(w,
K),
whe e
we
can
apply
(III),
and
so
*
(S)l
*6
(S)
<
I b(K)
* (S)
<
a
b
(K)*6(S)
<
ab(K)(-Y+
1/(2p))
<
,
whe e
=
Ab(K)(-y
-F-
1/p)
.

Since
* (S)
<
,
he e
exis s c
p
E
1
°°
(w,
IK)
such
ha
S
C
B[c
p
,
]
and
cp
E
l-(w,
K)
=
l-(I',
K)
sa is ies

cp
ll
<
,
o
aEI,a>a
p
.
Bea ing
in
mind
ha
1°°(I',IK)
E
Pj(IK),
conside
he
na u al
embedding
1°°(I',
IK)
-->
Í'
(I',
VK)
;
hen, he e
exis s
a
p ojec ion
o
no m
1,
P
:
(I',
IK)
->
1
-
( ,
K),
and x
=
P(c
p
) is
an
elemen
o
1'(I',OK)`which
sa is ies
lix«
-
xjj
=
IIP(xa)
T
P(cjj
C
j1xa
-
Ql
<
,
a
E
I,
a > ap
.
Thus,
o
e e y
p
>
Ab(K),
aking
a
p
E
N
g ea e
han
he
eal
numbe
Ab(K)/y(P
-
Ab(K)),
we
Na e
IIx«
-
XII
<
ab(K)(y
+p)
<
P7,
a
E
I,
a
>_
ap,
ha
is,
x
is
ap
-
y
-
limi
o
(xj,,
in
1°°(I',
Bá)
.

Since
his
is
alid o
e e y
y-
Cauchy
ne
in 100(I',IK)
and
o
any
-y
>
0
we
conclude
by
Lemma
4 ha
ab(1'( ,K))
_<
ab(o-c)
.
(V)
Now,
we
can
embed
E
linea ly
and
isome ically in o
1°°(E,
IK)
by
means
o
he
applica ion
0
:E
->
1
00
(E,
K)
x->¢(x)
:E-~6á
y

O(x)(y)
=
y(x)
whe e
y
is
a
con inuous
linea
unc ional
om
E
o
K
which
sa is ies
jj
y
jj
=
1
and
y
(y)
=
11yII,
he
exis en e
o
which
is
gua an eed
by
he
Hahn-Banach
heo em
.
So,
he e
exis s
a
p ojec ion
o
no m
1,
1°°(E,
IK)
->
E,
which
pe mi s
us
o
deduce
he
inequali y
Ab(E)
<
Ab(1°°(E,IK)),
and,
om
he
esul
in
he
p eceding
pa ag aph,
Ab(E)
<
Ab(K)
.
Re e en es
[1]
BAYOD,
J
.M
.,
MASA,
M
.C
.,
Coe icien es
de
Chebyshe
en
espacios
de
unciones
con inuas,
o
appea
in
he
Ac as
de
las
XIV
Jo nadas
His-
pano-lusas
de
Ma emá icas,
La
Laguna,
Spain
(1989)
.
[2J
LACEY
.
H
.E
.,
"The
Isome ic heo y
o
classicál
Banach
spaces,"
Sp in-
ge -Ve lag,
Be lin,
Heidelbe g,
New
Yo k,
1974
.
CHEBYSHEV
COEFFICIENTS
FOR
L
1
-PREDUALS

347
[3]
STROYAN,
K
.D
.,
BAYOD,
J
.M
.,
"Founda ions
o
in ini esimal
s ochas ic
analysis,"
No h-Holland,
Ams e dam,
1986
.
[4]
WARD
HENSON,
C
.,
The
isomo ism p ope y
in
nons anda d
analysis
and
i s
use
in
he heo y
o
Banach
spaces,
The
Jou nal
o
Symbolic
Logic
39,
4
(1974)
.
1980
Ma hema ics
subjec
classi icaiions
:
46B20,
46B25
Depa amen o
de
Ma emá icas,
Es adís ica
y
Compu ación
Uni e sidad
de
Can ab ia
39071-
San ande
SPAIN
Rebu
el
18
de
Desemb e
de
1989