We apply the Chebyshev coefficients λf and λb, recently introduced by the authors, to obtain some results related to certain geometric properties of Banach spaces. We prove that a real normed space E is an L1 predual if and only if λf (E) = 1/2, and that if a (real or complex) normed space E is a P1 space, then λb(E) equals λb(K), where K is the ground field of E

Bayod, José M.

Masa, Concepción

Masa, M. C.

Revistes Catalanes amb Accés Obert

Publicacions Matemütiques, Vol 34 (1990), 341-347 .
Abstract
CHEBYSHEV COEFFICIENTS FOR
L1-PREDUALS AND FOR SPACES
WITH THE EXTENSION PROPERTY
JOSÉ M . BAYOD AND M . CONCEPCIÓN MASA
We apply the Chebyshev coefficients Af and Ab, recently introduced by
the authors, to obtain some results related to certain geometric properties
of Banach spaces . We prove that a real normed space E is an Ll-predual
if and only if \ f(E) = 1/2, and that if a (real or complex) normed space
E is a 'P, space, then Ab(E) equals Ab(K), where oá is the ground field of
E.
In this note, IK will be the real or complex field, and E a normed space over
IK ; when we want to state a result only for the real case or the complex case,
we will indícate it specificaly. We will use the notations of [2], to which we
refer for all concepts of the theory of normed spaces which may appear without
defining them here .
If S is a non empty subset of E, the number
r(S) = inf sup lix - y¡¡
YEE-ES
is called the Chebyshev radius of S, and b(S) denotes the diameter of S .
Deflnition . We will call the finite Chebyshev coeficient of E the real number
Af(E) = sup{r(S)/b(S) : S C E, S finite, b(S) > 0},
and the bounded Chebyshev coeficient of E the real number
Ab(E) = sup{r(S)/b(S) : S C E, 0 < b(S) < oo} .
It is easy to prove that, in general,
1/2 < Af(E) < Ab(E) < 1.
342
	
J.M . BAYOD, M .C . MASA
Moreover, when E is finite dimensional, we have Af(E) = ab(E).
Specifically, the Chebyshev coefficients associated to the scalar fields are
Af(R) = Ab(R) = 1/2
,\f(C) = Ab(C) = 1/4.
Let us recall that a P,,(aá) space, where a is a real number greater than or
equal to 1, is a Banach space E for which any of following equivalent condition
holds :
(i) Given two Banach spaces, F and G, a linear isometry into, : F --> G,
and a bounded linear operator, L : F -> E, there exists a bounded linear
operator L : G -+ E, which extends L, in the sense of L o 0 = L, and
such that JILII < aJILI1 (a-extension property) .
(ii) Given a Banach space F, and a linear isometry, : E -+ F, there exists
a projection, P : F-+ O(E), such that JIPI¡ < a (a-projection property) .
It is said that a Banach space E is a Na space, where a is a real number
greater than or equal to 1, when there exists a collection (E.y ) .,Er of finite
dimensional subspaces of E, which is upwards directed, their union is dense in
E and every one of them is a Pa.(K) space. Note that a . Banach . space is .an
Ll -predual space if and only if it is a N,, space for every a > 1 ([2, theorem 2,
pg . 232]) .
Theorem 1 . If ¡he Banach space E is an L1 -predual, then
af (E) = a f(K) .
Proof- We fix an a > 1 . Given that E is an Ll -predual, it is a Na space,
and, so, there exists a collection (Ej-,Er of subspaces of E according to the
definition above . We put F= U E,, which, because it is dense in E, satisfiesyEr
af(E) = af(F).
Let S be a finite subset of F with more than one point . There exists y E P
such that S C E.y , and, if we indícate with subíndices the Chebyshev radii in
subspaces of E, we have
r(S) = r, (S) :5 re, (S) :5 b(S).Af(E-,) :5 b(S) .a.af(K),
where the last inequality is due to E, E P,,(IK) .
Therefore, Af(E) = af(F) < a.Af(K), for every a > 1, so Af(E) < Af(K) .
On the other hand, by the Hanh-Banach theorem, there exists a projection
of norm 1 from E to K, so af(K) < af(E) .
If *E is a non-standard enlargement of E, then, over the set fin*E = {x E
*E : 3y E E, lix - y¡¡ is a finite hyperreal number} of the finite elements of *E,
consider the equivalence relation "x is infinitely close to y", denoted by x - y,
and defined by "l[x - y¡¡ is infinitesimal" . In the quotient set, denoted E, the
norm ¡IxII = stlixil, í E E, is defined, and the resulting normed space is called
an infinitesimal hull of E.
CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS
	
343
Lemma 2. Af(É) = Af(E).
Proof. Let S be a finite subset of E with more than one point . Then, S is a
finite subset of fin*E without infinitely Glose points .
It is obvious that 6(S) = 6(S) and r(S) < r(S) . We suppose that r(S) <
r(S), and take a real number t such that r(S) < t < r(S) . Then, there exists
c E fin*E such that S C B[c, t], and so, lix - cil < r(S) for every x E S . Since
S is finite, *S = S, and we have a c E *E such that lix - cil < r(S) for every
x E *S . Applying the Transfer Principle, there exists a standard element c E E
such that lix - cil < r(S) for every x E S, and again because S is finite, this
would imply S C B[c, p], with p = mS lix - cil < r(S) . Therefore, it is true
r(S) = r(S) and we conclude A f(E) < Af (E) .
Let S be now a finite subset of fin*E with some points not infinitely Glose .
Then, S is a *-finite subset of *E with some points not infinitely Glose and S is
a finite subset of E such that 6(S) -*6(S) .
Since the relation r(T) < 6(T) .Af(E) is true for every finite subset T of E,
by the Transfer Principle, we have *r(T) <_*6(T).Af(E) for every *-finite T,
and, in particular, *r(S) <*6(S) .Af(E) - 6(S) .Af(E).
Let t be a hiperreal number such that t >*r(S), t - 6(S),Af(E) . There
exists a c E *E such that S C B[c, t], and then,
Iix - ¿Ii = stjjx - cil - lIx - cil < t'- 6(S) .Xf(E), tlx E S.
Since the first and last members are standard, we have 11 i-c11 < 6(S) .Af(E),
for every x E S, so that r(S) < 6(S).Af(E), and we conclude Af(E) < Af(E).
Theorem 3 . If E is a real Banach &pace such that Af(E) = 1/2, then E is
an Ll -predual.
Proof: We consider an infinitesimal hull E of E. Then, E has the radial
intersection property (2,4), that is, given four closed balls in E with the &ame
radius, which intersect in pairs, the total intersection is non empty.
Indeed, let p be a positive real number, and let xl, í2, i3, 24 E E, such
that II .ii - xi il < 2p, for i,j = 1, 2, 3, 4 . We take S = {xi, i2, ,i3, ¡J, a finite
subset of E with 6(S) _< 2p, and, so, r(S) _< p. For every natural number
p, there exists cp E E such that Iixi - Q¡ < p + (1/2p), for i = 1, 2, 3, 4,
and, consequently, there exists cp E fin*E such that lixi - CP11 < p + (1/p),
for i = 1, 2, 3, 4 . We consider now the sequence (cp)PEN in *E, which can be
enlarged to an internal sequence (cp)pE*rN in *E . The set of index p E *h1 such
that lixi -cp jj < p+(1/p), for i = 1, 2, 3,4 is an internal subset of *N containing
all standard natural numbers, and, so, if we work in a suitably saturated model
(cf . [3]), it contains an infinite index, w E *NI . Then, the element c,, E *E is
344
	
J.M . BAYOD, M.C . MASA
finite, because Ilxi - cw ll <_ p, so that we can take c, E E thus verifying that
j1x i - c u, 11< p, for i = 1, 2, 3, 4.
Therefore, E is an L1 -predual ([2, theorem 6, pg . 212]), that is, E' is an Ll
space . Projecting E' over E' by means of the functiori T E É' -> TSE E E', we
have that E' is also an Li space ([2, theorem 3, pg . 1620, and then E is an
Ll -predual .
Lemma 4. ab(E) is the infmum of the positive real numbers r such that for
every -y > 0, whenever (xcJaEI C E is a y-Cauchy net (that is, given e > 0,
there exists ceo E I such that lixa - xpll < -y + e, for every pair of subíndices
a, ,p E I greater ¡han or equal to ao), (xcjcrEI has some ry-limit x in E (that
is, given e > 0, there exists ao E I such that Iix a - xjj <_ ry -I- e, for every
subindex a E I greater than or equal to ao) .
Proof.. Let S be a bounded subset of E, with more than one point . For every
natural númber n, we consider S(n) = S x {n} and the bijection
x r-r x(n) = (x, n) .
We take now I = U S(n) with the order relation
nEN
a < p a = /0 V (a = x(n), ,Q = y(-), n < m),
which makes I a directed set . Over it, we build the net (xa)aC-I defined by
x, = x if a E I is such that a = x(n), for some n E N . Thus, (xa)aEI ls a
6(S)-Cauchy net in E, with range S, and such that for every y E S and every
a E I, there exists a /3 E I, /i >_ a, which satisfies xp = y, that is, for every
y E S there exists and infinite index 0 which satisfies xp = y .
We call A'(E) the infimum of the positive real numbers r such that for every
y > 0, every y-Cauchy net has an ry-limit in E, and let t be greater than
a'(E) . The previously builded net (xa)aEI has a t6(S)-limit x E E, and, so,
Ilx a - xil ;5 tó(S) for every infinite índex a . Hence, we have ¡¡y - xjj S t6(S)
for every y E S, and, because both members are standard, lix - y¡¡ < t6(S) for
every y E S, that is, S C E[s,tb(S)j . Thus, r(S) <_ t6(S) for every t > A'(E),
and ab(E) < A'(E) .
Conversely, if (xa)arj is a
, .
-y-Cauchy net in E for some y > 0, we put S« _
{xp : 3 E I, fl >_ a}, for every a E I . Thus, every set SI is bounded and we
can suppose that it has more than one point (otherwise the proof is trivial) ;
therefore, given e > 0, there exists a E I such that ó(Sa ) < y+e/Ab(E) . Then,
r(Sa) :5 ó(Sa) .Aj(E) < (y + e/Ab(E))-ME) = - Y- ,\b(E) + e,
and, so, there exists, cE E E such that Iixp-ce¡¡ <_ y,Ab(E)+e for every ,Q > a .
Hence, cE is a (y.Ab(E) -I- e)-limit of (xa)aEI, and, since this is true for every
e > 0 and for every y-Cauchy net in E, it follows that A'(E) < Ab(E) .
CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS
	
345
Theorem 5. If E is a P1(K) space, then Ab(E) = Ab(K) .
Proof. If we embed K into E by means of a linear isometry, identifying it
with a onedimensional subspace of E, the Hahn-Banach theorem assures the
existence of a projection of norm 1, P : E ---> K, whence we deduce Ab(K) <
Ab(E) .
We will prove the reciproca¡ inequality in several stages :
(I) In the first place, we observe that if E E Pl(K) and E is an infinitesi-
mal hull of E, then Ab(E) <_ Ab(E), because, considering the canonical linear
isometry E --+ E, there exists a contractive projection E -+ E.
(II) Let F be a non empty set . We denote by P0(I', K) the set of all bounded
functions from F to K, with the uniform norm . Giving to r the discrete to-
pology, we know that 1°°(r, K) is linearly isometric to the space C(fI', K) of
continuous functions with values in K defined over the Stone-Cech compactifi-
cation of r ; so, l00(17,K) E PI(K) ([2]), and Ab(100(r , K)) < ab(¡-(r,K)), by
(I) .
(III) We suppose that F is a finite set . We will prove that, in this case,
Ab(1'(17, K)) < Ab(K) .
Because 100(17, K) is a finite dimensional, we know that its bounded and finite
Chebyshev coefficients are equal, as are those of K .
Let p be a real number, p > Af(K), and let S = {xl. . . . . . xx,,} be a finite
subset of 1 0° (17, K) . Fixed ^y E F, we consider the finite subset of K S7 =
{xl (y), . . . , x .(-y)} ; then, r(Sy)/S(S7) < Af(K) < p, and
r(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, there exists a centre c, E K such that S 7 C B[c7 , p.b(S)] C K . We
define thus a function from F in K which associates c ., to every -y, and which
satisfies
Ilxi - cil = sup Ixi( -y) - cl C p.b(S), i = 1, . . . , n,
YEr
so that S C B[c, p.b(S)] C 100 (17, K) . Therefore, r(S) < p.b(S), and we conclude
A f(l'(r
K))
< Af (K) .
(IV) The inequality Ab(100(F , K)) < Ab(K) is valid also when F is an infinite
set .
Indeed, let (xa)aEI be a y-Cauchy net in 1 00(F,K), for y > 0 . We take
Xo = K U F U I in order to build a superstructure X with base Xo and over
it a polysaturated nonstandard model satisfying the Ro-isomorphism property
(cf . [3, sec . 0.4 .]) . In this case, we can identify ¡00 (17, K) to ¡00(w, K), for
every infinite natural number w, since ¡'(w, K) is isometrically isomorphic to
¡00 (INI,K), and this is so to ¡00(I',K) ([4, theorem 2.11]) .
Let p be a natural number . There exists an index ap E I such that
Ilxa - xp11 < y + 1/(2p), when a0 E `I, a,/l > ap . We consider the set
346
	
J.M . BAYOD, M .C . MASA
S = {x,, : a E *I,a >_ ap }, an internal *-bounded subset of 1°° (w, K), where
we can apply (III), and so
*r(S)l*6(S) < I\b(K)
*r(S) < a b (K)*6(S) < ab(K)(-Y+ 1/(2p)) < t,
where t = Ab(K)(-y -F- 1/p) . Since *r(S) < t, there exists cp E 1°°(w, IK) such
that S C B[cp , t] and cp E l-(w, K) = l-(I', K) satisfies cp ll < t, for
aEI,a>a p .
Bearing in mind that 1°°(I',IK) E Pj(IK), consider the natural embedding
1°°(I', IK) --> Í'(I', VK) ; then, there exists a projection of norm 1, P : (I', IK) ->
1-(r, K), and x = P(cp ) is an element of 1'(I',OK)`which satisfies
lix« - xjj = IIP(xa) T P(cjj C j1xa - Ql < t, a E I, a > ap .
Thus, for every p > Ab(K), taking a p E N greater than the real number
Ab(K)/y(P - Ab(K)), we Nave
IIx« - XII < ab(K)(y +
p)
< P7, a E I, a >_ ap,
that is, x is a p -y-limit of (xj,, in 1°°(I', Bá) . Since this is valid for every y-
Cauchy net in 100(I',IK) and for any -y > 0 we conclude by Lemma 4 that
ab(1'(r,K)) _< ab(o-c) .
(V) Now, we can embed E linearly and isometrically into 1°°(E, IK) by means
of the application
0 :E -> 100(E, K)
x->¢(x) :E-~6á
y O(x)(y) = fy(x)
where fy is a continuous linear functional from E to K which satisfies jjf y jj = 1
and fy (y) = 11yII, the existente of which is guaranteed by the Hahn-Banach
theorem . So, there exists a projection of norm 1, 1°°(E, IK) -> E, which permits
us to deduce the inequality Ab(E) < Ab(1°°(E,IK)), and, from the result in the
preceding paragraph, Ab(E) < Ab(K) .
Referentes
[1] BAYOD, J.M ., MASA, M.C ., Coeficientes de Chebyshev en espacios de
funciones continuas, to appear in the Actas de las XIV Jornadas His-
pano-lusas de Matemáticas, La Laguna, Spain (1989) .
[2J LACEY. H.E ., "The Isometric theory of classicál Banach spaces," Sprin-
ger-Verlag, Berlin, Heidelberg, New York, 1974 .
CHEBYSHEV COEFFICIENTS FOR L1 -PREDUALS
	
347
[3] STROYAN, K.D ., BAYOD, J .M ., "Foundations of infinitesimal stochastic
analysis," North-Holland, Amsterdam, 1986 .
[4] WARD HENSON, C ., The isomorfism property in nonstandard analysis
and its use in the theory of Banach spaces, The Journal of Symbolic Logic
39, 4 (1974) .
1980 Mathematics subject classificaiions : 46B20, 46B25
Departamento de Matemáticas, Estadística y Computación
Universidad de Cantabria
39071- Santander
SPAIN
Rebut el 18 de Desembre de 1989


Chebyshev coefficients for $L^1$-preduals and for spaces with the extension property

J. M. Bayod

M. C. Masa

Crossref

Publicacions Matemàtiques

Diposit Digital de Documents de la UAB

Publicacions Matemütiques, Vol 34 (1990), 341-347 .AbstractCHEBYSHEV COEFFICIENTS FORL1-PREDUALS AND FOR SPACESWITH THE EXTENSION PROPERTYJOSÉ M . BAYOD AND M . CONCEPCIÓN MASAWe apply the Chebyshev coefficients Af and Ab, recently introduced bythe authors, to obtain some results related to certain geometric propertiesof Banach spaces . We prove that a real normed space E is an Ll-predualif and only if \ f(E) = 1/2, and that if a (real or complex) normed spaceE is a 'P, space, then Ab(E) equals Ab(K), where oá is the ground field ofE.In this note, IK will be the real or complex field, and E a normed space overIK ; when we want to state a result only for the real case or the complex case,we will indícate it specificaly. We will use the notations of [2], to which werefer for all concepts of the theory of normed spaces which may appear withoutdefining them here .If S is a non empty subset of E, the numberr(S) = inf sup lix - y¡¡YEE-ESis called the Chebyshev radius of S, and b(S) denotes the diameter of S .Deflnition . We will call the finite Chebyshev coeficient of E the real numberAf(E) = sup{r(S)/b(S) : S C E, S finite, b(S) > 0},and the bounded Chebyshev coeficient of E the real numberAb(E) = sup{r(S)/b(S) : S C E, 0 < b(S) < oo} .It is easy to prove that, in general,1/2 < Af(E) < Ab(E) < 1.342	J.M . BAYOD, M .C . MASAMoreover, when E is finite dimensional, we have Af(E) = ab(E).Specifically, the Chebyshev coefficients associated to the scalar fields areAf(R) = Ab(R) = 1/2,\f(C) = Ab(C) = 1/4.Let us recall that a P,,(aá) space, where a is a real number greater than orequal to 1, is a Banach space E for which any of following equivalent conditionholds :(i) Given two Banach spaces, F and G, a linear isometry into, : F --> G,and a bounded linear operator, L : F -> E, there exists a bounded linearoperator L : G -+ E, which extends L, in the sense of L o 0 = L, andsuch that JILII < aJILI1 (a-extension property) .(ii) Given a Banach space F, and a linear isometry, : E -+ F, there existsa projection, P : F-+ O(E), such that JIPI¡ < a (a-projection property) .It is said that a Banach space E is a Na space, where a is a real numbergreater than or equal to 1, when there exists a collection (E.y ) .,Er of finitedimensional subspaces of E, which is upwards directed, their union is dense inE and every one of them is a Pa.(K) space. Note that a . Banach . space is .anLl -predual space if and only if it is a N,, space for every a > 1 ([2, theorem 2,pg . 232]) .Theorem 1 . If ¡he Banach space E is an L1 -predual, thenaf (E) = a f(K) .Proof- We fix an a > 1 . Given that E is an Ll -predual, it is a Na space,and, so, there exists a collection (Ej-,Er of subspaces of E according to thedefinition above . We put F= U E,, which, because it is dense in E, satisfiesyEraf(E) = af(F).Let S be a finite subset of F with more than one point . There exists y E Psuch that S C E.y , and, if we indícate with subíndices the Chebyshev radii insubspaces of E, we haver(S) = r, (S) :5 re, (S) :5 b(S).Af(E-,) :5 b(S) .a.af(K),where the last inequality is due to E, E P,,(IK) .Therefore, Af(E) = af(F) < a.Af(K), for every a > 1, so Af(E) < Af(K) .On the other hand, by the Hanh-Banach theorem, there exists a projectionof norm 1 from E to K, so af(K) < af(E) .If *E is a non-standard enlargement of E, then, over the set fin*E = {x E*E : 3y E E, lix - y¡¡ is a finite hyperreal number} of the finite elements of *E,consider the equivalence relation "x is infinitely close to y", denoted by x - y,and defined by "l[x - y¡¡ is infinitesimal" . In the quotient set, denoted E, thenorm ¡IxII = stlixil, í E E, is defined, and the resulting normed space is calledan infinitesimal hull of E.CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS	343Lemma 2. Af(É) = Af(E).Proof. Let S be a finite subset of E with more than one point . Then, S is afinite subset of fin*E without infinitely Glose points .It is obvious that 6(S) = 6(S) and r(S) < r(S) . We suppose that r(S) <r(S), and take a real number t such that r(S) < t < r(S) . Then, there existsc E fin*E such that S C B[c, t], and so, lix - cil < r(S) for every x E S . SinceS is finite, *S = S, and we have a c E *E such that lix - cil < r(S) for everyx E *S . Applying the Transfer Principle, there exists a standard element c E Esuch that lix - cil < r(S) for every x E S, and again because S is finite, thiswould imply S C B[c, p], with p = mS lix - cil < r(S) . Therefore, it is truer(S) = r(S) and we conclude A f(E) < Af (E) .Let S be now a finite subset of fin*E with some points not infinitely Glose .Then, S is a *-finite subset of *E with some points not infinitely Glose and S isa finite subset of E such that 6(S) -*6(S) .Since the relation r(T) < 6(T) .Af(E) is true for every finite subset T of E,by the Transfer Principle, we have *r(T) <_*6(T).Af(E) for every *-finite T,and, in particular, *r(S) <*6(S) .Af(E) - 6(S) .Af(E).Let t be a hiperreal number such that t >*r(S), t - 6(S),Af(E) . Thereexists a c E *E such that S C B[c, t], and then,Iix - ¿Ii = stjjx - cil - lIx - cil < t'- 6(S) .Xf(E), tlx E S.Since the first and last members are standard, we have 11 i-c11 < 6(S) .Af(E),for every x E S, so that r(S) < 6(S).Af(E), and we conclude Af(E) < Af(E).Theorem 3 . If E is a real Banach &pace such that Af(E) = 1/2, then E isan Ll -predual.Proof: We consider an infinitesimal hull E of E. Then, E has the radialintersection property (2,4), that is, given four closed balls in E with the &ameradius, which intersect in pairs, the total intersection is non empty.Indeed, let p be a positive real number, and let xl, í2, i3, 24 E E, suchthat II .ii - xi il < 2p, for i,j = 1, 2, 3, 4 . We take S = {xi, i2, ,i3, ¡J, a finitesubset of E with 6(S) _< 2p, and, so, r(S) _< p. For every natural numberp, there exists cp E E such that Iixi - Q¡ < p + (1/2p), for i = 1, 2, 3, 4,and, consequently, there exists cp E fin*E such that lixi - CP11 < p + (1/p),for i = 1, 2, 3, 4 . We consider now the sequence (cp)PEN in *E, which can beenlarged to an internal sequence (cp)pE*rN in *E . The set of index p E *h1 suchthat lixi -cp jj < p+(1/p), for i = 1, 2, 3,4 is an internal subset of *N containingall standard natural numbers, and, so, if we work in a suitably saturated model(cf . [3]), it contains an infinite index, w E *NI . Then, the element c,, E *E is344	J.M . BAYOD, M.C . MASAfinite, because Ilxi - cw ll <_ p, so that we can take c, E E thus verifying thatj1x i - c u, 11< p, for i = 1, 2, 3, 4.Therefore, E is an L1 -predual ([2, theorem 6, pg . 212]), that is, E' is an Llspace . Projecting E' over E' by means of the functiori T E É' -> TSE E E', wehave that E' is also an Li space ([2, theorem 3, pg . 1620, and then E is anLl -predual .Lemma 4. ab(E) is the infmum of the positive real numbers r such that forevery -y > 0, whenever (xcJaEI C E is a y-Cauchy net (that is, given e > 0,there exists ceo E I such that lixa - xpll < -y + e, for every pair of subíndicesa, ,p E I greater ¡han or equal to ao), (xcjcrEI has some ry-limit x in E (thatis, given e > 0, there exists ao E I such that Iix a - xjj <_ ry -I- e, for everysubindex a E I greater than or equal to ao) .Proof.. Let S be a bounded subset of E, with more than one point . For everynatural númber n, we consider S(n) = S x {n} and the bijectionx r-r x(n) = (x, n) .We take now I = U S(n) with the order relationnENa < p a = /0 V (a = x(n), ,Q = y(-), n < m),which makes I a directed set . Over it, we build the net (xa)aC-I defined byx, = x if a E I is such that a = x(n), for some n E N . Thus, (xa)aEI ls a6(S)-Cauchy net in E, with range S, and such that for every y E S and everya E I, there exists a /3 E I, /i >_ a, which satisfies xp = y, that is, for everyy E S there exists and infinite index 0 which satisfies xp = y .We call A'(E) the infimum of the positive real numbers r such that for everyy > 0, every y-Cauchy net has an ry-limit in E, and let t be greater thana'(E) . The previously builded net (xa)aEI has a t6(S)-limit x E E, and, so,Ilx a - xil ;5 tó(S) for every infinite índex a . Hence, we have ¡¡y - xjj S t6(S)for every y E S, and, because both members are standard, lix - y¡¡ < t6(S) forevery y E S, that is, S C E[s,tb(S)j . Thus, r(S) <_ t6(S) for every t > A'(E),and ab(E) < A'(E) .Conversely, if (xa)arj is a, .-y-Cauchy net in E for some y > 0, we put S« _{xp : 3 E I, fl >_ a}, for every a E I . Thus, every set SI is bounded and wecan suppose that it has more than one point (otherwise the proof is trivial) ;therefore, given e > 0, there exists a E I such that ó(Sa ) < y+e/Ab(E) . Then,r(Sa) :5 ó(Sa) .Aj(E) < (y + e/Ab(E))-ME) = - Y- ,\b(E) + e,and, so, there exists, cE E E such that Iixp-ce¡¡ <_ y,Ab(E)+e for every ,Q > a .Hence, cE is a (y.Ab(E) -I- e)-limit of (xa)aEI, and, since this is true for everye > 0 and for every y-Cauchy net in E, it follows that A'(E) < Ab(E) .CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS	345Theorem 5. If E is a P1(K) space, then Ab(E) = Ab(K) .Proof. If we embed K into E by means of a linear isometry, identifying itwith a onedimensional subspace of E, the Hahn-Banach theorem assures theexistence of a projection of norm 1, P : E ---> K, whence we deduce Ab(K) <Ab(E) .We will prove the reciproca¡ inequality in several stages :(I) In the first place, we observe that if E E Pl(K) and E is an infinitesi-mal hull of E, then Ab(E) <_ Ab(E), because, considering the canonical linearisometry E --+ E, there exists a contractive projection E -+ E.(II) Let F be a non empty set . We denote by P0(I', K) the set of all boundedfunctions from F to K, with the uniform norm . Giving to r the discrete to-pology, we know that 1°°(r, K) is linearly isometric to the space C(fI', K) ofcontinuous functions with values in K defined over the Stone-Cech compactifi-cation of r ; so, l00(17,K) E PI(K) ([2]), and Ab(100(r , K)) < ab(¡-(r,K)), by(I) .(III) We suppose that F is a finite set . We will prove that, in this case,Ab(1'(17, K)) < Ab(K) .Because 100(17, K) is a finite dimensional, we know that its bounded and finiteChebyshev coefficients are equal, as are those of K .Let p be a real number, p > Af(K), and let S = {xl. . . . . . xx,,} be a finitesubset of 1 0° (17, K) . Fixed ^y E F, we consider the finite subset of K S7 ={xl (y), . . . , x .(-y)} ; then, r(Sy)/S(S7) < Af(K) < p, andr(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<nThen, there exists a centre c, E K such that S 7 C B[c7 , p.b(S)] C K . Wedefine thus a function from F in K which associates c ., to every -y, and whichsatisfiesIlxi - cil = sup Ixi( -y) - cl C p.b(S), i = 1, . . . , n,YErso that S C B[c, p.b(S)] C 100 (17, K) . Therefore, r(S) < p.b(S), and we concludeA f(l'(rK))< Af (K) .(IV) The inequality Ab(100(F , K)) < Ab(K) is valid also when F is an infiniteset .Indeed, let (xa)aEI be a y-Cauchy net in 1 00(F,K), for y > 0 . We takeXo = K U F U I in order to build a superstructure X with base Xo and overit a polysaturated nonstandard model satisfying the Ro-isomorphism property(cf . [3, sec . 0.4 .]) . In this case, we can identify ¡00 (17, K) to ¡00(w, K), forevery infinite natural number w, since ¡'(w, K) is isometrically isomorphic to¡00 (INI,K), and this is so to ¡00(I',K) ([4, theorem 2.11]) .Let p be a natural number . There exists an index ap E I such thatIlxa - xp11 < y + 1/(2p), when a0 E `I, a,/l > ap . We consider the set346	J.M . BAYOD, M .C . MASAS = {x,, : a E *I,a >_ ap }, an internal *-bounded subset of 1°° (w, K), wherewe can apply (III), and so*r(S)l*6(S) < I\b(K)*r(S) < a b (K)*6(S) < ab(K)(-Y+ 1/(2p)) < t,where t = Ab(K)(-y -F- 1/p) . Since *r(S) < t, there exists cp E 1°°(w, IK) suchthat S C B[cp , t] and cp E l-(w, K) = l-(I', K) satisfies cp ll < t, foraEI,a>a p .Bearing in mind that 1°°(I',IK) E Pj(IK), consider the natural embedding1°°(I', IK) --> Í'(I', VK) ; then, there exists a projection of norm 1, P : (I', IK) ->1-(r, K), and x = P(cp ) is an element of 1'(I',OK)`which satisfieslix« - xjj = IIP(xa) T P(cjj C j1xa - Ql < t, a E I, a > ap .Thus, for every p > Ab(K), taking a p E N greater than the real numberAb(K)/y(P - Ab(K)), we NaveIIx« - XII < ab(K)(y +p)< P7, a E I, a >_ ap,that is, x is a p -y-limit of (xj,, in 1°°(I', Bá) . Since this is valid for every y-Cauchy net in 100(I',IK) and for any -y > 0 we conclude by Lemma 4 thatab(1'(r,K)) _< ab(o-c) .(V) Now, we can embed E linearly and isometrically into 1°°(E, IK) by meansof the application0 :E -> 100(E, K)x->¢(x) :E-~6áy O(x)(y) = fy(x)where fy is a continuous linear functional from E to K which satisfies jjf y jj = 1and fy (y) = 11yII, the existente of which is guaranteed by the Hahn-Banachtheorem . So, there exists a projection of norm 1, 1°°(E, IK) -> E, which permitsus to deduce the inequality Ab(E) < Ab(1°°(E,IK)), and, from the result in thepreceding paragraph, Ab(E) < Ab(K) .Referentes[1] BAYOD, J.M ., MASA, M.C ., Coeficientes de Chebyshev en espacios defunciones continuas, to appear in the Actas de las XIV Jornadas His-pano-lusas de Matemáticas, La Laguna, Spain (1989) .[2J LACEY. H.E ., "The Isometric theory of classicál Banach spaces," Sprin-ger-Verlag, Berlin, Heidelberg, New York, 1974 .CHEBYSHEV COEFFICIENTS FOR L1 -PREDUALS	347[3] STROYAN, K.D ., BAYOD, J .M ., "Foundations of infinitesimal stochasticanalysis," North-Holland, Amsterdam, 1986 .[4] WARD HENSON, C ., The isomorfism property in nonstandard analysisand its use in the theory of Banach spaces, The Journal of Symbolic Logic39, 4 (1974) .1980 Mathematics subject classificaiions : 46B20, 46B25Departamento de Matemáticas, Estadística y ComputaciónUniversidad de Cantabria39071- SantanderSPAINRebut el 18 de Desembre de 1989

Chebyshev coefficients for L1-preduals and for spaces with the extension property

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Chebyshev coefficients for L1-preduals and for spaces with the extension property

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