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STUDIA MATHEMATICA 196 (1) (2010)
Schauder bases and the bounded approximation
property in separable Banach spaces
by
Jorge Mujica and Daniela M. Vieira (Campinas)
Abstract. Let E be a separable Banach space with the λ-bounded approximation
property. We show that for each > 0 there is a Banach space F with a Schauder basis
such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover,
the sequence (Tn ) of canonical projections in F has the properties
sup kTn k ≤ λ + and
lim sup kTn k ≤ λ.
n→∞
n∈N
This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.
Introduction. Pełczyński [12] and Johnson, Rosenthal and Zippin [7]
have independently shown that every separable Banach space with the bounded approximation property is topologically isomorphic to a complemented
subspace of a Banach space with a Schauder basis.
If E is a separable Banach space with the λ-bounded approximation
property, then the proof of Pełczyński [12], which has been reproduced in
the book of Lindenstrauss and Tzafriri [8, p. 38], shows that E is isometrically isomorphic to a 1-complemented subspace of a Banach space F with a
Schauder basis, whose basis constant is at most 5λ.
In this paper we show that if E is a separable Banach space with the
λ-bounded approximation property, then for each > 0 there is a Banach
space F with a Schauder basis, whose basis constant is at most λ + , such
that E is isometrically isomorphic to a 1-complemented subspace of F . Our
proof is a refinement of the proof of Pełczyński [12].
We establish our main result in Section 1. In Section 2 we use that result
to extend some recent results for holomorphic functions on Banach spaces
with Schauder bases (see [11]) to the realm of separable Banach spaces with
the bounded approximation property.
2010 Mathematics Subject Classification: Primary 46B15, 46B28, 46G20.
Key words and phrases: Banach space, Schauder basis, bounded approximation property.
DOI: 10.4064/sm196-1-1
[1]
c Instytut Matematyczny PAN, 2010
2
J. Mujica and D. M. Vieira
1. Schauder bases and the bounded approximation property. If
λ ≥ 1, then a Banach space E is said to have the λ-bounded approximation
property if for each compact set K ⊂ E and > 0, there is a finite rank
operator T ∈ L(E; E) such that kT k ≤ λ and kT x − xk ≤ for every x ∈ K;
E is said to have the bounded approximation property if E has the λ-bounded
approximation property for some λ ≥ 1; and E is said to have the metric
approximation property if E has the 1-bounded approximation property.
A sequence (en ) ⊂ E is said to be a Schauder
P basis if each x ∈ E can be
uniquely written as a convergent series x = ∞
n=1 ξn en with (ξn ) ⊂ K. Let
(Tn ) denote the sequence of canonical projections, that is,
n
∞
X
X
Tn x =
ξj ej for every x =
ξj ej ∈ E.
j=1
j=1
Let c denote the basis constant, and let ca denote the asymptotic basis constant, that is,
c = sup kTn k, ca = lim sup kTn k.
n→∞
n∈N
A Schauder basis is said to be monotone if c = 1, and asymptotically monotone if ca = 1. Clearly every monotone basis is asymptotically monotone,
but while monotone bases are rather special, asymptotically monotone bases
can be found everywhere. Indeed, as pointed out in [11], an examination of
the proof of a classical result of Mazur (see [8, Theorem 1.a.5]) shows the
following theorem.
Theorem 1.1. Every infinite-dimensional Banach space contains a
closed, infinite-dimensional subspace with an asymptotically monotone Schauder basis.
We remark that Theorem 1.1 was first proved by Day [2]. His proof is
based on a generalization of the Borsuk–Ulam antipodal theorem.
We also remark that Szarek [13] has given an example of a separable
reflexive Banach space with the metric approximation property which does
not have a Schauder basis.
For background information on Schauder bases and approximation properties we refer to the book of Lindenstrauss and Tzafriri [8].
Our main result is the following theorem.
Theorem 1.2. Let E be a separable Banach space with the λ-bounded
approximation property. Then for each > 0 there is a Banach space F with
a Schauder basis such that E is isometrically isomorphic to a 1-complemented
subspace of F and, moreover, the sequence (Tn ) of canonical projections in
F has the properties
c = sup kTn k ≤ λ + n
and
ca = lim sup kTn k ≤ λ.
n→∞
Schauder bases and bounded approximation property
3
Corollary 1.3. Every separable Banach space with the metric approximation property is isometrically isomorphic to a 1-complemented subspace
of a Banach space with an asymptotically monotone Schauder basis.
To prove the theorem we need the following lemma, a refinement of a
lemma due to Pełczyński [12].
Lemma 1.4. Let E be a Banach space of finite dimension n. Then for
each > 0, there are m ∈ N and operators B1 , . . . , Bmn ∈ L(E; E) of rank
one such that
mn
X
Bj x = x for all x ∈ E
j=1
and
k
mn
X
X
B
B
+
j
j ≤ 1 + j=1
for all 1 ≤ k ≤ mn.
j=k+1
Proof. By Auerbach’s lemma (see [8, Proposition 1.c.3]) there are normone vectors e1 , . . . , en ∈ E and φ1 , . . . , φn ∈ E 0 such that φj (ek ) = δjk for
j, k = 1, . . . , n. Let A1 , . . . , An ∈ L(E; E) be defined by Aj x = φj (x)ej .
Then each Aj has norm one and rank one, and
n
X
(1)
Aj x =
j=1
n
X
φj (x)ej = x
for every x ∈ E.
j=1
Fix m ∈ N. Then each integer k with 1 ≤ k ≤ mn can be uniquely written
in the form
k = rn + s
(2)
with 0 ≤ r ≤ m − 1, 1 ≤ s ≤ n.
If k is given by (2) we define Bk = m−1 As . By using (1), it follows that
k
X
Bj x =
j=1
rn
X
Bj x +
j=1
rn+s
X
Bj x
j=rn+1
n
s
s
r X
1 X
r
1 X
=
At x +
At x =
x+
At x.
m
m
m
m
t=1
t=1
t=1
Similarly we have
mn
X
j=k+1
(r+1)n
Bj x =
X
j=rn+s+1
Bj x +
mn
X
Bj x
j=(r+1)n+1
n
n
n
1 X
m−r−1X
1 X
m−r−1
=
At x +
At x =
At x +
x.
m
m
m
m
t=s+1
t=1
t=s+1
4
J. Mujica and D. M. Vieira
Since mn = (m − 1)n + n, it follows that
mn
X
1
m−1
x + x = x.
Bj x =
m
m
j=1
On the other hand, if 1 ≤ k ≤ mn, then
k
mn
r + s (n − s) + (m − r − 1)
X
X
m+n−1
B
+
+
=
.
B
j
j ≤
m
m
m
j=1
j=k+1
To complete the proof it suffices to choose m such that (n − 1)/m < .
Proof of Theorem 1.2. Let (xn ) be a countable dense subset of E. Let
(Un ) be a sequence of finite rank operators in L(E; E) such that kUn k ≤ λ
for every n and kUn xj − xj k ≤ 1/n whenever 1 ≤ j ≤ n. It follows that (Un )
converges to the identity uniformly on compact subsets of E. If we define
V1 = U1 and Vn = Un − Un−1 for every n P
≥ 2, then (Vn ) is a sequence of
finite rank operators in L(E; E) such that nj=1 Vj x = Un x, and therefore
P∞
j=1 Vj x = x for every x ∈ E.
Let Mn = Vn (E) for every n, and let (n ) be a strictly decreasing sequence
of positive numbers tending to zero, with 1 = /λ. By Lemma 1.4 for each
n ∈ N there are operators B1n , . . . , Bpnn ∈ L(Mn ; Mn ), of rank one, such that
pn
X
Bjn x = x
for every x ∈ Mn
j=1
and
pn
k
X
X
Bjn + Bjn ≤ 1 + n
j=1
for every 1 ≤ k ≤ pn .
j=k+1
Set p0 = 0. Then each k ∈ N can be uniquely written in the form
k = p0 + p1 + · · · + pn−1 + s
(3)
with 1 ≤ s ≤ pn .
If k is given by (3), then we define δk = n and Ak = Bsn ◦ Vn . Then
k
X
p0 +···+pn−1
Aj x =
j=1
X
p0 +···+pn−1 +s
j=1
=
pr
n−1
XX
X
Aj x +
Aj x
j=p0 +···+pn−1 +1
s
X
Btr ◦ Vr x +
r=1 t=1
Btn ◦ Vn x =
t=1
= Un−1 x +
s
X
n−1
X
Vr x +
r=1
Btn ◦ (Un − Un−1 )x
t=1
=
s
X
t=1
Btn ◦ Un x +
pn
X
t=s+1
Btn ◦ Un−1 x.
s
X
t=1
Btn ◦ Vn x
Schauder bases and bounded approximation property
5
On the one hand, it follows that
k
s
X
X
Aj x − x ≤ kUn−1 x − xk + Btn k(Un − Un−1 )xk
t=1
j=1
≤ kUn−1 x − xk + (1 + n )k(Un − Un−1 )xk,
and the last expression tends to zero when n tends to infinity. Since each
Aj has rank one, we may write Aj x = φj (x)aj with φj ∈ E 0 and aj ∈ E,
kaj k = 1. Thus
x=
∞
X
Aj x =
j=1
∞
X
for every x ∈ E.
φj (x)aj
j=1
On the other hand, it follows that
pn
k
s
X
X
X
n
Aj ≤ Bt kUn k + Btn kUn−1 k ≤ λ(1 + n ),
j=1
t=1
t=s+1
and therefore
k
X
Aj ≤ λ(1 + δk )
for every k ∈ N.
j=1
Let F P
be the vector space of all sequences y = (ηj ) ⊂ K such that the
series ∞
j=1 ηj aj converges in E. Then a standard argument shows that F ,
endowed with the norm
k
X
kyk = sup ηj aj ,
k
j=1
is a Banach space whose unit vectors form a monotone Schauder basis. Let
(Tk ) denote the sequence of canonical projections in F , that is,
Tk y = (η1 , . . . , ηk , 0, 0, . . .)
for every y = (ηj ) ∈ F.
Let A ∈ L(E; F ) and B ∈ L(F ; E) be defined by
Ax = (φj (x)) and By =
∞
X
η j aj .
j=1
Then
k
k
X
X
kAxk = sup φj (x)aj = sup Aj x k
j=1
k
j=1
≤ sup λ(1 + δk )kxk = λ(1 + 1 )kxk,
k
6
J. Mujica and D. M. Vieira
and
∞
k
X
X
kByk = ηj aj ≤ sup ηj aj = kyk.
k
j=1
j=1
Since B ◦ Ax = x for every x ∈ E, E is isomorphic to a complemented
subspace of F .
Next we will renorm F so as to obtain all the stated properties. Consider
the set
∞
k
X
X
n
o
W = y = (ηj ) ∈ F : ηj aj ≤ 1, ηj aj ≤ λ(1 + δk ) for every k .
j=1
j=1
Clearly W is convex and balanced. Since
W ⊂ {y = (ηj ) ∈ F : kyk ≤ λ(1 + 1 )},
it follows that W is bounded in F . And since
W ⊃ {y = (ηj ) ∈ F : kByk ≤ 1 and kyk ≤ λ},
it follows that W is a 0-neighborhood in F , and is in particular absorbing.
Let FW denote the vector space F , seminormed by the Minkowski functional
pW of W . Since W is bounded in F , pW is a norm and the identity mapping
FW → F is continuous. And since W is a 0-neighborhood in F , the identity
mapping FW → F is actually a homeomorphism. In particular, FW is a
Banach space whose unit vectors form a Schauder basis.
If x ∈ BE , then Ax = (φj (x)) ∈ W , since
∞
k
X
X
φj (x)aj ≤ 1 and φj (x)aj ≤ λ(1 + δk )
j=1
for every k.
j=1
P
Conversely, if Ax = (φj (x)) ∈ W , then x ∈ BE , since kxk = k ∞
j=1 φj (x)aj k
≤ 1. Thus A : E → FW is an isometric embedding.
Clearly A ◦ B is a projection from FW onto A(E). To show that A(E) is
a 1-complemented subspace of FW we will show that A ◦ B(y) ∈ W for each
y ∈ W . Let y = (ηj ) ∈ W . Then A ◦ B(y) = (φj (By)). On the one hand,
∞
∞
X
X
φj (By)aj = kB(A ◦ B(y))k = kByk = ηj aj ≤ 1.
j=1
j=1
On the other hand,
k
k
k
X
X
X
φj (By)aj = Aj (By) ≤ Aj kByk ≤ λ(1 + δk )
j=1
j=1
for every k, proving that A ◦ B(y) ∈ W .
j=1
Schauder bases and bounded approximation property
7
To complete the proof of the theorem we will prove that
Tk y ∈ λ(1 + δk )W
(4)
for all y ∈ W and k ∈ N.
To see this write y = (ηj ) and Tk y = (σj ), where σj = ηj if 1 ≤ j ≤ k and
σj = 0 if j > k. On the one hand,
∞
k
X
X
σj aj = ηj aj ≤ λ(1 + δk ).
j=1
j=1
On the other hand, if 1 ≤ l ≤ k, then
l
l
X
X
σj aj = ηj aj ≤ λ(1 + δl ) ≤ λ(1 + δk )λ(1 + δl ).
j=1
j=1
Finally, if l > k, then
l
k
X
X
σ
a
=
η
a
j j
j j ≤ λ(1 + δk ) ≤ λ(1 + δk )λ(1 + δl ).
j=1
j=1
Thus (4) shows that kTk k ≤ λ(1 + δk ) if we regard Tk as a member of
L(FW ; FW ). It follows that
sup kTk k ≤ λ(1 + 1 ) = λ + and
k
lim sup kTk k ≤ λ,
k→∞
thus completing the proof.
Johnson [6] has proved that every separable reflexive Banach space with
the bounded approximation property is isomorphic to a complemented subspace of a reflexive Banach space with a Schauder basis, but he gives no
estimate of the size of the basis constant. On the other hand, Grothendieck
[5, Chapitre I, p. 181] (see also [8, Theorem 1.e.15]) has shown that every reflexive Banach space with the approximation property already has the metric
approximation property.
Motivated by these results, we would like to prove that every separable
reflexive Banach space with the (metric) approximation property is isometrically isomorphic to a 1-complemented subspace of a reflexive Banach space
with an asymptotically monotone Schauder basis. But we have been unable
to use the method of proof of Theorem 1.2 to prove this.
2. Weakly continuous holomorphic functions. In a recent paper
we have made a detailed study of certain algebras of weakly continuous
holomorphic functions on a Banach space with a Schauder basis (see [11]).
With the aid of Theorem 1.2 we can extend some of those results to the realm
of separable Banach spaces with the bounded approximation property.
Before stating our results we have to introduce some notation and terminology. For background information on infinite-dimensional complex analysis
we refer to the books of Dineen [3] or Mujica [10].
8
J. Mujica and D. M. Vieira
Let E be a complex Banach space, let U be an open subset of E, and let
H(U ) denote the algebra of all complex-valued holomorphic functions on U .
Given A ⊂ U and F ⊂ H(U ), the set ÂF is defined by
ÂF = {x ∈ U : |f (x)| ≤ sup |f | for every f ∈ F}.
A
For each x ∈ U let dU (x) denote the distance from x to the boundary
of U . For each A ⊂ U we define dU (A) = inf x∈A dU (x). The open set U is
said to be pseudoconvex if the function − log dU is plurisubharmonic on U .
Let F be a separable Banach space with the λ-bounded approximation
property. By Theorem 1.2 we may identify F with a 1-complemented subspace of a Banach space E with a Schauder basis, whose asymptotic constant
is at most λ. Let (Tn ) denote the sequence of canonical projections in E, and
let π be a norm-one projection from E onto F .
If V is a pseudoconvex open subset of F , then U = π −1 (V ) is a pseudoconvex open subset of E and U ∩ F = V (see [10, Proposition 37.8]).
Consider the following sequences of open subsets of U :
Aj (U ) = {x ∈ U : sup kTn x − xk < dU (x)},
n≥j
Bj (U ) = {x ∈ Aj (U ) : kxk < j and dAj (U ) (x) > 1/j},
Cj (U ) = {x ∈ Bj (U ) : sup kTn x − xk < dBj (U ) (x)}.
n≥j
Let C(U ) denote the sequence (Cj (U )), and let Hwu (C(U )) denote the subalgebra of all f ∈ H(U ) that are weakly uniformly continuous on each Cj (U ).
It is a Fréchet algebra for the topology of uniform convergence on the sets
Cj (U ).
Next consider the following sequence of open subsets of V :
Cj (V ) = Cj (U ) ∩ F ⊂ U ∩ F = V.
Let C(V ) denote the sequence (Cj (V )), and let Hwu (C(V )) denote the subalgebra of all f ∈ H(V ) that are weakly uniformly continuous on each Cj (V ).
It is a Fréchet algebra for the topology of uniform convergence on the sets
Cj (V ).
With this terminology we have the following theorem.
Theorem 2.1. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. Then U is the domain of existence
of a function f ∈ Hwu (C(U )).
(b) Let F be a separable Banach space with the bounded approximation
property, and let V be a pseudoconvex open subset of F . Then V is the
domain of existence of a function g ∈ Hwu (C(V )).
Schauder bases and bounded approximation property
9
Proof. (a) was proved in [11, Theorem 1.2], and the proof rests on the
following property of the sequence (Cj (U )), established in [11, Lemma 1.4]:
(5)
\
dU (C
j (U )Hwu (C(U )) ) > 0
for every j.
(b) The proof of (b) is a straightforward adaptation of the proof of [11,
Theorem 1.2], and the proof rests on the following property of the sequence
(Cj (V )):
(6)
\
dV (C
j (V )Hwu (C(V )) ) > 0
for every j.
\
We can readily see that (6) follows from (5). Indeed, if C
j (U )Hwu (C(U )) +
BE (0; r) ⊂ U , then, since Hwu (C(U )) ⊂ Hwu (C(V )), it follows that
\
\
C
j (V )Hwu (C(V )) +BF (0; r) ⊂ Cj (U )Hwu (C(U )) ∩F +BE (0; r)∩F ⊂ U ∩F = V.
Let 0 < α ≤ 1, and let Hαwud (U ) (resp. Hαd (U )) denote the algebra of
all f ∈ H(U ) that are weakly uniformly continuous (resp. bounded) on each
ball B(x; r) with x ∈ U and 0 < r < αdU (x). It follows from an argument of
Aron and Prolla [1, Lemma 2.2] that Hαwud (U ) ⊂ Hαd (U ). If the underlying
space is separable, then Hαd (U ) is a Fréchet space for the topology of uniform
convergence on the balls B(x; r) with x ∈ U and 0 < r < αdU (x), and
Hαwud (U ) is a closed subalgebra of Hαd (U ). The algebra Hαd (U ) was studied
by Matos [9] a long time ago, and has recently been rediscovered by Dineen
and Venkova [4] when α = 1. With this terminology we have the following
theorem.
Theorem 2.2. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. If α = (3ca )−2 , then U is the
domain of existence of a function f ∈ Hαwud (U ).
(b) Let F be a separable Banach space with the λ-bounded approximation
property, and let V be a pseudoconvex open subset of F . If β = (3λ)−2 , then
V is the domain of existence of a function g ∈ Hβwud (V ).
Proof. (a) was proved in [11, Theorem 1.5], and the proof rests on the
following property of the sequence (Cj (U )), established in [11, Lemma 1.6]:
(7)
For each x ∈ U and 0 < r < (3ca )−2 dU (x), the ball BE (x, r) is contained in some Cj (U ).
If α = (3ca )−2 , then it follows from (7) that Hwu (C(U )) ⊂ Hαwud (U ), and
then (a) follows from Theorem 2.1(a).
(b) The proof of (b) is similar. Indeed, it follows from Theorem 1.2 that
ca ≤ λ, and therefore (7) implies the following:
(8)
For each y ∈ V and 0 < r < (3λ)−2 dV (y), the ball BF (y; r) is contained in some Cj (V ).
10
J. Mujica and D. M. Vieira
If β = (3λ)−2 , then it follows from (8) that Hwu (C(V )) ⊂ Hβwud (V ), and
then (b) follows from Theorem 2.1(b).
Theorem 2.3. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. Let (xp ) be a sequence of distinct
points of U such that dU (xp ) → 0. Then there exists f ∈ Hwu (C(U )) such
that |f (xp )| → ∞ and f (xp ) 6= f (xq ) whenever p 6= q.
(b) Let F be a separable Banach space with the bounded approximation
property, and let V be a pseudoconvex open subset of F . Let (yp ) be a sequence of distinct points of V such that dV (yp ) → 0. Then there exists
g ∈ Hwu (C(V )) such that |g(yp )| → ∞ and g(yp ) 6= g(yq ) whenever p 6= q.
Proof. (a) was proved in [11, Theorem 2.1], and the proof rests on property (5) of the sequence (Cj (U )).
(b) The proof of (b) is a straightforward adaptation of the proof of [11,
Theorem 2.1], and rests on property (6) of (Cj (V )).
Theorem 2.4. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. Let (xp ) be a sequence of distinct
points of U such that dU (xp ) → 0. If α = (3ca )−2 , then there exists f ∈
Hαwud (U ) such that |f (xp )| → ∞ and f (xp ) 6= f (xq ) whenever p 6= q.
(b) Let F be a separable Banach space with the λ-bounded approximation
property, and let V be a pseudoconvex open subset of F . Let (yp ) be a sequence
of distinct points of V such that dV (yp ) → 0. If β = (3λ)−2 , then there exists
g ∈ Hβwud (V ) such that |g(yp )| → ∞ and g(yp ) 6= g(yq ) whenever p 6= q.
Proof. (a) was proved in [11, Theorem 2.3]. Actually, it follows from
property (7) of the sequence (Cj (U )) that Hwu (C(U )) ⊂ Hαwud (U ), and
then (a) follows from Theorem 2.3(a).
(b) Likewise, it follows from property (8) of the sequence (Cj (V )) that
Hwu (C(V )) ⊂ Hβwud (V ), and then (b) follows from Theorem 2.3(b).
If G is a Banach space, then the space Hαwud (U ; G) is defined in the
obvious way.
Theorem 2.5. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. Let (xp ) be a sequence of distinct
points of U such that dU (xp ) → 0, and let (zp ) be an arbitrary sequence in
a Banach space G. If α = (3ca )−2 , then there exists f ∈ Hαwud (U ; G) such
that f (xp ) = zp for every p.
(b) Let F be a separable Banach space with the λ-bounded approximation property, and let V be a pseudoconvex open subset of F . Let (yp ) be a
sequence of distinct points of V such that dV (yp ) → 0, and let (zp ) be an
arbitrary sequence in a Banach space G. If β = (3λ)−2 , then there exists
g ∈ Hβwud (V ; G) such that g(yp ) = zp for every p.
Schauder bases and bounded approximation property
11
Proof. (a) was proved in [11, Theorem 2.6]. Actually, it follows from
Theorem 2.4(a) and [11, Lemma 2.5].
(b) Likewise (b) follows from Theorem 2.4(b) and [11, Lemma 2.5].
τ0 denotes the compact-open topology on H(U ).
Theorem 2.6. (a) Let E be a Banach space with a Schauder basis, and
let U be a pseudoconvex open subset of E. If α = (3ca )−2 , then Hαwud (U ) is
sequentially dense in (H(U ), τ0 ).
(b) Let F be a separable Banach space with the λ-bounded approximation
property, and let V be a pseudoconvex open subset of F . If β = (3λ)−2 , then
Hβwud (V ) is sequentially dense in (H(V ), τ0 ).
Proof. (a) was proved in a more general form in [11, Theorem 3.2].
(b) By Theorem 1.2 we may identify F with a 1-complemented subspace
of a Banach space E with a Schauder basis, with ca ≤ λ. Let π be a normone projection from E onto F , and let U = π −1 (V ). If g ∈ H(V ), then
g ◦ π ∈ H(U ). By (a) there is a sequence (fn ) ⊂ Hαwud (U ) which converges
to g ◦ π in (H(U ), τ0 ). If we define gn = fn |U ∩ F = fn |V , then it follows
that (gn ) ⊂ Hαwud (V ) and (gn ) converges to g in (H(V ), τ0 ). Since ca ≤ λ,
it follows that β ≤ α, and the proof is complete.
In a similar manner we could extend other results from [11] to the realm
of separable Banach spaces with the bounded approximation property.
Acknowledgements. This research was supported by FAPESP, Brazil,
Project 06/02378-7.
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Jorge Mujica, Daniela M. Vieira
IMECC-UNICAMP
Caixa Postal 6065
13083-970 Campinas, SP, Brazil
E-mail: [email protected]
[email protected]
Current address of Daniela M. Vieira:
Instituto de Matemática e Estatística
Universidade de São Paulo
Caixa Postal 66281
05315-970 São Paulo, SP, Brazil
E-mail: [email protected]
Received August 30, 2007
Revised version November 26, 2008
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