Rev. R. Acad. Cien. Serie A. Mat.
SAM
VOL.99 (1), 2005, pp. 15–22
An´alisis Matem´atico / Mathematical Analysis
The structure of nonseparable Banach spaces with uncountable unconditional bases
Carlos Finol and Marek W ´ ojtowicz
Abstract. LetXbe a Banach space with an uncountable unconditional Schauder basis, and letY be an arbitrary nonseparable subspace ofX. IfX contains no isomorphic copy of`1(J)withJuncountable then (1) the density ofY and the weak*-density ofY∗are equal, and (2) the unit ball ofX∗is weak*
sequentially compact. Moreover, (1) implies thatY contains large subsets consisting of pairwise disjoint elements, and a similar property holds for uncountable unconditional basic sets inX.
La estructura de los espacios de Banach no separables que tienen bases incondicionales no numerables
Resumen. SeaX un espacio de Banach con una base incondicional de Schauder no numerable, y seaY un subespacio arbitrario no separable deX. SiXno contiene una copia isomorfa de`1(J)con J no numerable entonces (1) la densidad deY y la d´ebil*-densidad deY∗son iguales, y (2) la bola unidad deX∗es d´ebil* sucesionalmente compacta. Adem´as, (1) implica queY contiene subconjuntos grandes formados por elementos disjuntos dos a dos, y una propiedad similar se verifica para las bases incondicionales no numerables deX.
1 Introduction
Throughout this paperX will denote a Banach space with an unconditional basis(xγ)γ∈Γ, whereΓis an uncountable set, andY will be its nonseparable closed linear subspace. The best known examples of such spacesXare`p(Γ),1≤p <∞, andc0(Γ)(another examples are addressed in [10, 15, 20, 24]). By`1(ℵ1) we denote the space`1(Γ)withcard(Γ) =ℵ1.
This paper deals with the structure of nonseparable subspaces ofX whose study is motivated by the result below, included implicitly in the proofs of two results by Rodriguez-Salinas ([15, Proposition 2]) and Granero ([5, Proposition 1]):
(RSG) LetY be a nonseparable subspace ofXwithχ(Y) =χ∗(Y∗). ThenY contains a set of the cardi- nality ofχ(Y)consisting of elements of norm one with pairwise disjoint supports.
(Here and in what followsχ(Y)andχ∗(Y∗), respectively, denote the density and weak*-density character ofY andY∗, respectively, whose definition is given below.) Since the conditionχ(Y) = χ∗(Y∗)holds true forY reflexive or weakly compactly generated, the above result gives almost immediately a description of complemented subspaces of`p(Γ), with1< p <∞, andc0(Γ)(see [15, 5]), generalizing Pełczy´nski’s
Presentado por Jerzy Kakol el 2 de febrero de 2005.
Recibido: 31 de enero de 2005.Aceptado: 2 de marzo de 2005.
Palabras clave / Keywords: closed subspaces, complemented subspaces, unconditional basic sets, symmetric basic sets, spaces c0(Γ),`p(Γ)
Mathematics Subject Classifications: 46B25, 46B26 c
2005 Real Academia de Ciencias, Espa˜na.
classical theorem asserting thatevery infinite dimensional complemented subspace of`p, with1≤p≤ ∞, (respectively,c0) is isomorphic to`p(respectively,c0) (see [13, Theorem 2.a.3]). One should mention here that in 1966 a similar result for`1(Γ)was obtained by K¨othe [12].
In the next section we give a characterization of those spacesX where the conditionχ(Y) = χ∗(Y∗) holds forallsubspacesY ofX (Theorem 1); this appears to be equivalent to the non-containment (byX) of an isomorphic copy of the space`1(ℵ1)or, under the continuum hypothesis, to the weak* sequential compactness of the unit ball ofX∗(Proposition 1). The latter equivalence relates to the problem posed in 1977 by Rosenthal [17]:Suppose that the dual unit ballBW∗ of a Banach spaceW is not weak* sequen- tially compact; can we then conclude thatW contains a subspace isomorphic to`1?, which was answered in 1978 in the negative by Hagler and Odell [7] (cf. [6, 11]). Moreover, in 1977 Haydon constructed a Banach spaceZ (of the typeC(K)) not containing isomorphic copies of`1(ℵ1)such thatBZ∗ is not weak* sequentially compact [9]. This shows that our equivalence does depend on the structure of the given Banach space (for other results concerning the embeddability of`1(ℵ1)into Banach spaces see [19] and the references given therein). The characterization given in Theorem 1 allows us to generalize, by (RSG), the cited results of Rodriguez-Salinas and Granero (Theorem 2); it also shows that ifX contais no copy of`1(ℵ1)thenY, containing large (unconditional) basic set consisting of pairwise disjoint elements, has
“big” unconditional structure (see the comment in ([4, p. 396]) on atomic Banach lattices). In Section 3, complementing the previous theorems, we show that every uncountable unconditional basic set(yj)j∈J inX contains a subset of the same cardinality asJ consisting of pairwise disjoint elements provided that (yj)j∈J has no uncountable subsets of the`1-type (Theorem 3).
The restrictive role of`1(J), withJuncountable, in Theorems 1, 2, and 3 explains the following result obtained in 1975 by Troyanski [21]
(T) Let the basis(xγ)γ∈ΓofX be symmetric. IfXhas a subspace isomorphic to`1(J) [resp.,c0(J)]for some uncountable setJ, then the basis is equivalemt to the natural basis of`1(Γ) [resp.,c0(Γ)], and generalized in 1988 by Drewnowski [3] who showed that if the basis in(T)is merely unconditional then it contains “large” subbases of the`1-[resp.,c0-]type. Therefore, in the context of Troyanski’s result, the last section is devoted only to the structure of nonseparable subspaces ofX=`1(Γ), and the basic tool we use in our studies is the notion ofε-disjoint systems. In Theorem 4 we prove the existence, for every ε >0, of such systems inX, which allows one to strengthen the above-cited result of K¨othe (Corollary 6) and to give its shorter proof (Corollary 7).
Our terminology and notation is that of [13] and [20]. All subspaces are assumed to be linear and closed.
Recall that a family(xγ)γ∈ΓinXis said to be an (unconditional) basis ofX if, for everyx∈Xthere is a unique family of scalars(tγ)γ∈Γ such thatx=P
γ∈Γtγxγ (unconditional convergence). By(x∗γ)γ∈Γwe denote the dual family, biorthogonal to(xγ)γ∈Γ; then
x=X
γ∈Γ
x∗γ(x)xγ for everyx∈X, (1)
and the support ofx ∈ X is defined as supp(x) := {γ ∈ Γ : x∗γ(x) 6= 0}. We say that two elements u, v∈X aredisjointif their supports,supp(u)andsupp(v), are disjoint subsets ofΓ. From (1) it follows that every elementx∗∈X∗has the representation
x∗=X
γ∈Γ
x∗(xγ)x∗γ (weak*-convergence), (2)
which allows one to define the support ofx∗assupp(x∗) :={γ ∈Γ :x∗(xγ)6= 0}. The basis(xγ)γ∈Γ
is calledsymmetric if, for every sequence(γn)inΓthe basic sequence(xγn)is symmetric in the usual sense ([13, p. 113]). We say that a family(vj)j∈J is a basic set inX if it is a basis of the closed linear span of this family (denoted by[vj]j∈J). Two basic sets(uj)j∈J,(vj)j∈J in a Banach spaceW are said to be equivalent if the linear operatorG: [uj]j∈J →[vj]j∈J of the formG(uj) =vj is an isomorphism.
An isomorphismT between two Banach spacesV andW is said to be an(1 +ε)-isometry provided that kTk kT−1k ≤1 +ε.
A subspaceW0of a Banach spaceW is said to be complemented [k-complemented for somek ≥1, resp.] inW if it is the range of a continuous projectionP [withkPk=k, resp.]. IfFis a nonempty subset ofΓ, thenXF denotes the subspace ofX consisting of the elements with supports included inF, andPF
denotes the continuous projection fromXontoXFof the formPFx=x·1F, where1Fis the characteristic function ofF. Notice that ifX =`p(Γ), then the spacesXF and`p(F)are isometric. From (2) it easily follows that for everyx∗∈X∗the elementx∗F :=P
γ∈Fx∗(xγ)x∗γ (weak*-convergence) is well defined, and hence the operatorPbFonX∗of the formPbF(x∗) :=x∗Fis a continuous projection (in fact,PbF =PF∗).
If Y is a subspace of X thenSY denotes its unit sphere, supp(Y) stands for the support ofY (=
∪y∈Y supp(y)), andχ(Y)[resp., χ∗(Y∗)] denotes the density character of Y [resp., the weak*-density character ofY∗], i.e., the smallest cardinalαsuch thatY [resp.,Y∗] contains a subsetAwithcard(A) =α and such thatAis linearly norm-dense inY [resp., weak*-dense inY∗]. Recall thatχ∗(Y∗)equals also min{card(F) :F ⊂Y∗and Fis total overY}, and thatχ∗(Y∗)≤χ(Y)([20, p. 599]).
2 The weak* cardinality property and weak* sequential com- pactness
Following Vaˇsak [22], we say that a Banach spaceW has the weak* cardinality property (W*CP, for short) if, for every subspaceV of W we haveχ(V) = χ∗(V∗). In [22, Corollary 2] Vaˇsak proved that every weakly countably determined (in particular, every weakly compactly generated (WCG)) Banach space possesses this property. In the theorem below we give a characterization of the class of thoseX’s which have the W*CP.
Theorem 1 LetX be a Banach space with an uncountable unconditional basis(xγ)γ∈Γ. ThenXhas the W*CP if and only if one of the following equivalent conditions is satisfied:
(i) Xcontains no isomorphic copy of the space`1(ℵ1).
(ii) No subbasis(xγ)γ∈J, withcard(J) =ℵ1, is equivalent to the standard basis of`1(ℵ1).
(iii) Every element ofX∗has countable support.
PROOF. (i)⇒(ii). Obvious.
non-(iii)⇒non-(ii). LetJ := supp(x∗)be uncountable. From (2) it follows there exist an uncountable subsetJ0 of J andε0 > 0 such that|x∗(xγ)| > ε0 for allγ ∈ J0, whence we obtain (since the basis (xγ)γ∈Γ is unconditional) that the seriesP
γ∈J0x∗γ(x)converges unconditionally for everyx ∈ X and defines an elementx∗0ofX∗. PutW := [xγ]γ∈J0. For everyw∈Wof the formw=P
γ∈J0tγxγ we thus havex∗0(w) =P
γ∈J0tγ(unconditional convergence), and soP
γ∈J0|tγ|<∞. It follows that the basis of W and the standard basis of`1(J)are equivalent, and henceW is an isomorphic copy of`1(J0), withJ0
uncountable.
(iii) implies X has W*CP. We may assume that a subspaceY ofX is nonseparable. LetΛdenote the class{F ⊂X∗:Fis total overY}. By the Hahn-Banach theorem,χ∗(Y∗) = min{card(F) : F ∈Λ}= card(F0)for someF0∈Λ. Now we define two subsets ofΓ:
A= [
x∗∈F0
supp(x∗), ΓY = supp(Y).
From(iii)we obtain that
card(A)≤card(F0) =χ∗(Y∗), (3)
and from the defintion of the setsAandΓY we getx∗(PΓY\Ay) = 0, and hencex∗(PAy) = x∗(y)for everyy∈Y andx∗∈ F0. It follows that the setF0is total overPA(Y); thus the operatorPArestricted to Y is injective which, together with (3), gives
χ(Y)≤card(Y) = card(PA(Y)) = card(ΓY ∩A)≤χ∗(Y∗).
Finally,χ(Y) =χ∗(Y∗), as claimed.
If X has W*CP then (i) holds. Assume thatXcontains an isomorphic copy of`1(J)withcard(J) =ℵ1. The remaining part of the proof depends on the observation thatif W is a separable Banach space then for every infinite dimensional subspaceY ofW∗ we haveχ∗(Y∗) = ℵ0, which we apply to the space W =C[0,1]whose dual containsY :=`1([0,1]).
As a by-product of the equivalence of(i)and(ii)in Theorem 1 we obtain the Troyanski’s result(T)(see Introduction) which immediately gives
Corollary 1 LetX be a Banach space with an uncountable symmetric basis(xγ)γ∈Γ. ThenX has the W*CP if and only if the basis is not equivalent to the standard basis of`1(Γ).
The above Corollary applies to “big” Orlicz spaceshϕ(Γ), whereϕis an Orlicz function (for exact definition ofhϕ(Γ)see e.g. [10]), giving thathϕ(Γ)has the W*CP if and only ifϕis not equivalent to the linear functionψ(t) =tat0.
The next theorem is an immediate consequence of Theorem 1 and the result (RSG); it applies to the spaceshϕ(Γ), in particular to`p(Γ),1 < p < ∞, andc0(Γ)(cf. [5, 15]). It also complements a similar result obtained in [15, Proposition 2] forY reflexive.
Theorem 2 IfX contains no isomorphic copies of`1(ℵ1), then every nonseparable subspace Y ofX contains a set of the cardinality ofχ(Y)consisting of pairwise disjoint elements of norm one.
The proposition below deals with weak* sequential compactness of the dual unit ball ofX∗. The proof of the first implication is a discrete version of the proof given in 1968 by Lozanovskii [14] for a class of Banach lattices (cf. [23, Theorem 4.4]), and is included here for the convenience of the reader who is not familiar with the theory of Banach lattices (one should also note that the original proof works forreal Banach lattices).
Proposition 1 LetX be a Banach space with an uncountable unconditional basis. Then statement (iii) in Theorem 1 implies that
(iv) the dual unit ballBX∗ ofX∗is weak* sequentially compact.
Moreover, under the continuum hypothesis(CH)statements (i) and (iv) are equivalent.
PROOF. (iii)⇒(iv)Let(x∗n)be a sequence inBX∗, and putV :={x∗ ∈ X∗ : supp(x∗)⊂A}, where A=S∞
n=1supp(x∗n). We obviously haveV =PbA(X∗), andx∗n ∈V for alln’s. We setY :=PA(X).
SinceAis countable, the spaceY is separable. It is easy to check that the annihilatorY⊥ofY inX∗equals PbΓ\A(X∗), and henceY∗can be identified withPbA(X∗)(= V). The separability ofY implies that the ballBY∗ isσ(Y∗, Y)-sequentially compact, and using the above identification ofY∗andV, we can find a σ(X∗, X)-convergent subsequence(x∗n
k)of(x∗n).
(iv)⇒(i)(under CH; cf [23, pp. 78–79]). It is known that condition(iv) impliesX cannot contain isomorphic copies of`1(R), whereRdenotes the set of all real numbers (see e.g. [1, p. 226]), and hence, under CH, the spaceXcannot contain any copy of`1(ℵ1).
From Corollary 1 and Proposition 1 we immediately obtain
Corollary 2 Let the basis(xγ)γ∈ΓofXbe symmetric. Under the continuum hypothesis, the dual unit ball ofXis weak* sequentially compact if and only if the basis is not equivalent to the standard basis of`1(Γ).
3 Uncountable unconditional basic sets in X
In the theorem below we show that large unconditional basic sets inX have “nice” structure (the conclu- sion(i)below was obtained in [15, Proposition 6] under more restrictive assumption); its application is given in three corollaries following it.
Theorem 3 LetX be a Banach space with an uncountable unconditional basis, and let(yj)j∈J be an uncountable unconditional normalized basic set inX. Then the following alternative holds:
(i) There is a subsetJ0ofJ withcard(J0) = card(J)such that the elements of(yj)j∈J0are pairwise disjoint.
(ii) For every infinite cardinal numberα0 <card(J)there exists a subsetJ0ofJ withcard(J0)> α0
such that(yj)j∈J0 is equivalent to the unit vector basis of`1(J0).
In particular, the conclusion of part (i) holds if(yj)j∈J is equivalent to the unit vector basis ofc0(J)or
`p(J), with1< p <∞.
We would like to comment on the above property(ii)in Theorem 3. One should note that it is impossi- ble, in general, to choose a pairwise disjoint subsequence even from asequence(yn)inXequivalent to the unit vector basis of`1: it is enough to take anyγ0 ∈Γ\S∞
n=1supp(yn)and consider the sequence(y0n), equivalent to(yn), of the formyn0 =yn+xγ0,n= 1,2, . . .. On the other hand, it is known that a Banach space with an unconditional Schauder basis contains a copy of`1iff it contains a normalized block basic sequence of the basis equivalent to the unit vector basis of`1(see e.g. [13, Theorem 1.c.9]).
The proof of Theorem 3 depends essentially on Lemma 1 below and it is a modification of the arguments used in the proof of [2, Lemma 3]. To shorten the text we say that a family(yj)j∈Jof non-null elements of a Banach spaceW is totally non-`1(α), whereαis an infinite cardinal number withα≤card(J)(TN`1(α), for short) if, for every subsetCofJ withcard(C) = αthere is a family(tj)j∈C of scalars such that the seriesP
j∈Ctjyjconverges unconditionally, butP
j∈C|tj|=∞. (Forα=ℵ0this notion coincides with the notion of a totally non-`1family considered by Drewnowski in [2].) If(yj)j∈Jis a basic set inX then it is totally non-`1(α)whenever, for every subsetC ofJ withcard(C) = α, the basic set(yj)j∈C isnot equivalentto the standard basis of`1(C). We have that ifα1 < α2, then TN`1(α1)implies TN`1(α2);
thus, if(yj)j∈Jis totally non-`1then it is TN`1(α)for every infiniteα≤card(J).
Lemma 1 LetXbe a Banach space with an uncountable unconditional basis, letα0be an infinite cardinal number, and letJ be a set withcard(J)> α0. If, for every cardinalαwithα0 < α ≤card(J)a family (yj)j∈Jof non-null elements ofXisT N `1(α), then there exists a subsetJ0ofJwithcard(J0) = card(J) such that the elements of the subfamily(yj)j∈J0 are pairwise disjoint.
PROOF. It is an immediate consequence of the following combinatorial fact, the proof of which is similar to the proof of [2, Lemma 2] and therefore omitted:
LetJbe an uncountable set, and letmbe an infinite cardinal number withm<card(J). Let (Sj)j∈Jbe a family of subsets of a setΓsuch that:
(a)for everyj ∈Jwe havecard(Sj)≤m, and
(b)for everyγ∈Γwe havecard{j∈J :γ∈Sj} ≤m.
Then there exists a subsetJ0ofJwithcard(J0) = card(J)such that the elements of the family (Sj)j∈J0 are pairwise disjoint.
THE PROOF OFTHEOREM3. Assume condition(ii)is false. Then(yj)j∈Jis TN`1(α)for all cardinal numbersαwithα0< α≤card(J). Now we apply Lemma 1.
The two below corollaries of Theorem 3 show that uncountable unconditional basic sets in the spaces
`p(Γ)andc0(Γ)contain long symmetric subsets. (One should note here that subspaces with symmetric uncountable bases in Orlicz spaces`ϕ(Γ)were described by Rodriguez-Salinas [16]; see also [10].)
The first corollary is now obvious (the caseX =c0(Γ)andX =`p(Γ), with1< p <∞, was studied in [5] and [15], respectively).
Corollary 3 LetX be a Banach space with an uncountable unconditional basis, and let(yj)j∈J be an uncountable unconditional normalized basic set inX. If X contains no isomorphic copy of the space
`1(ℵ1), then there exists a subsetJ0ofJ withcard(J0) = card(J)such that the elements of(yj)j∈J0 are pairwise disjoint.
Each of the either cases of Theorem 3 proves the next corollary.
Corollary 4 Let(yj)j∈J be an uncountable unconditional normalized basic set in`1(Γ). Then for every infinite cardinal numberα0<card(J)there exists a subsetJ0ofJ withα0<card(J0)and such that the basic subset(yj)j∈J0 is equivalent to the natural symmetric basis of`1(J0).
It is known that every symmetric basic sequence in the sequence space`p (orc0) is equivalent to the unit vector basis of the given space [13, Remark following Proposition 3.b.5]. From Corollaries 3 and 4 we immediately obtain a similar property for the spaces`p(Γ)andc0(Γ).
Corollary 5 LetX(Γ)denote the space`p(Γ),1≤p <∞, orc0(Γ). Every uncountable, normalized and symmetric basic set(yj)j∈JinX(Γ)is equivalent to the natural basis ofX(J).
4 ε-disjoint systems in X
The main result of this section is motivated by the remark following Theorem 3 (see also the proof of Theorem 1 in [3]). Here we show that the structure of infinite dimensional subspaces ofX can also be studied effectively by the use of “almost” disjoint elements.
Letε∈[0,1), and letY be a subspace ofX. We say that two elementsy1, y2∈X\ {0}areε-disjoint if there exist disjoint elementsu1, u2∈X\ {0}such thatkxi−uik ≤ε,i= 1,2. A system(yj)j∈J⊂SY
is said to beε-disjoint provided that there exists a system(uj)j∈J of pairwise disjoint elements ofX with kyj−ujk ≤εfor allj∈J. A concreteε-disjoint system(yj)j∈Jwith the corresponding pairwise disjoint system(uj)j∈Jwill be denoted by(yj, uj)j∈J.
Remark 1 It is obvious that every0-disjoint system is pairwise disjoint.
Remark 2 From inequalitymin{|a|+|b|,|c|} ≤min{|a|,|c|}+ min{|b|,|c|}, for all scalarsa, b, c(see [18, Corollary, p. 53]), we easily obtain thatmin{|a|,|b|} ≤ 2|a−u|+|b−v|+ min{|u|,|v|}for all a, b, u, v. Hence, if the elementsyi = P
γ∈Γt(i)γ xγ,i = 1,2, areε-disjoint, with corresponding disjoint elementsui=P
γ∈Γs(i)γ xγ,i= 1,2, thenkP
γ∈Γmin{|tγ|,|sγ|}xγk ≤2Kky1−u1k+Kky2−u2k ≤ 3Kε, whereK is the basis constant. It follows that the supports of two ε-disjoint elements intersect at
“norm-min-small” subsets.
Remark 3 Letε∈(0,1/2), and let a system(yj, uj)j∈Jbeε-disjoint in`1(Γ). Then(yj)j∈Jis equivalent to the standard basis of`1(J), withkujk ∈(1−ε,1 +ε)for allj’s, and similarly for(yj)j∈J:
X
j∈J
|tj| ≥ kX
j∈J
tjyjk ≥(1−2ε)X
j∈J
|tj|,
for all(tj)j∈J ∈ `1(J). Proving as in [13, Proposition 1.a.9 and Theorem 2.a.3], we obtain that for ε∈(0,√
2−1), the spaces[yj]j∈Jand[uj]j∈Jare(1 +ε)-isometric andδ-complemented in`1(Γ), where δ≤1 +1−2ε−ε2ε 2.
The main result of this section reads as follows.
Theorem 4 LetY be an infinite dimensional subspace ofX. Then for everyε ∈ (0,1) the spaceY contains anε-disjoint system(yj, uj)j∈J withcard(J) = χ(Y)and such thatsupp(uj)⊂supp(yj)for allj∈J.
PROOF. PutF = supp(Y). We first consider the caseχ(Y) =ℵ0. ThenFis countable, and henceY is a subspace of the spaceXF with the countable unconditional basis(xγ)γ∈F. By [13, Proposition 1.a.11], Y contains anε-disjoint countable infinite system(yn, un)n≥1withsupp(un)⊂supp(yn)for alln’s.
Now assumeχ(Y)>ℵ0, and letEbe the class of allε-disjoint systems(yj, vj)j∈Jwithcard(J)≥ ℵ0
andsupp(uj)⊂supp(yj)for allj ∈J. By the previous case,E 6=∅. We introduce the following partial ordering inE: (y0j, u0j)j∈J (yl00, u00l)l∈L iffJ ⊂ Landy0j = yj00 andu0j = u00j for all j ∈ J, and let (yMj , uMj )j∈JM be a maximal element inE. We define the cardinal numberλM := card(JM), and we put IM :=S
j∈JMsupp(uMj ). Then we have
λM = card(IM)≤χ(Y). (4)
We claim we have two equalities in (4). Assume this is not so, i.e.,λM < χ(Y). Then we must have:
for everyη >0there isyη ∈SY withkPIMyηk< η (*) (in the oposite case the numberinfy∈SY kPIMykwere positive, and hence the operatorPIM restricted toY would be injective; this and (4) would then imply thatχ(Y) ≤card(Y) = card(PIM(Y)) = card(F∩ IM) ≤ λM < χ(Y), a contradiction). Now chooseyη ∈ SY fulfilling (*) withη = ε, and putwη = PF\IMyη; we see that supp(wη) ⊂ supp(yη). Next, from (*) we obtain kyη −wηk < ε, and since supp(wη)∩IM =∅, we also have that for everyj ∈ JM the elementsuη anduj are disjoint. It follows that for the setJη:=JM∪ {η}the system(yj, uj)j∈Jηisε-disjoint, and it strictly dominates(yj, uj)j∈J. This contradiction proves our claim and finishes the proof.
From Theorem 4 and Remark 3 we get
Corollary 6 Let Y be a nonseparable subspace of `1(Γ). Then for every ε ∈ (0,√
2−1) the space Y contains an(1 +ε)-isometric and δ-complemented copy of`1(J), wherecard(J) = χ(Y)and δ ≤ 1 +1−2ε−ε2ε 2.
Consequently, from Corollary 6 and Pełczy´nski’s decomposition method we obtain K¨othe’s result ([12, Theorem (6), p. 187]):
Corollary 7 Every complemented subspace of`1(Γ)is isomorphic to`1(J)for someJ ⊂Γ.
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Marek W´ojtowicz Carlos Finol
Instytut Matematyki Escuela de Matem´atica
Akademia Bydgoska Universidad Central de Venezuela
Pl. Weyssenhoffa 11 Caracas 1020
85-072 Bydgoszcz, Poland Venezuela