ISRAEL JOURNAL OF MATHEMATICS 184 (2011), 79–91 DOI: 10.1007/s11856-011-0060-2
UNIQUENESS OF UNCONDITIONAL BASES IN NONLOCALLY CONVEX c0 -PRODUCTS∗ BY
´noz F. Albiac∗∗ and C. Lera Departamento de Matem´ aticas, Universidad P´ ublica de Navarra Pamplona 31006, Spain e-mail:
[email protected] and
[email protected] We would like to dedicate this paper to the memory of Professor Nigel Kalton.
ABSTRACT
We show that the c0 -product (X ⊕ X ⊕ · · · ⊕ X ⊕ · · · )0 of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c0 -products of the nonlocally convex classical quasi-Banach spaces.
1. Introduction Suppose that X is a quasi-Banach space (in particular, a Banach space) with a quasi-norm · and a normalized unconditional basis (xn )∞ n=1 . The space X is said to have a unique unconditional basis (up to a permutation) if whenever ∞ (yn )∞ n=1 is another normalized unconditional basis of X, then (yn )n=1 is equivalent to (a permutation of) (xn )∞ n=1 , i.e., there exists an automorphism of X which takes one basis to (a permutation of) the other. ∗ The authors acknowledge support from the Spanish Research Grant Estructuras y
Complejidad en Espacios de Banach II, reference number MTM2010-20190-C0202. ∗∗ The first-named author was supported by the Spanish Research Grant Operadores, reticulos, y geometria de espacios de Banach, reference number MTM200802652/MTM. Received June 9, 2009
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Lindenstrauss and Zippin [15] proved in 1969 that the list of Banach spaces having a unique unconditional basis reduces to c0 , 1 , and 2 . The elegance of this result encouraged further research in this direction that culminated in 1985 with the work of Bourgain et al. [5]. The authors of this Memoir embarked on a comprehensive study aimed at classifying those Banach spaces with unique unconditional basis up to permutation. They showed that the spaces c0 (1 ), c0 (2 ), 1 (c0 ), and 1 (2 ) all have unique unconditional bases up to permutation, while 2 (1 ) and 2 (c0 ) do not. However, the hopes of attaining a satisfactory classification were shattered when they found a nonclassical Banach space, namely the 2-convexification of Tsirelson space T , having uniqueness of unconditional basis up to a permutation. Their work also left many open questions, most of which remain unsolved as of today. One of the questions that were raised in [5] was: does the space c0 (X) = (X ⊕ X ⊕ · · · ⊕ X ⊕ · · · )0 have a unique unconditional basis up to permutation whenever X does? ([5, Problem 11.1]). Casazza and Kalton [7] solved this problem in the negative in 1999 by showing that T has a unique unconditional basis up to permutation while c0 (T ) does not. They also gave a shorter proof of the uniqueness of unconditional bases up to permutation in both c0 (1 ) and 1 (c0 ) using that these two spaces are not “sufficiently Euclidean.” In this paper, we translate the above-mentioned problem from [5] into the nonlocally convex setting. We show that, at least for an ample class of quasi-Banach spaces X which are not sufficiently Euclidean, any complemented unconditional basic sequence of c0 (X) = (X ⊕ X ⊕ · · · ⊕ X ⊕ · · · )0 must be equivalent to a subset of the canonical basis of the space. As a consequence we obtain that c0 (X) has a unique unconditional basis up to permutation even without knowing whether X has a unique unconditional basis itself. Our arguments provide also an alternative proof of the fact that c0 (p ) has a unique unconditional basis up to permutation for 0 < p < 1; the original proof of this result in [13] was overly technical. We plan to tackle unconditional bases in p -products for 0 < p < 1 and complete the study of nonlocally convex 1 -products (see [4]) and 2 -products in later publications. For an up-to-date account on what is known on the subject of uniqueness of unconditional bases up to permutation in infinite direct sums of quasi-Banach spaces, see [2].
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Throughout this article we use standard Banach space theory terminology and notation, as may be found in [1, 12]. Other more specific references will be provided in context. 2. Preliminaries Suppose X is an infinite-dimensional quasi-Banach space. A basis (xn )∞ n=1 of X is said to be strongly absolute (see [11]) if, given ε > 0, there exists a constant Cε > 0 so that ∞ ∞ (2.1) |αn | ≤ Cε sup |αn | + ε αn xn , n
n=1
n=1
X
for any (αn )∞ n=1 ∈ c00 . If X is a quasi-Banach space with a separating dual, its Banach envelope ˆ (i.e., the smallest Banach space containing X, see [10]) will be denoted by X. It follows readily from (2.1) that if an infinite-dimensional quasi-Banach space ˆ is isomorphic to 1 . X has a strongly absolute basis, then X Quasi-Banach spaces with a strongly absolute basis are abundant and amongst them we find most of the nonlocally convex classical spaces, like the sequence spaces p , spaces of analytic functions Hp , and spaces of harmonic functions hp for 0 < p < 1, as well as many Lorentz and Orlicz sequence spaces. All these spaces have a 1-unconditional basis which induces a p-convex lattice structure for some p > 0. Recall that a quasi-Banach lattice X is said to be p-convex, where 0 < p < ∞, if there is a constant M such that for any {yi }ni=1 in X and n ∈ N we have n n 1/p 1/p |yi |p yi p . (2.2) ≤M i=1
n
i=1
The procedure to define the element i=1 |yi |p )1/p is exactly the same as for Banach lattices ([14]). If a quasi-Banach space X is isomorphic to a closed subspace of a p-convex quasi-Banach lattice, then it is called natural; if X is also a lattice then it is r-convex for some r > 0 (see [9]). The other notion that we will need is that of a lattice anti-Euclidean quasiBanach lattice X. A quasi-Banach lattice is called sufficiently lattice Euclidean if there is a constant Γ so that for any n there are operators S : X −→ n2 and T : n2 −→ X so that ST = In2 and ST ≤ Γ, and S is a lattice homomorphism. This is equivalent to asking that 2 is finitely representable as a
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complemented sublattice of X. We will say that X is lattice anti-Euclidean if it is not sufficiently lattice Euclidean. We refer the reader to [6] for more details on this definition and the role it has played in the classification of complemented basic sequences in Banach lattices which are lattice anti-Euclidean. Our proofs hinge on the next Lemma, which summarizes several results and ideas contained in [3], mainly Corollary 3.6. Theorem 2.1: Let Y and Z be quasi-Banach sequence spaces. Suppose Z is p-convex for some p > 0 and that Y is isomorphic to a complemented subspace of Z. Suppose Yˆ is lattice anti-Euclidean. Then there exists N ∈ N such that Y is a lattice isomorphic to a complemented sublattice of Z N . More specifically, there exists a complemented disjoint positive sequence (vn ) in Z N equivalent to the unit vector basis (un ) in Y . Furthermore, the projection P of Z N onto [vn ] may be given in the form P (z) =
∞
vn∗ (z)vn
n=1
where vn∗ ≥ 0 and supp (vn∗ ) ⊆ supp (vn ) for all n. Another technique that has become crucial to determine the uniqueness of unconditional basis in quasi-Banach spaces is the so-called “large coefficient technique.” It was introduced by Kalton in [8] to prove the uniqueness of unconditional basis in nonlocally convex Orlicz sequence spaces, and was extended to the framework of quasi-Banach lattices in [11]. Lemma 2.2 ([11, Theorem 2.3]): Let Z be a p-convex quasi-Banach lattice (0 < p < 1) with normalized unconditional basis (en )∞ n=1 and let Y be a complemented subspace of Z with a normalized unconditional basis (un )n∈ S (S ⊆ N). ∗ Let (e∗n )∞ n=1 and (un )n∈ S be the sequences of biorthogonal linear functionals associated to (en )∞ n=1 and (un )n∈ S , respectively. Suppose that there is a constant ν > 0 and an injective map σ : S −→ N so that |e∗σ(n) (un )u∗n (eσ(n) )| > ν for all n ∈ S. Then, the basic sequences (un )n∈S and (eσ(n) )n∈S are equivalent. The following generalization of Lemma 2.2 will be also used. The proofs are similar.
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Lemma 2.3: Let Z be a p-convex quasi-Banach lattice (0 < p < 1) with normalized unconditional basis (en )∞ n=1 and let Y be a complemented subspace with a normalized unconditional basis (un )n∈ S (S ⊆ N) so that supp (u∗n ) ⊆ supp (un ) and the sets supp (un ) are disjoint for all n ∈ S. Suppose that there is a constant ν > 0 (independent of n) such that to each n ∈ S corresponds a subset Tn ⊆ supp (un ) for which ∗ ∗ ek (un )un (ek ) > ν. k∈Tn
Then, the basic sequence vn =
e∗k (un )ek
(n ∈ S)
k∈Tn
is equivalent to (un )n∈S and the subspace [vn ]n∈S is complemented in Z. Furthermore, supp (vn∗ ) ⊆ supp (vn ), and vn∗ ≥ 0 if u∗n ≥ 0 (n ∈ S). Finally, we remark that there is an important Cantor–Bernstein-type principle which helps determine whether two unconditional bases are permutatively equivalent. We will use this principle in the form in which it was reinterpreted by Wojtaszczyk in [16, Proposition 2.11]. ∞ Proposition 2.4: Suppose (un )∞ n=1 and (vn )n=1 are two unconditional basic sequences of a quasi-Banach space X. Then (un ) and (vn ) are permutatively equivalent if and only if (un ) is equivalent to a permutation of a subbasis of (vn ) and (vn ) is equivalent up to permutation to a subbasis of (un ).
3. Uniqueness of unconditional basis in c0 (X) Throughout this section (X, · X ) will be a quasi-Banach space with a normalized strongly absolute 1-unconditional basis (xk )∞ k=1 which induces on X a p-convex lattice structure for some 0 < p < 1. In particular, equation (2.2) yields the existence of a constant M such that n n yi ≤ M yi X , i=1
for any Let
{yi }N i=1
X
i=1
⊂ X.
∞ c0 (X) = {z = (zl )∞ l=1 : zl ∈ X for each l and (zl X )l=1 ∈ c0 }
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endowed with the quasi-norm z = sup zl X . l∈N
∞
For each l ∈ N, we can write zl = k=1 αl,k xk , and then identify c0 (X) with the space of infinite real matrices A = (αl,k )∞ l,k=1 such that ∞ αl,k xk < ∞. A = sup l∈N k=1
X
The space c0 (X) has a canonical 1-unconditional basis that will be denoted ∞ by (el,k )∞ l,k=1 so that the lattice structure induced by (el,k )l,k=1 in c0 (X) is p-convex. ˆ ·c ) of (c0 (X), ·) is isomorphic to (c0 (1 ), · The Banach envelope (c0 (X), 1 ). More precisely, if we let C1 be the strongly absolute constant of (xk )∞ k=1 for ε = 1 in equation (2.1), we have
∞
∞
k=1
k=1
1 sup |αl,k | ≤ zc ≤ sup |αl,k |, (C1 + 1) l∈N l∈N
for any z = l,k αl,k el,k ∈ c0 (X). In turn, the dual of c0 (X) is isomorphic to 1 (∞ ) since for any f ∈ (c0 (X))∗ , ∞ l=1
sup |f (el,k )| ≤ f ≤ (C1 + 1) k∈N
∞ l=1
sup |f (el,k )|. k∈N
Suppose Q is a bounded linear projection from c0 (X) onto a subspace Y with normalized unconditional basis (un )n∈S ; the cardinality of S can be finite or infinite. We will denote by (e∗l,k )l,k∈N and (u∗n )n∈S the sequences in 1 (X ∗ ) of the biorthogonal linear functionals associated to (el,k )l,k∈N and (un )n∈S , for which ∞ z= e∗l,k (z)el,k and Q(z) = u∗n (z)un , n∈S
l,k=1
for all z ∈ c0 (X). Also, for each n ∈ S we can write un =
∞
e∗l,k (un )el,k ,
l,k=1
and u∗n =
∞ l,k=1
u∗n (el,k )e∗l,k ,
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where the convergence of this last series is understood in the weak∗ -sense. Then, we have (3.3)
u∗n ≤ KQ (n ∈ S).
We also recall that (un )n∈S is a K-unconditional basis of Yˆ , the Banach envelope ˆ and from (3.3) we easily obtain of Y , which is complemented in c0 (X), (QK)−1 ≤ un c ≤ 1 (n ∈ S). We aim to prove the following theorem. Theorem 3.1: Let Q be a bounded linear projection from c0 (X) onto a subspace Y with a normalized K-unconditional basis (un )n∈S . Then (un )n∈S is equivalent to a permutation of a subbasis of the canonical basis (el,k )∞ l,k=1 . To that end we need to establish a few reduction lemmas that will allow us to unravel the form in which any complemented unconditional basic sequence in c0 (X) can be written in terms of the canonical basis of the space. Let us introduce first the notation we will be using in the sequel. Given ε = 1/4KQ, the strong absoluteness of the basis (xk )∞ k=1 of X yields a constant C1/4KQ that we will denote simply by C. Put δ0 =
1 . 4CKQ
For each n ∈ S we single out the sets Tn = {(l, k) ∈ supp (un ) : |e∗l,k (un )| > δ0 } and Ωn = {l : (l, k) ∈ Tn for some k}. Let us start estimating, for each vector un , the “size” of the coefficients e∗l,k (un ) that remain outside the set Tn for each row l. Lemma 3.2: For each n ∈ S and l ∈ N we have k (l,k)∈Tn
|e∗l,k (un )| ≤
1 . 2KQ
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Proof. The strong absoluteness of (xk )∞ k=1 yields
|e∗l,k (un )| ≤ C
k (l,k)∈Tn
≤C
sup |e∗l,k (un )| +
k (l,k)∈Tn
1 ∗ el,k (un )xk 4K X k (l,k)∈Tn
1 1 1 + un = . 4CKQ 4KQ 2KQ
We continue with an important simplification on the support of the unconditional bases of complemented subspaces of c0 (X). Lemma 3.3: Under the hypotheses of Theorem 3.1, we can assume (by taking a sequence equivalent to (un )n∈S ) that for each n ∈ S, (i) e∗l,k (un ) ≥ 0 and u∗n (el,k ) ≥ 0 for all l, k ∈ N; (ii) for each l ∈ Ωn there exists kn (l) such that (l, kn (l)) ∈ Tn and kn (l) = km (l) whenever n = m and l ∈ Ωn ∩ Ωm ; (iii) supp (u∗n ) ⊆ supp (un ) = {(l, kn (l)), l ∈ Ωn } ⊆ Tn . In particular, e∗l,kn (l) (un ) > δ0 for all (l, kn (l)) ∈ supp (un ). Proof. From Theorem 2.1 we can assume that e∗l,k (un ) ≥ 0 and u∗n (el,k ) ≥ 0 for all n ∈ S and l, k ∈ N, that supp (u∗n ) ⊆ supp (un ) for all n ∈ S, and that the sets supp (un ) are disjoint. Notice that the conditions of Theorem 2.1 are satisfied because, since X is p-convex for some p > 0, Z = c0 (X) is p-convex for that p, and also c0 (X) = Z ≈ Z N for any N ∈ N. Furthermore, c0 (1 ) is lattice anti-Euclidean ([7, Corollary 2.5]) and so are all its complemented subspaces. For each n ∈ S, an elementary duality arguments gives ∗ ∗ ∗ ∗ el,k (un )un (el,k ) ≤ sup un (el,k ) sup el,k (un ) l
(l,k)∈Tn
≤ u∗n
k (l,k)∈Tn
l
k (l,k)∈Tn
1 1 ≤ , 2KQ 2
hence (l,k)∈Tn
e∗l,k (un )u∗n (el,k ) =
e∗l,k (un )u∗n (el,k ) −
l,k∈N
1 1 ≥1− ≥ . 2 2
(l,k)∈Tn
e∗l,k (un )u∗n (el,k )
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UNIQUENESS OF UNCONDITIONAL BASES IN c0 (X)
Moreover, for each n ∈ S, ∗ un ≥ el,k (un )el,k sup > δ0 l∈Ω
n
(l,k)∈Tn
≥
87
xk k (l,k)∈Tn
X
δ0 sup |{k : (l, k) ∈ Tn }|, C1 + 1 l∈Ωn
where C1 is the 1-strongly absolute constant of the basis (xk )∞ k=1 of X. Thus the size of the set {k : (l, k) ∈ Tn } is uniformly bounded by (C1 + 1)/δ0 , and for each n ∈ S and l ∈ Ωn , we can choose kn (l) such that (l, kn (l)) ∈ supp (un ) and δ0 . e∗l,kn (l) (un )u∗n (el,kn (l) ) ≥ 2(C1 + 1) l∈Ωn
The result now follows from Lemma 2.3. Since the supp (un )’s are disjoint, we obtain that kn (l) = km (l) whenever n = m and l ∈ Ωn ∩ Ωm . Lemma 3.4: There exists an integer N0 such that for any collection of basis vectors {xkj }N j=1 of X with N ≥ N0 we have N 1 2(C1 + 1)QK xkj > . N j=1 δ0 X
Proof. Given ε > 0, using the the strong-absoluteness of the basis (xk )∞ k=1 we obtain a constant Cε such that N 1 Cε 1 ≤ + xkj , ε εN N j=1
X
for every N ∈ N. We are now ready to prove Theorem 3.1. Proof of Theorem 3.1. We assume that (un )n∈S satisfies the three properties of Lemma 3.3. We are going to cut up S in two disjoint subsets according to the size of the coefficients of the biorthogonal functionals (u∗n )n∈S . For that, let us fix δ0 , δ1 = 2(C1 + 1)N0 and put
A=
n ∈ S : sup u∗n (el,kn (l) ) > δ1 l∈Ωn
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and B = S \ A. First notice that by Lemma 2.2, (un )n∈A is equivalent to a subbasis of (el,k )∞ l,k=1 . We turn to (un )n∈B , which we show is equivalent to the canonical c0 -basis. Suppose there exists C ⊆ B such that n∈C Ωn = ∅. We will pick different n1 , . . . , nN in C and disjoint sets ω1 ⊆ Ωn1 , . . . , ωN ⊆ ΩnN so that δ1 <
l∈ωk
u∗nj (el,knj (l) ) ≤ 2δ1 ,
for 1 ≤ j ≤ N . The argument that allows us to do that goes as follows: • Take n1 ∈ C and ω1 ⊆ Ωn1 such that δ1 < l∈ω1 u∗n1 (el,kn1 (l) ) ≤ 2δ1 and l∈ω1 u∗n (el,kn (l) ) ≤ 2δ1 for all n ∈ C \ {n1 }. • Next take n2 ∈ C \ {n1 }, and ω2 ⊆ Ωn2 disjoint with ω1 , such that ∗ ∗ δ1 < l∈ω2 un2 (el,kn2 (l) ) ≤ 2δ1 and l∈ω2 un (el,kn (l) ) ≤ 2δ1 for all n ∈ C \ {n1 , n2 }. • · · · And so on. Since for all n ∈ S, l∈Ωk
u∗n (el,kn (l) ) ≥
1 2δ1 N0 u∗n ≥ = ≥ 2δ1 N0 , C1 + 1 C1 + 1 δ0
we can repeat the above steps up to N = min{|C|, N1 }, with N1 ≥ N0 , times. If l0 ∈ n∈C Ωn , then N N Q ≥ Q el,knj (l) = u∗nj (el,knj (l) ) unj j=1 l∈ωj j=1 l∈ωj N δ1 > unj K j=1 N δ0 δ1 xknj (l0 ) ≥ K j=1 X N δ0 1 = xknj (l0 ) , 2(C1 + 1)K N0 j=1
and we conclude that N = |C| < N0 .
X
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Now, given any finitely nonzero scalars (an )n∈B we have 1/p ∗ a u a e (u )e sup |an |, = n n n l,kn (l) n l,kn (l) ≤ M N0 n∈B
and, since
n∈B
n∈B l∈Ωn
1 sup |an | ≤ an u n , K n∈B n∈B
we conclude that (un )n∈B is equivalent to the unit vector basis of c0 . Corollary 3.5: Let X be a natural quasi-Banach space with strongly absolute unconditional basis (xk )∞ k=1 . For each l ∈ N, let Xl = [xk ]k∈Nl with |Nl | ≤ ∞. Then any unconditional basis (un )∞ n=1 of a complemented subspace ∞ of c0 (Xl )l=1 = (X1 ⊕ X2 ⊕ · · · ⊕ Xl ⊕ · · · )0 is equivalent to a permutation of a subbasis of the canonical basis of c0 (Xl )∞ l=1 . Proof. We just note that c0 (Xl )∞ l=1 is a complemented subspace of c0 (X), which makes it lattice anti-Euclidean. Now the proof would follow the same steps as the proof of Theorem 3.1 and its preparatory lemmas, with exactly the same constants. We omit the details. Theorem 3.6: Suppose that X is a natural quasi-Banach space with stronglyabsolute unconditional basis. Then, the space c0 (X) has a unique unconditional basis up to permutation. Proof. Suppose that (un )∞ n=1 is a normalized unconditional basis of c0 (X). By ∞ Theorem 3.1, (un )n=1 is equivalent to a permutation of a subbasis (el,k )(l,k)∈M of (el,k )∞ l,k=1 , the canonical basis of c0 (X). In order to obtain the equivalence ∞ up to permutation of (un )∞ n=1 and (el,k )l,k=1 , appealing to Proposition 2.4 it suffices to show the converse, i.e., that (el,k )∞ l,k=1 is permutatively equivalent to a subbasis of (el,k )(l,k)∈M . Clearly, (el,k )(l,k)∈M is the canonical basis of a space c0 (Xl )∞ l=1 , where each Xl = [xk ]k∈Nl and Nl is a subset of integers of cardinality |Nl | ≤ ∞. Since ∞ ∞ c0 (Xl )∞ l=1 is isomorphic to c0 (X), there exists a basis (vn )n=1 in c0 (Xl )l=1 equiv∞ alent to (el,k )∞ l,k=1 . Now, Corollary 3.5 yields that (vn )n=1 must be equivalent to a permutation of a subbasis of (el,k )(l,k)∈M and the proof is over. Corollary 3.7: The following spaces have a unique unconditional basis up to permutation:
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(i) c0 (p ) for 0 < p < 1; (ii) c0 (kpn ), where (kn )∞ n=1 is any increasing sequence of positive integers and 0 < p < 1; (iii) c0 (p (q )) for 0 < p, q < 1; (iv) c0 (p (kq n )), where (kn )∞ n=1 is any increasing sequence of positive integers and 0 < p < 1; (v) c0 (Hp ) for 0 < p < 1; (vi) c0 (hp ) for 0 < p < 1; (vii) c0 (d(w, p)) for 0 < p < 1, where d(w, p) is a Lorentz sequence space1 1/p for which w = (wn )∞ /n = ∞; n=1 satisfies limn→∞ (w1 + · · · + wn ) (viii) c0 (F ), where F is a nonlocally convex Orlicz sequence space2 for which limx→0 F (x)/x = ∞. Proof. As the reader can check in a straighforward manner, the canonical basis of each of the spaces whose infinite c0 -product are considered in (i)–(viii) is strongly absolute. Remark 3.8: There are nonlocally convex quasi-Banach spaces X whose Banach envelope is isomorphic to 1 but c0 (X) does not have unique unconditional basis up to permutation. Indeed, let X = F be a nonlocally convex Orlicz sequence space such that F ≈ F ⊕ 1 . Kalton gave in [8, Theorem 7.5] a sufficient condition on F for this to happen, which implies that F does not have a unique unconditional basis up to permutation. Since c0 (F ) ≈ c0 (F ⊕ 1 ), we deduce that c0 (F ) cannot have a unique unconditional basis up to permutation. 1 If w = (wn ) n∈N ∈ ∞ \ 1 is a decreasing nonnegative sequence and 0 < p < ∞, the Lorentz space d(w, p) is defined to be the space of all sequences a = (an )n∈N such that aw,p = sup
π∈Π
∞
1/p |aπ(n) |p wn
< ∞,
n=1
where Π is the group of permutations of N. 2 If F (x) is a nondegenerate Orlicz function satisfying the Δ2 -condition, the Orlicz sequence space F is defined to be the space of all sequences (an )n∈N such that ∞ n=1
F (|an |) < ∞.
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References [1] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006, [2] F. Albiac, N. J. Kalton and C. Ler´ anoz, Uniqueness of unconditional bases in quasiBanach spaces, in Trends in Banach Spaces and Operator Theory, Memphis, TN, (2001), Contemporary Mathematics V. 321, Amer. Math. Soc. Providence, RI, 2003, pp. 15–27. [3] F. Albiac, N. J. Kalton and C. Ler´ anoz, Uniqueness of the unconditional basis of l1 (lp ) and lp (l1 ), 0 < p < 1, Positivity 8 (2004), 443–454. [4] F. Albiac and C. Ler´ anoz, Uniqueness of unconditional bases in nonlocally convex 1 products, Journal of Mathematical Analysis and Applications 374 (2011), 394–401. [5] J. Bourgain, P. G. Casazza, J. Lindenstrauss and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Memoires of the American Mathematical Society 54 (1985). [6] P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in Banach spaces, Israel Journal of Mathematics 103 (1998), 141–175. [7] P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in c0 -products, Studia Mathematica 133 (1999), 275–294. [8] N. J. Kalton, Orlicz sequence spaces without local convexity, Mathematical Proceedings of the Cambridge Philosophical Society 81 (1977), 253–277. [9] N. J. Kalton, Convexity conditions on non-locally convex lattices, Glasgow Mathematical Journal 25 (1984), 141–152. [10] N. J. Kalton, Banach envelopes of non-locally convex spaces, Canadian Journal of Mathematics 38 (1986), 65–86. [11] N. J. Kalton, C. Ler´ anoz and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel Journal of Mathematics 72 (1990), 299–311. [12] N. J. Kalton, N. T. Peck and J. W. Roberts, An F -space sampler, London Mathematical Society Lecture Note Series, Vol. 89, Cambridge University Press, Cambridge, 1984. [13] C. Ler´ anoz, Uniqueness of unconditional bases of c0 (lp ), 0 < p < 1, Studia Mathematica 102 (1992), 193–207. [14] J. Lindenstrauss and L. Tzafriri, ‘Function spaces’, Classical Banach Spaces. II, Vol. 97, Springer-Verlag, Berlin, 1979. [15] J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, Journal of Functional Analysis 3 (1969), 115–125. [16] P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, II, Israel Journal of Mathematics 97 (1997), 253–280.