# Norm Attaining Arens Extensions onℓ1

Authors: Javier Falcó; Domingo García; Manuel Maestre; Pilar Rueda
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form onℓ1 with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.


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## Body

## 1. Introduction

The Bishop-Phelps theorem [1] states that the set of norm attaining forms on a real or complex Banach space is norm dense in the set of linear and continuous forms. Bishop and Phelps raised the question of extending their results to operators between Banach spaces. This question was answered in the negative by Lindenstrauss in his seminal paper [2], where he gave an example of a Banach space X such that the identity mapping on X cannot be approximated by norm attaining operators. However, if one considers the adjoint T*:Y*→X* of an operator T:X→Y between Banach spaces, given by T*(y*)(x)=y*(T(x)), for all x∈X, y*∈Y*, Lindenstrauss proved the denseness of those operators whose second adjoints attain their norms.The theory of norm attaining operators has spread to the nonlinear setting. The denseness of the set of norm attaining multilinear mappings has been deeply studied in the last decades. Assuming the Radon-Nikodým property, this density has been established for multilinear forms (see [3]). However, a general result for multilinear mappings cannot be expected. The first counterexample was given in [4] for bilinear forms. Based on Lindenstrauss result and making use of the Arens extensions to the second duals (see next section for the definitions), Acosta [5] proved a Lindenstrauss type result for bilinear forms whose third Arens transpose attains its norm. Afterwards, in [6] the denseness of bilinear forms whose Arens extensions to the biduals attain their norms at the same point was established. It is worth mentioning that in [6, Example 2] an example of a bilinear mapping is given such that only one of their Arens extensions attains its norm. This asymmetry between the two Arens extensions reveals the importance of the stronger condition of attaining their norms simultaneously. The generalization of Lindenstrauss result to n-linear vector-valued mappings was finally obtained in [7] in its strongest form; that is, the space formed by those n-linear mappings whose Arens extensions attain their norms simultaneously at the same point is dense in the space of all n-linear mappings.The aim of this paper is to study the norm attaining properties of the Arens extensions of multilinear forms onℓ1. On one hand, inspired by [6, Example 2], several examples of multilinear forms whose extensions suffer different kinds of asymmetries from the point of view of norm attainment are provided. These examples are built using multilinear forms on ℓ1, which is the classical example of a non-Arens regular Banach space. For instance, if we fix a priori some of the Arens extensions, we can construct a multilinear form on ℓ1 with the property that only these extensions attain their norms. Moreover, by undertaking a detailed study of the procedure used to generate such examples, we also get examples with stronger properties that allow a better understanding of the norm attaining behavior of the Arens extensions. These examples are presented as general results on existence of multilinear forms that fulfill the required norm attaining properties. On the other hand, we also deal with general Banach spaces and study when Arens extensions attain their norms in terms of convergence of sequences.The paper is organized as follows. Next section is devoted to fix the notation and to recall some of the basics on Arens extensions. Section3 is involved with the norm attaining behavior of the Arens extensions of multilinear forms on general Banach spaces. We prove that if an extension of a multilinear form attains its norm at a point then the norm is achieved just considering sequential limits. As a converse, we prove that if the norm of an extension is achieved with limits of subsequences of a normalized Schauder basis, then such extension attains its norm at a point whose coordinates are in the bidual. In Section 4 we deal with multilinear forms on ℓ1. We strengthen the results from the former section by proving a characterization of norm attaining extensions of bilinear forms at points with coordinates in Bℓ1**∖ℓ1 in terms of sequential limits of the images of subsequences of the canonical sequence {en}n=1∞. It is also proved that, fixing a number of Arens extensions, there exists an n-linear form on (ℓ1)n of norm one such that only these extensions fixed a priori are norm attaining. Finally, we show that such n-linear forms are dense in the set of all n-linear forms of norm one that fulfill a condition in terms of sequential limits.


## 2. Background and Notation

In this paperX, Y, X1, …, Xn are real or complex Banach spaces. Let ℒ(X1,…,Xn) denote the space of continuous n-linear forms A from X1×⋯×Xn into 𝕂 (𝕂=ℝ or ℂ) with the usual norm
(1)∥A∥=sup⁡⁡{|A(x1,…,xn)|:xi∈Xi,hhhhhh∥xi∥≤1,i=1,…,n}.
When X1=⋯=Xn=X, we just write ℒ(nX). We denote by BX the closed unit ball of X, by SX the unit sphere, by X* the strong dual, and by X** the bidual of X.We say thatA isnorm attaining (or A attains its norm) if there exist xi∈Xi,∥xi∥=1, i=1,…,n, such that ∥A∥=|A(x1,…,xn)|.Arens [8] found a natural way to extend a continuous bilinear mapping A:X1×X2↦Y to a continuous bilinear mapping from X1**×X2** into Y**. His method consists in applying three times the operation defined as
(2)At:Y*×X1⟼X2*,(y*,x1)⟿At(y*,x1)(x2)=y*(A(x1,x2)),x1∈X1, x2∈X2, and y*∈Y*. The first extension is defined as Attt:X**×Y**↦Z** and the second one is ATtttT, where BT(x1,x2)=B(x2,x1) for any bilinear mapping B. These extensions, which are in general different, are known as Arens products. This procedure was generalized by Aron and Berner [9] to arbitrary multilinear mappings.For our purposes we will use an alternative approach due to Davie and Gamelin [10]. The key of such approach is Goldstine theorem as it is based on limits in the weak-startopology, denoted by w(X**,X*). Consider Σn the group of all permutations of the set {1,…,n}. Given σ∈Σn they defined the extension Aσ associated with σ of an n-linear form A defined on X1×⋯×Xn, by
(3)Aσ(x1**,…,xn**)=lim⁡dσ(1)⋯lim⁡dσ(n)A(xd1,…,xdn),
where {xdi}di is a bounded net (∥xdi∥≤∥xi**∥, for all di) w(X*,X) convergent to xi**∈Xi**, for i=1,…,n. The mapping Aσ is called anArens extension of A and the n! Arens extensions may be different from each other. When convenient, we will write Aσ(1),…,σ(n) instead of Aσ. In particular, for n=2, AId=A1,2=Attt and A2,1=ATtttT, where Id is the identity permutation of the set {1,2}.Note that the use of thew(X*,X) topology prevents us in general from using sequences in the above limits. However we will show that in the study of norm attaining multilinear forms one can reduce such iterated limits to sequential ones.In [7] Lindenstrauss theorem is extended to multilinear forms by using the Arens extensions.Theorem 1 ([7, Theorem 2.1]).
LetXi be Banach spaces (1≤i≤n). Then the set of n-linear forms on X1×⋯×Xn such that all their Arens extensions to X1**×⋯×Xn** attain their norms at the same n-tuple is dense in the space ℒ(X1,…,Xn).Letℓ1 denote the space of all absolutely summing sequences in 𝕂 with its usual norm. In [6] the following example is provided. It illustrates that, although all Arens extensions have the same norm, the fact that one of them attains its norm does not imply that the other extensions should attain their norms too. More precisely, it shows a bounded bilinear form A whose first extension A1,2 is not norm attaining, whereas the second one A2,1 is norm attaining. This example brings into relief that the extensions of a bilinear form may have different behaviors from the point of view of attaining their norms and is the core of our study.Example 2 ([6, Example 2]).
The bilinear formA∈ℒ(2ℓ1), defined by
(4)A(x1,x2):=∑t1=1∞‍x1(t1)(∑t2=1t1‍t2t2+1x2(t2)),
is such that neither A nor AId=A1,2 is norm attaining, but A2,1 is norm attaining.


## 3. Norm Attaining Extensions of Multilinear Forms on General Banach Spaces

It is well known that, under the first axiom of separability, nets can be replaced with sequences, which turns out to be an advantage when dealing with limits. Our first result is just a lemma that will clarify how to pass from nets to sequences in the context of several indexes that will be helpful in the context of multilinear mappings. We give the proof for the sake of completeness.Lemma 3.
Letn∈ℕ. For each j=1,…,n, let Dj be an infinite directed set. Consider a family {aα1,…,αn}(α1,…,αn)∈D1×⋯×Dn of real or complex numbers. If the iterated limit a:=lim⁡α1∈D1⋯lim⁡αn∈Dnaα1,…,αn is finite then there exist strictly increasing sequences {αj(m)}m=1∞ in Dj, 1≤j≤n, such that lim⁡m1→∞⋯lim⁡mn→∞aα1(m1),…,αn(mn)=a.Proof.
We proceed by induction onn. For n=1, since lim⁡α1∈D1aα1=a, for each k∈ℕ there exists α1(k)∈D1 such that |aα1-a|<1/k for all α1≥α1(k). Besides, by the condition on D1, we can choose the sequence {α1(k)}k∈ℕ strictly increasing.
Assume that the result is true forn-1 and let us prove it for n. So, if we assume that a=lim⁡α1∈D1⋯lim⁡αn∈Dnaα1,…,αn is finite, define
(5)bα1,…,αn-1:=lim⁡αn∈Dnaα1,…,αn-1,αn.
By the assumption applied to the family{bα1,…,αn-1}(α1,…,αn-1)∈D1×⋯×Dn-1, for each j=1, …, n-1 there exists a strictly increasing sequence {αj(mj)}mj∈ℕ such that
(6)a=lim⁡m1→∞⋯lim⁡mn-1→∞bα1(m1),…,αn-1(mn-1).
Let us construct the sequence {αn(k)}k∈ℕ by induction on k.
Sincebα1(1),…,αn-1(1)=lim⁡αn∈Dnaα1(1),…,αn-1(1),αn, there exists αn(1)∈Dn such that
(7)|bα1(1),…,αn-1(1)-aα1(1),…,αn-1(1),αn|<1
for all αn≥αn(1). Assume that we have found αn(1), …, αn(k-1)∈Dn with αn(1)<⋯<αn(k-1) such that |bα1(m1),…,αn-1(mn-1)-aα1(m1),…,αn-1(mn-1),αn|<(1/l) for all αn≥αn(l), all 1≤m1,…,mn-1≤l, and all l=1, …, k-1.
Fix1≤m1, …, mn-1≤k. Since bα1(m1),…,αn-1(mn-1)=lim⁡αn∈Dnaα1(m1),…,αn-1(mn-1),αn, there exists αn(m1,…,mn-1)∈Dn, with αn(m1,…,mn-1)≥αn(k-1), such that
(8)|bα1(m1),…,αn-1(mn-1)-aα1(m1),…,αn-1(mn-1),αn|<1k
for all αn≥αn(m1,…,mn-1). Take αn(k)>αn(m1,…,mn-1) for all 1≤m1, …, mn-1≤k. Then
(9)|bα1(m1),…,αn-1(mn-1)-aα1(m1),…,αn-1(mn-1),αn|<1k
whenever αn≥αn(k). Hence the limit lim⁡k→∞aα1(m1),…,αn-1(mn-1),αn(k) exists and is equal to bα1(m1),…,αn-1(mn-1). Now,
(10)a=lim⁡m1→∞⋯lim⁡mn-1→∞bα1(m1),…,αn-1(mn-1)=lim⁡m1→∞⋯lim⁡mn-1→∞lim⁡mn→∞aα1(m1),…,αn-1(mn-1),αn(mn)
and the proof is over.Theorem 4.
LetX1,…,Xn be infinite dimensional Banach spaces, C∈ℒ(X1,…,Xn), and σ∈Σn. If the extension Cσ attains its norm then there exist sequences {xm11}m1=1∞,…,{xmnn}mn=1∞ with each xmkk∈BXk, mk∈ℕ, and k=1,…,n, such that
(11)lim⁡mσ(1)→∞…lim⁡mσ(n)→∞|C(xm11,…,xmnn)|=∥C∥.Proof.
For simplicity we assume thatσ=Id. Let (x1**,…,xn**) be a point in BX1**×⋯×BXn**, where Cσ attains its norm. Let K={k:xk**∈X**∖X}. By density, each xk** is the weak-star limit of a net {xαkk}αk∈Dk in BXk, k∈K. For k∉K, set Dk=ℕ and xαkk:=xk**∈X for all αk∈Dk. Then
(12)∥C∥=∥Cσ∥=|Cσ(x1**,…,xn**)|=lim⁡α1∈D1⋯lim⁡αn∈Dn|C(xα11,…,xαnn)|.
By Lemma 3 applied to aα1,…,αn:=|C(xα11,…,xαnn)|, we obtain the desired sequences {xmkk}mk=1∞, for every 1≤k≤n.Proposition 5.
LetX be an infinite dimensional Banach space, and let {xn}n=1∞ be a basic sequence. Then, any nonzero weak-star cluster point of {xn}n=1∞ belongs to X**∖X.Proof.
LetZ be the closed linear span of {xn}n=1∞ and let {xn*}n=1∞ be the orthogonal functionals in Z* associated with {xn}n=1∞. By the Hahn-Banach extension theorem, we can consider each xn* in X*.
Letx**∈X** be a nonzero cluster point of {xn}n=1∞, and let {xd}d∈D be a subnet of {xn}n=1∞ weak-star converging to x**.
We first prove thatx** is none of the vectors xn. Assume that this is not the case; that is, x**=xn0 for some n0. Since {xd}d∈D weak-star converges to x**, the net {〈xd,xn0*〉}d∈D converges to 〈x**,xn0*〉=1. Then, there is d~∈D such that
(13)|〈xd,xn0*〉|>12
for all d≥d~. Since D is cofinal, there is d1∈D such that d1≥d~ and d1≥n0+1>n0. By the biorthogonality of {xn*}n=1∞ it follows that 〈xd1,xn0*〉=0, which contradicts (13).
We prove now thatx**∉Z. Let us assume that x**∈Z. Then there is a unique sequence of scalars {an}n=1∞ so that x**=∑n=1∞‍anxn. Let ϵ>0 and take n1:=1. Since {〈xd,x1*〉}d∈D converges to 〈x**,x1*〉=a1, there is d~~∈D so that |〈xd,x1*〉-a1|<ϵ for all d≥d~~. Since D is cofinal, there is d~1∈D such that d~1≥2. Let d~2≥d~1,d~~. Then n2:=d~2≥d~1>1=n1. Therefore 〈xd2,x1*〉=〈xn2,xn1*〉=0 and |〈xd2,x1*〉-a1|<ϵ. Hence, |a1|<ϵ. This shows that a1=0. Reiterating this process we can prove that an=0 for all n∈ℕ, which contradicts the fact that x**≠0.
To finish the proof, sincex** belongs to the w(X**,X*) closure of Z, if we assume that x**∈X, then x** actually belongs to the w(X,X*) closure of Z. This closure coincides with the norm closure, that is, with Z. As we have already proved, this is impossible. Therefore, x**∉X.Theorem 6.
Letn∈ℕ. For each 1≤j≤n let Xj be a Banach space with a normalized Schauder basis {xnj}n=1∞. Let C∈ℒ(X1,…,Xn) and σ∈Σn. If there exist strictly increasing sequences of natural numbers {k(j,mj)}mj=1∞, j=1, …, n, such that
(14)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|C(xk(1,m1)1,…,xk(n,mn)n)|=∥C∥
then Cσ attains its norm at a point in (BX1**∖X1)×⋯×(BXn**∖Xn).Proof.
Consider any1≤j≤n. Let xj** be a cluster point of the subsequence {xk(j,mj)j}mj=1∞ and hence of the sequence {xnj}n=1∞. As the Schauder basis is normalized, xj**∈BXj** and by Proposition 5   xj**∉Xj. Let {xk(j,dj)}dj∈Dj be a subnet of {xk(j,mj)}mj=1∞ that weak-star converges to xj**. Then
(15)∥Cσ∥=∥C∥=lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|C(xk(1,m1)1,…,xk(n,mn)n)|=lim⁡dσ(1)∈Dσ(1)⋯lim⁡dσ(n)∈Dσ(n)|C(xk(1,d1)1,…,xk(n,dn)n)|=|Cσ(x1**,…,xn**)|.


## 4. Norm Attaining Extensions of Multilinear Forms onℓ1

Our aim in this section is to show that, when working with the spaceℓ1, one can strengthen the results in Section 3. But before, let us recall some well known facts on ℓ1 that we need to use later. First is that, since ℓ1** is the third dual of c0, then ℓ1 is a complemented subspace of ℓ1**. Actually, ℓ1**=c0*⊕c0⊥=ℓ1⊕c0⊥, where a linear form belongs to c0⊥ if it vanishs on c0. Moreover, ℓ1** is 1-sum of ℓ1 and c0⊥ [11, page 158]; that is, if we denote by π:ℓ1**→ℓ1 the projection of ℓ1** onto ℓ1, we have that ∥x**∥=∥π(x**)∥+∥x⊥∥ for every x** in ℓ1**, where x⊥=x**-π(x**). If A is in ℒ(nℓ1**), then
(16)∥A∥=sup⁡k1,…,kn∈ℕ|A(ek1,…,ekn)|,
where {ek}k=1∞ is the canonical basis of ℓ1.We have seen that, even if the norm of an extension of a multilinear functional is attained in points of the bidual, we can deal with sequential limits of points in the unit ball of the space. We now prove that, when dealing with bilinear forms defined onℓ1×ℓ1, sequences in the unit ball of ℓ1 can be replaced with subsequences of the canonical basis of ℓ1, and so a full characterization works.Lemma 7.
LetA∈ℒ(nℓ1**) with ∥A∥=1, x1**,…,xn**∈Bℓ1**∖ℓ1, and xi⊥=xi**-π(xi**), i=1,…,n. If A attains its norm at (x1**,…,xn**) then A attains its norm at (x1⊥/∥x1⊥∥,…,xn⊥/∥xn⊥∥) too.Proof.
Let us prove it first forn=1, that is, for A being linear. If we assume that |A(x1⊥)|<∥x1⊥∥ then for some ɛ∈𝕂 with |ɛ|=1(17)1=A(ɛx1**)=A(ɛx1⊥)+A(ɛπ(x1**))<∥x1⊥∥+∥π(x1**)∥=∥x1**∥=1
which is a contradiction.
Assume now thatA is bilinear. The associated linear mapping A1(y):=A(y,x2**), y∈ℓ1**, attains its norm at x1**∈Bℓ1**∖ℓ1 and so, by the linear case, A1 attains its norm at x1⊥/∥x1⊥∥. Now, if we consider the other associated linear mapping A2(y):=A(x1⊥/∥x1⊥∥,y), y∈ℓ1**, it attains its norm at x2**. Then, A2 also attains its norm at x2⊥/∥x2⊥∥. That is, |A(x1⊥/∥x1⊥∥,x2⊥/∥x2⊥∥)|=1.
An easy induction yields the general case.Lemma 8.
LetM and N be subsets of ℕ, 0<β<1, and for each n∈ℕ let an≥0 be such that ∑n=1∞‍an=1. If   ∑t∈M‍at+∑t∈N‍at>2-β then ∑t∈M∩N‍at>1-β.Proof.
Since(18)1=∑n=1∞‍an≥∑t∈M∖Nat‍+∑t∈N∖Mat‍+∑t∈M∩N‍at
it follows that
(19)∑t∈M∖Nat‍+∑t∈N∖Mat≤1‍-∑t∈M∩Nat‍.
Combining this with the hypothesis we finally get that
(20)2-β<∑t∈M∖N‍at+∑t∈M∩Nat‍+∑t∈N∖Mat‍+∑t∈M∩Nat‍≤2∑t∈M∩Nat‍-∑t∈M∩Nat+1‍=∑t∈M∩Nat+1‍.Theorem 9.
Given a bilinear formA∈ℒ(2ℓ1) of norm one, the following are equivalent:(a)
lim⁡ilim⁡j|A(emi,enj)|=1 for some strictly increasing sequences of natural numbers (mi)i=1∞ and (nj)j=1∞;(b)
there existx1**,x2**∈ℓ1**∖ℓ1 of norm one such that |AId(x1**,x2**)|=1.Proof.
(
a
)
⇒
(
b
) is a consequence of Theorem 6.
(
b
)
⇒
(
a
): notice that ℓ1 is an L-summand space in its bidual so ∥xs**∥=∥π(xs**)∥+∥xs**-π(xs**)∥, for s=1,2, where π is the projection from ℓ1** onto ℓ1. For each n∈ℕ let πn denote the projection from ℓ1** onto ℓ1n. Note that πn is weak-star continuous.
By Lemma7 we can assume that π(x1**)=π(x2**)=0. Consider the linear form AId(·,x2**) of ℓ1* with norm one defined by AId(x,x2**)=lim⁡d2A(x,xd2) for all x in ℓ1, whenever {xd2}d2∈D2 is a net in the unit ball of ℓ1 weak-star convergent to x2**.
Let us see that there exists a strictly increasing sequence of natural numbers{mi}i=1∞ with lim⁡i|AId(emi,x2**)|=1. If this is not the case, then there exists ϵ>0 and there exists a natural number r with |AId(ek,x2**)|≤1-ϵ for all k>r. Let {xd1}d1∈D1 be a net in the unit ball of ℓ1 weak-star convergent to x1**. Since π(x1**)=0 and πr(xd1) converges to πr(x1**)=0 then {xd1-πr(xd1)}d1∈D1 weak-star converges to x1**. Moreover, ∥xd1-πr(xd1)∥≤∥xd1∥≤1 and so by replacing xd1 with xd1-πr(xd1) we can assume that πr(xd1)=0; that is, xd1(t)=0 for all t=1,…,r.
Therefore for alld1∈D1(21)|AId(xd1,x2**)|=|∑t=1∞xd1(t)AId(et,x2**)‍|≤∑t=r+1∞|xd1(t)||AId(et,x2**)|‍≤1-ϵ,
contradicting the fact that |lim⁡d1AId(xd1,x2**)|=|AId(x1**,x2**)|=1.
Without loss of generality assume that for alli∈ℕ(22)1-|AId(emi,x2**)|≤2-(2i+2).
By using induction, let us find a strictly increasing sequence of natural numbers{nj}j=1∞ such that |A(emi,enj)|≥1-2-i for all 1≤i≤j.
Let{xd2}d2∈D2 be a net in the unit ball of ℓ1 weak-star convergent to x2**. Since |lim⁡d2AId(em1,xd2)|=|AId(em1,x2**)|>1-2-4, there exists d0 in D2 with |A(em1,xd0)|>2-1. Then
(23)2-1<|A(em1,xd0)|≤∑t∈ℕ‍|xd0(t)||A(em1,et)|≤sup⁡t∈ℕ{|A(em1,et)|}∑t∈ℕ‍|xd0(t)|≤sup⁡t∈ℕ{|A(em1,et)|}.
Letn1 be a natural number with |AId(em1,en1)|>2-1. Now, assume that we have found n1<⋯<nr with |A(emi,enj)|>1-2-i for 1≤i≤j≤r and let us find nr+1. Considering that π(x2**)=0, by replacing xd2 with xd2-πnr(xd2), we can assume that πnr(xd2)=0; that is, xd2(t)=0 for all t=1,…,nr and all d2∈D2.
By (22), consider x0 an element of the net {xd2}d2∈D2 such that
(24)|A(emi,x0)|≥1-2-2ifori=1,…,r+1.
For eachi=1,…,r+1 define the sets
(25)Ti:={t∈ℕ:t>nr,|AId(emi,et)|≥1-2-i}.
Therefore, for everyi=1,…,r+1,
(26)1-2-2i≤|A(emi,x0)|≤∑t∈Ti|x0(t)||A(emi,et)|‍+∑t∉Ti‍|x0(t)||A(emi,et)|≤∑t∈Ti‍|x0(t)|+(1-2-i)∑t∉Ti‍|x0(t)|≤∑t∈Ti|x0(t)|‍+(1-2-i)(1-∑t∈Ti|x0(t)|‍)=(1-2-i)+2-i∑t∈Ti|x0(t)|‍,
where in the first inequality we have used (24). Thus 2-i∑t∈Ti‍|x0(t)|≥2-i-2-2i and so
(27)∑t∈Ti|x0(t)|‍≥1-2-i.
We use now finite induction and Lemma 8 to see that ⋂i=1r+1‍Ti≠∅. Indeed, by (27)
(28)∑t∈T1‍|x0(t)|+∑t∈T2|x0(t)|‍>2-(12+122).
Lemma 8 yields that ∑t∈T1∩T2‍|x0(t)|>1-((1/2)+(1/22)). If for some 1≤l<r+1 we assume that
(29)∑t∈∩j=1lTj|x0(t)|‍>1-(12+122+⋯+12l),
then
(30)∑t∈∩j=1lTj|x0(t)|‍+∑t∈Tl+1|x0(t)|‍>2-(12+122+⋯+12l+12l+1).
Once more, Lemma 8 yields that
(31)∑t∈∩j=1l+1Tj|x0(t)|‍>1-(12+122+⋯+12l+1).
Therefore, we can conclude that
(32)∑t∈∩j=1r+1Tj|x0(t)|‍>1-(12+122+⋯+12r+1),
and so ∩j=1r+1Tj≠∅. We define nr+1:=min⁡(∩j=1r+1Tj). Note that nr+1>nr.
From (25) it follows that
(33)|A(emi,enr+1)|≥1-2-i
for all i=1,…,r+1.
By (33)
(34)1≥liminf⁡j|A(emi,enj)|≥1-2-i.
Then
(35)lim⁡iliminf⁡j|A(emi,enj)|=1.
To finish the proof, we show that the liminf⁡ can be replaced with lim⁡ by just choosing a suitable subsequence of {enj}j=1∞.
Let us proceed once more by induction. By (33), |A(em1,enj)|≥1-(1/2) for all j≥1. Then, there exists a subsequence {enjk}k=1∞ of {enj}j=1∞ such that lim⁡k|A(em1,enjk)| exists and it is greater than or equal to 1-(1/2). To make the notation clear, we write n(1,k):=njk and so
(36)lim⁡k|A(em1,en(1,k))|≥1-12.
Assume that we have a chain of sequences{en(1,j)}j=1∞,…,{en(p,j)}j=1∞ with each of them being a subsequence of the previous one, such that lim⁡j|A(emi,en(i,j))|≥1-(1/2i), for all i=1,…,p. Let us construct a subsequence {en(p+1,j)}j=1∞ of {en(p,j)}j=1∞ such that |A(emp+1,en(p+1,j))|≥1-(1/2p+1) for all j∈ℕ. Indeed, since |A(emp+1,en(p,j))|≥1-(1/2p+1) for all j≥p+1, there exists a subsequence {en(p,j)l}l=1∞ of {en(p,j)}j=1∞ such that lim⁡l|A(emp+1,en(p,j)l)| exists and it is greater than or equal to 1-(1/2p+1). We write n(p+1,l):=n(p,j)l and so
(37)lim⁡l|A(emp+1,en(p+1,l))|≥1-12p+1.
So we have countably many sequences {en(1,j)}j=1∞, {en(2,j)}j=1∞,…, with each of them being a subsequence of the previous one, such that lim⁡j|A(emi,en(i,j))|≥1-(1/2i), for all i=1,2,…. The diagonal sequence {en(j,j)}j=1∞ is the one we were looking for. Note that {en(j,j)}j=i∞ is a subsequence of {en(i,j)}j=1∞ and then there exists
(38)lim⁡j|A(emi,en(j,j))|≥1-12i,
for all i∈ℕ.
Therefore, we have found sequences{emi}i=1∞ and {en(j,j)}j=1∞, with {mi}i=1∞ and {n(j,j)}j=1∞ strictly increasing, for which there exists
(39)lim⁡ilim⁡j|A(emi,en(j,j))|=1.
This concludes the case withπ(x1**)=π(x2**)=0.
Ifπ(x1**)≠0 or π(x2**)≠0, then y1**=x1**-π(x1**) and y2**=x2**-π(x2**) are nonzero points of ℓ1**∖ℓ1 with |A(y1**/∥y1**∥,y2**/∥y2**∥)|=1, and the former case gives us the desired result.Corollary 10.
Given a bilinear formA∈ℒ(2ℓ1) of norm one and σ∈Σ2, the following are equivalent:(a)
lim⁡(i,σ(1),σ)lim⁡(i,σ(2),σ)|A(em(i,1,σ),em(i,2,σ))|=1 for some strictly increasing sequences of natural numbers (m(i,1,σ))i=1∞ and (m(i,2,σ))i=1∞;(b)
there existx1,σ**,x2,σ**∈ℓ1**∖ℓ1 of norm one such that |Aσ(x1,σ**,x2,σ**)|=1.Remark 11.
We do not know if Theorem9 is valid for n-linear mappings with n>2. Our conjecture is the following. Let n∈ℕ, A∈ℒ(nℓ1), and σ∈Σn. If Aσ attains its norm on ℓ1** but only in n-tuples that belong to (Bℓ1**∖ℓ1)n, then there exist increasing sequences of natural numbers {k(j,mj)}mj=1∞, j=1,…,n, such that
(40)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1),…,ek(n,mn))|=∥A∥.Next we give the following lemma.Lemma 12.
Let{k1(h),…,kn(h)}h=1∞ be a sequence in ℕn such that each {kj(h)}h=1∞ is strictly increasing, j=1,…,n, and let the n-linear mapping A:ℓ1n→ℝ be defined by
(41)A(ek1,…,ekn)={(k1(h)k1(h)+1)nif(k1,…,kn)=(k1(h),…,kn(h))hhhhhhhhhhhhhforsomeh∈ℕ.0otherwise.
One has that ∥A∥=1 and there is no permutation σ such that Aσ attains its norm (at any n-tuple of Bℓ1**×⋯×Bℓ1**).Proof.
Note first that, for arbitraryxi:=∑k=1∞‍ak,iek∈Bℓ1, i=1,…,n, if we fix 1≤j≤n then
(42)|A(x1,…,xn)|≤∥x1∥⋯∥xj-1∥·∥xj+1∥⋯∥xn∥×∑h=1∞‍k1(h)k1(h)+1|akj(h),j|≤∑h=1∞‍k1(h)k1(h)+1|akj(h),j|.
Thus, for anyσ∈Σn, Aσ, any x1**, …, xk-j**, xj+1**, …, xn**∈Bℓ1**, and any xj∈Bℓ1, by taking nets in Bℓ1 weak-star convergent if necessary, we get
(43)|Aσ(x1**,…,xj-1**,xj,xj+1**,…,xn**)|≤∑h=1∞‍k1(h)k1(h)+1|akj(h),j|<1.
Hence if there exist a permutation σ∈Σn and x1**, …, xn** in Bℓ1** such that ∥xj**∥=1 for every j and
(44)|Aσ(x1**,…,xn**)|=1,
we have that x1**, …, xn**∈Bℓ1**∖Bℓ1. Moreover, by Lemma 7 it can also be assumed that xj** belongs to c0⊥ for j=1,…,n. Finally, by making a rearrangement of coordinates, if necessary, we can assume that σ is the identity permutation.
We defineB:ℓ1×ℓ1→ℝ by B(x,y)=AId(x,y,x3**,…,xn**). Clearly
(45)|BId(x1**,x2**)|=|AId(x1**,…,xn**)|=1.
By Theorem 9, there exist two sequences (enj) and (eml) such that
(46)lim⁡j→∞lim⁡l→∞|B(enj,eml)|=1.
Thus there exist j, l such that
(47)|B(enj,eml)|>12.
But there exists h0 such that nj<k1(h) and ml<k2(h) for every h≥h0 and we get that
(48)(nj,ml,k3,…,kn)∉{(k1(h),…,kn(h)):h≥h0},
for every k3, …, kn∈ℕ with k3>k3(h0), …, kn>kn(h0). Now consider a net {xdj}dj∈Dj in Bℓ1 weak-star convergent to xj** for j=3, …, n. Since xj** belongs to c0⊥, as in the proof of Theorem 9, we can assume additionally that, for every dj∈Dj, the first kj(h0) components of xdj are 0. Hence
(49)A(enj,eml,ek3,…,ekn-1,xdn)=0,
for every dn∈Dn. Hence
(50)AId(enj,eml,ek3,…,ekn-1,xn**)=0,
for every k3>k3(h0),…,kn>kn(h0). By induction we obtain the contradiction
(51)B(enj,eml)=AId(enj,eml,x3**,…,xn**)=0.Theorem 13.
Given a subsetP⊆Σn, there exists an n-linear form A(P)∈ℒ(nℓ1) with ∥A(P)∥=1 such that A(P)σ is norm attaining if and only if σ∈P.Proof.
The proof will be divided into two cases.
IfP is the empty set, consider A(P)∈ℒ(nℓ1)(52)A(P)(ek1,ek2,…,ekn)={(k1k1+1)nifk1=k2=⋯=kn,0otherwise.
By Lemma 12, A(P) does not attain its norm at any point of the unit ball of ℓ1**.
IfP is not empty, consider
(53)A(P)(ek1,ek2,…,ekn)={∏i=1n‍kiki+1if∃σ∈P,kσ(1)≤kσ(2)≤⋯≤kσ(n),0otherwise.
Clearly,∥A(P)∥=1. A similar argument to the one given in (43) shows that, for any σ∈Σn, Aσ does not attain its norm at any n-tuple in Bℓ1**×⋯×Bℓ1** with at least a coordinate j belonging to Bℓ1. If σ∈P then
(54)lim⁡kσ(1)→∞⋯lim⁡kσ(n)→∞A(P)(ek1,…,ekn)=1.
Hence, considering x** a weak-star cluster point of the sequence {ek}k=1∞ we obtain
(55)A(P)σ(x**,…,x**)=1.
Thus, A(P)σ is norm attaining.
Now we see thatA(P)σ does not attain its norm whenever σ is not in P. For simplicity we will assume that σ is the identity permutation. Let us assume that A(P)Id does attain its norm at (x1**,…,xn**)∈Bℓ1**×⋯×Bℓ1**. By the above observation, xi** is a point in Bℓ1**∖ℓ1 for i=1,…,n. By Lemma 7 we can assume that π(xi**)=0 for i=1,…,n. Let {xdi}di∈Di be nets in the unit sphere of ℓ1 weak-star convergent to xi**, for i=1,…,n.
Letl0=0. Since |A(P)Id(x1**,…,xn**)|=1 there exists d10∈D1 with
(56)|A(P)Id(xd10,x2**,…,xn**)|>1-2-n.
Let l1 be such that ∥πl1(xd10)∥>1/2. Now, using (56) and since π(x2**)=0 we can find d2∈D2 and a natural number l2 with |A(P)Id(xd10,xd20,x3**,…,xn**)|>1-2-n and ∥πl2(xd20)∥-∥πl1(xd20)∥>1/2. In general, by using finite induction over i, we can find di0∈Di and a natural number li such that |A(P)Id(xd10,…,xdi0,xi+1**,…,xn**)|>1-2-n and ∥πli(xdi0)∥-∥πli-1(xdi0)∥>1/2, for i=2,…,n.
But then, if we denote by(57)C:={(t1,…,tn)∈ℕn:li-1<ti≤lifori=1,…n}
since Id is not in P, we have A(P)(et1,…,etn)=0 for all (t1,…,tn)∈C. Therefore,
(58)1-2-n<|A(P)(xd10,…,xdn0)|≤∑(t1,…,tn)‍∏i=1n‍|xdi0(ti)|A(P)(et1,…,etn)=∑(t1,…,tn)∉C‍∏i=1n‍|xdi0(ti)|A(P)(et1,…,etn)<∑(t1,…,tn)∉C‍∏i=1n‍|xdi0(ti)|≤1-∑(t1,…,tn)∈C‍∏i=1n‍|xdi0(ti)|=1-∏i=1n‍(∥πki(xdi0)∥-∥πki-1(xdi0)∥)<1-2-n
which is a contradiction. Hence A(P)Id does not attain its norm.Theorem 14.
LetA∈ℒ(nℓ1) of norm one such that, for every ϵ>0 and every σ∈Σn, there exist subsequences {ek(i,mi,σ)}mi=1∞(that depend on σ and ϵ) of the sequence {ek}k=1∞ so that
(59)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-ϵ.
Then for every ϵ>0 and each subset P⊆Σn, there exists A(P,ϵ)∈ℒ(nℓ1) with ∥A(P,ϵ)∥=1 such that ∥A(P,ϵ)-A∥≤ϵ, and A(P,ϵ)σ is norm attaining if and only if σ∈P.Proof.
Consider then-linear form
(60)B(x1,…,xn)=∑k1,…,kn∈ℕ‍B(ek1,…,ekn)∏i=1n‍xi(ki)
for x1,…,xn∈ℓ1, where
(61)B(ek1,…,ekn)={A(ek1,…,ekn)if1-ϵ2≥|A(ek1,…,ekn)|(1-ϵ2)sign⁡(A(ek1,…,ekn)),if|A(ek1,…,ekn)|>1-ϵ2.
We have ∥B∥≤1-(ϵ/2).
Fix a nonempty subset of permutationsP, consider the n-linear form A(P) from Theorem 13, and define the n-linear form A(P,ϵ) as follows:
(62)A(P,ϵ)(ek1,…,ekn)≔B(ek1,…,ekn)+sign⁡(B(ek1,…,ekn))×ϵ2A(P)(ek1,…,ekn)
if k1=k(1,m1,σ),…,kn=k(n,mn,σ), for some σ∈P, with mσ(1)≤⋯≤mσ(n) and B(ek1,…,ekn) otherwise.
Clearly,(63)∥A(P,ϵ)∥≤∥B∥+ϵ2∥A(P)∥≤1-ϵ2+ϵ2=1,
and hence ∥A(P,ϵ)∥≤1.
By hypothesis, for eachσ∈Σn, there exist sequences {ek(i,mi,σ)}mi=1∞, with the property that {k(i,mi,σ)}mi=1∞ is strictly increasing, such that
(64)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-ϵ2.
From (64) there exists mσ(1)0 such that for every mσ(1)≥mσ(1)0(65)lim⁡mσ(2)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-ϵ2.
Taking mσ(1)≥mσ(1)0, there is mσ(2)0 that depends on mσ(1), such that for every mσ(2)≥mσ(2)0(66)lim⁡mσ(3)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-ϵ2.
Reiterating this process and assuming that we have fixed natural numbers mσ(1), mσ(1), …, mσ(n-1) with mσ(1)≥mσ(1)0, mσ(2)≥mσ(2)0,…,mσ(n-1)≥mσ(n-1)0, where mσ(n-1)0 depends on mσ(1),mσ(2),…,mσ(n-2), so that lim⁡mσ(n)|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-(ϵ/2), we can find mσ(n)0 that depends on mσ(1),mσ(2),…,mσ(n-1), such that for every mσ(n)≥mσ(n)0(67)|A(ek(1,m1,σ),…,ek(n,mn,σ))|>1-ϵ2.
Then,
(68)|B(ek(1,m1,σ),…,ek(n,mn,σ))|=1-ϵ2,
and then ∥B∥=1-(ϵ/2). Moreover, given σ∈P and δ>0 we can take mσ(1)≤⋯≤mσ(n) big enough so that A(P)(ek(1,m1,σ),…,ek(n,mn,σ))≥1-δ. Hence,
(69)|A(P,ϵ)(ek(1,m1,σ),…,ek(n,mn,σ))|≥1-ϵ2+ϵ2(1-δ)
and so ∥A(P,ϵ)∥≥1-(ϵ/2)+(ϵ/2)(1-δ). Thus, ∥A(P,ϵ)∥=1.
Notice that|A(P,ϵ)(x1,…,xn)-B(x1,…,xn)|≤(ϵ/2), for all x1,…,xn∈ℓ1; hence
(70)∥A(P,ϵ)-A∥≤∥A(P,ϵ)-B∥+∥B-A∥≤ϵ.
Now we show thatA(P,ϵ)σ is norm attaining if and only if σ∈P.
Ifσ∉P and we assume that there is (x1**,…,xn**)∈Bℓ1**×⋯×Bℓ1** such that A(P,ϵ)σ(x1**,…,xn**)=1, since A(P)σ(x1**,…,xn**)<1, then 1<|Bσ(x1**,…,xn**)|+(ϵ/2). Hence |Bσ(x1**,…,xn**)|>1-(ϵ/2), which is impossible.
Take nowσ∈P. From (68) we have for mσ(1)≤⋯≤mσ(n)(71)|A(P,ϵ)(ek(1,m1,σ),…,ek(n,mn,σ))|=1-ϵ2+ϵ2∏i=1n‍mimi+1.
Hence
(72)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|A(P,ϵ)(ek(1,m1,σ),…,ek(n,mn,σ))|=lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞1-ϵ2+ϵ2∏i=1n‍mimi+1=1
so A(P,ϵ)σ is norm attaining at a point (x1**,…,xn**), where each xj** is a weak-star cluster point of the sequence {ek(j,mj,σ)}mj=1∞,j=1,…,n.
IfP=∅, for every h, by taking ɛ=(1/h), the process above gives the existence of a sequence of n-tuples {(k1(h),…,kn(h))}h=1∞ in ℕn such that each {kj(h)}h=1∞ is strictly increasing, j=1,…,n, and
(73)|A(ek1(h),…,ekn(h)))|>1-1h.
We define
(74)C(ek1,…,ekn)={(k1(h)k1(h)+1)nif(k1,…,kn)=(k1(h),…,kn(h))hhhhhhhlhhhhhhforsomeh∈ℕ,0otherwise
and A(P,ϵ) at (ek1,…,ekn) as
(75)A(P,ϵ)(ek1,…,ekn):=B(ek1,…,ekn)+sign⁡(B(ek1,…,ekn))×ϵ2C(ek1,…,ekn).
Notice that as before ∥A(P,ϵ)∥≤∥B∥+(ϵ/2)∥C∥=1, and, since
(76)|A(P,ϵ)(ek1(h),…,ekn(h))|=|B(ek1(h),…,ekn(h))|+(k1(h)k1(h)+1)n=1-ɛ2+ɛ2(k1(h)k1(h)+1)n,
for every h such that (1/h)<(ϵ/2), we obtain that ∥A(P,ϵ)∥=1. Since, by Lemma 12, C does not attain the norm at any point of Bℓ1** neither can A(P,ϵ). To conclude the proof, notice ∥A(P,ϵ)-A∥≤∥A(P,ϵ)-B∥+∥B-A∥≤ϵ/2+ϵ/2=ϵ.Remark 15.
If the conjecture in Remark11 was true, we could get the following result. Let A∈ℒ(nℓ1) of norm one. The following are equivalent.(1)
For everyϵ>0 and every σ∈Σn, there exist subsequences {ek(i,mi)}mi=1∞ (that depend on σ and ϵ) of the sequence {ek}k=1∞ so that
(77)lim⁡mσ(1)→∞⋯lim⁡mσ(n)→∞|A(ek(1,m1),…,ek(n,mn))|>1-ϵ.(2)
For everyϵ>0 and each subset P⊆Σn, there exists A(P,ϵ)∈ℒ(nℓ1) with ∥A(P,ϵ)∥=1 such that ∥A(P,ϵ)-A∥≤ϵ, and A(P,ϵ)σ is norm attaining if and only if σ∈P.



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