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Norm Attaining Arens Extensions on
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 on 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.
The Bishop-Phelps theorem  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 , where he gave an example of a Banach space such that the identity mapping on cannot be approximated by norm attaining operators. However, if one considers the adjoint of an operator between Banach spaces, given by , for all , , 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 ). However, a general result for multilinear mappings cannot be expected. The first counterexample was given in  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  proved a Lindenstrauss type result for bilinear forms whose third Arens transpose attains its norm. Afterwards, in  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 -linear vector-valued mappings was finally obtained in  in its strongest form; that is, the space formed by those -linear mappings whose Arens extensions attain their norms simultaneously at the same point is dense in the space of all -linear mappings.
The aim of this paper is to study the norm attaining properties of the Arens extensions of multilinear forms on . 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 , 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 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. Section 3 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 . We strengthen the results from the former section by proving a characterization of norm attaining extensions of bilinear forms at points with coordinates in in terms of sequential limits of the images of subsequences of the canonical sequence . It is also proved that, fixing a number of Arens extensions, there exists an -linear form on of norm one such that only these extensions fixed a priori are norm attaining. Finally, we show that such -linear forms are dense in the set of all -linear forms of norm one that fulfill a condition in terms of sequential limits.
2. Background and Notation
In this paper , , , , are real or complex Banach spaces. Let denote the space of continuous -linear forms from into ( or ) with the usual norm When , we just write . We denote by the closed unit ball of , by the unit sphere, by the strong dual, and by the bidual of .
We say that is norm attaining (or attains its norm) if there exist , , such that .
Arens  found a natural way to extend a continuous bilinear mapping to a continuous bilinear mapping from into . His method consists in applying three times the operation defined as , , and . The first extension is defined as and the second one is , where for any bilinear mapping . These extensions, which are in general different, are known as Arens products. This procedure was generalized by Aron and Berner  to arbitrary multilinear mappings.
For our purposes we will use an alternative approach due to Davie and Gamelin . The key of such approach is Goldstine theorem as it is based on limits in the , denoted by . Consider the group of all permutations of the set . Given they defined the extension associated with of an -linear form defined on , by where is a bounded net (, for all ) convergent to , for . The mapping is called an Arens extension of and the Arens extensions may be different from each other. When convenient, we will write instead of . In particular, for , and , where is the identity permutation of the set .
Note that the use of the 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  Lindenstrauss theorem is extended to multilinear forms by using the Arens extensions.
Theorem 1 ([7, Theorem 2.1]). Let be Banach spaces (). Then the set of -linear forms on such that all their Arens extensions to attain their norms at the same -tuple is dense in the space .
Let denote the space of all absolutely summing sequences in with its usual norm. In  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 whose first extension is not norm attaining, whereas the second one 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 form , defined by is such that neither nor is norm attaining, but 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. Let . For each , let be an infinite directed set. Consider a family of real or complex numbers. If the iterated limit is finite then there exist strictly increasing sequences in , , such that .
Proof. We proceed by induction on . For , since , for each there exists such that for all . Besides, by the condition on , we can choose the sequence strictly increasing.
Assume that the result is true for and let us prove it for . So, if we assume that is finite, define
By the assumption applied to the family , for each , , there exists a strictly increasing sequence such that Let us construct the sequence by induction on .
Since , there exists such that for all . Assume that we have found , , with such that for all , all , and all , , .
Fix , , . Since , there exists , with , such that for all . Take for all , , . Then whenever . Hence the limit exists and is equal to . Now, and the proof is over.
Theorem 4. Let be infinite dimensional Banach spaces, , and . If the extension attains its norm then there exist sequences with each , , and , such that
Proof. For simplicity we assume that . Let be a point in , where attains its norm. Let . By density, each is the weak-star limit of a net in , . For , set and for all . Then By Lemma 3 applied to , we obtain the desired sequences , for every .
Proposition 5. Let be an infinite dimensional Banach space, and let be a basic sequence. Then, any nonzero weak-star cluster point of belongs to .
Proof. Let be the closed linear span of and let be the orthogonal functionals in associated with . By the Hahn-Banach extension theorem, we can consider each in .
Let be a nonzero cluster point of , and let be a subnet of weak-star converging to .
We first prove that is none of the vectors . Assume that this is not the case; that is, for some . Since weak-star converges to , the net converges to . Then, there is such that for all . Since is cofinal, there is such that and . By the biorthogonality of it follows that , which contradicts (13).
We prove now that . Let us assume that . Then there is a unique sequence of scalars so that . Let and take . Since converges to , there is so that for all . Since is cofinal, there is such that . Let . Then . Therefore and . Hence, . This shows that . Reiterating this process we can prove that for all , which contradicts the fact that .
To finish the proof, since belongs to the closure of , if we assume that , then actually belongs to the closure of . This closure coincides with the norm closure, that is, with . As we have already proved, this is impossible. Therefore, .
Theorem 6. Let . For each let be a Banach space with a normalized Schauder basis . Let and . If there exist strictly increasing sequences of natural numbers , , , , such that then attains its norm at a point in .
Proof. Consider any . Let be a cluster point of the subsequence and hence of the sequence . As the Schauder basis is normalized, and by Proposition 5 . Let be a subnet of that weak-star converges to . Then
4. Norm Attaining Extensions of Multilinear Forms on
Our aim in this section is to show that, when working with the space , one can strengthen the results in Section 3. But before, let us recall some well known facts on that we need to use later. First is that, since is the third dual of , then is a complemented subspace of . Actually, , where a linear form belongs to if it vanishs on . Moreover, is -sum of and [11, page 158]; that is, if we denote by the projection of onto , we have that for every in , where . If is in , then where is the canonical basis of .
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 on , sequences in the unit ball of can be replaced with subsequences of the canonical basis of , and so a full characterization works.
Lemma 7. Let with , , and , . If attains its norm at then attains its norm at too.
Proof. Let us prove it first for , that is, for being linear. If we assume that then for some with
which is a contradiction.
Assume now that is bilinear. The associated linear mapping , , attains its norm at and so, by the linear case, attains its norm at . Now, if we consider the other associated linear mapping , , it attains its norm at . Then, also attains its norm at . That is, .
An easy induction yields the general case.
Lemma 8. Let and be subsets of , , and for each let be such that . If then .
Proof. Since it follows that Combining this with the hypothesis we finally get that
Theorem 9. Given a bilinear form of norm one, the following are equivalent:(a) for some strictly increasing sequences of natural numbers and ;(b)there exist of norm one such that .
Proof. is a consequence of Theorem 6.
: notice that is an -summand space in its bidual so , for , where is the projection from onto . For each let denote the projection from onto . Note that is weak-star continuous.
By Lemma 7 we can assume that . Consider the linear form of with norm one defined by for all in , whenever is a net in the unit ball of weak-star convergent to .
Let us see that there exists a strictly increasing sequence of natural numbers with . If this is not the case, then there exists and there exists a natural number with for all . Let be a net in the unit ball of weak-star convergent to . Since and converges to then weak-star converges to . Moreover, and so by replacing with we can assume that ; that is, for all .
Therefore for all contradicting the fact that .
Without loss of generality assume that for all
By using induction, let us find a strictly increasing sequence of natural numbers such that for all .
Let be a net in the unit ball of weak-star convergent to . Since , there exists in with . Then
Let be a natural number with . Now, assume that we have found with for and let us find . Considering that , by replacing with , we can assume that ; that is, for all and all .
By (22), consider an element of the net such that
For each define the sets
Therefore, for every , where in the first inequality we have used (24). Thus and so We use now finite induction and Lemma 8 to see that . Indeed, by (27) Lemma 8 yields that . If for some we assume that then Once more, Lemma 8 yields that Therefore, we can conclude that and so . We define . Note that .
From (25) it follows that for all .
By (33) Then To finish the proof, we show that the can be replaced with by just choosing a suitable subsequence of .
Let us proceed once more by induction. By (33), for all . Then, there exists a subsequence of such that exists and it is greater than or equal to . To make the notation clear, we write and so
Assume that we have a chain of sequences with each of them being a subsequence of the previous one, such that , for all . Let us construct a subsequence of such that for all . Indeed, since for all , there exists a subsequence of such that exists and it is greater than or equal to . We write and so So we have countably many sequences , , with each of them being a subsequence of the previous one, such that , for all . The diagonal sequence is the one we were looking for. Note that is a subsequence of and then there exists for all .
Therefore, we have found sequences and , with and strictly increasing, for which there exists
This concludes the case with .
If or , then and are nonzero points of with , and the former case gives us the desired result.
Corollary 10. Given a bilinear form of norm one and , the following are equivalent:(a) for some strictly increasing sequences of natural numbers and ;(b)there exist of norm one such that .
Remark 11. We do not know if Theorem 9 is valid for -linear mappings with . Our conjecture is the following. Let , , and . If attains its norm on but only in -tuples that belong to , then there exist increasing sequences of natural numbers , , such that
Next we give the following lemma.
Lemma 12. Let be a sequence in such that each is strictly increasing, , and let the -linear mapping be defined by One has that and there is no permutation such that attains its norm at any -tuple of .
Proof. Note first that, for arbitrary , , if we fix then
Thus, for any , , any , , , , , , and any , by taking nets in weak-star convergent if necessary, we get Hence if there exist a permutation and , , in such that for every and we have that , , . Moreover, by Lemma 7 it can also be assumed that belongs to for . Finally, by making a rearrangement of coordinates, if necessary, we can assume that is the identity permutation.
We define by . Clearly By Theorem 9, there exist two sequences and such that Thus there exist , such that But there exists such that and for every and we get that for every , , with , , . Now consider a net in weak-star convergent to for , , . Since belongs to , as in the proof of Theorem 9, we can assume additionally that, for every , the first components of are 0. Hence for every . Hence for every . By induction we obtain the contradiction
Theorem 13. Given a subset , there exists an -linear form with such that is norm attaining if and only if .
Proof. The proof will be divided into two cases.
If is the empty set, consider By Lemma 12, does not attain its norm at any point of the unit ball of .
If is not empty, consider
Clearly, . A similar argument to the one given in (43) shows that, for any , does not attain its norm at any -tuple in with at least a coordinate belonging to . If then