Research Article Norm Attaining Arens Extensions on ℓ 1 Javier Falcó, 1 Domingo García, 2 Manuel Maestre, 2 and Pilar Rueda 2 1 Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA 2 Departamento de An´ alisis Matem´ atico, Universidad de Valencia, Doctor Moliner 50, Burjasot, 46100 Valencia, Spain Correspondence should be addressed to Pilar Rueda; pilar.rueda@uv.es Received 23 December 2013; Accepted 6 February 2014; Published 8 April 2014 Academic Editor: Miguel Mart´ ın Copyright © 2014 Javier Falc´ o et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study norm attaining properties of the Arens extensions of multilinear forms defned on Banach spaces. Among other related results, we construct a multilinear form on ℓ 1 with the property that only some fxed 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. 1. Introduction Te 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. Tis question was answered in the negative by Lindenstrauss in his seminal paper [2], 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. Te theory of norm attaining operators has spread to the nonlinear setting. Te denseness of the set of norm attaining multilinear mappings has been deeply studied in the last decades. Assuming the Radon-Nikod´ ym property, this density has been established for multilinear forms (see [3]). However, a general result for multilinear mappings cannot be expected. Te frst 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 defnitions), Acosta [5] proved a Lindenstrauss type result for bilinear forms whose third Arens transpose attains its norm. Aferwards, 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. Tis asymmetry between the two Arens extensions reveals the importance of the stronger condition of attaining their norms simultaneously. Te generalization of Lindenstrauss result to -linear vector-valued mappings was fnally obtained in [7] 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. Te aim of this paper is to study the norm attaining properties of the Arens extensions of multilinear forms on ℓ 1 . On one hand, inspired by [6, Example 2], several examples of multilinear forms whose extensions sufer diferent kinds of asymmetries from the point of view of norm attainment are provided. Tese examples are built using multilinear forms on ℓ 1 , which is the classical example of a non-Arens regular Banach space. For instance, if we fx a priori some of the Arens extensions, we can construct a multilinear form on ℓ 1 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. Tese examples are presented as general results on existence of multilinear forms that fulfll the required norm attaining properties. On the other hand, we also deal with general Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 315641, 10 pages http://dx.doi.org/10.1155/2014/315641