Denseness of norm attaining mappings.

Acosta, María D.

RACSAM, Tome 100 (2006), p. 9-30 / Harvested from Biblioteca Digital de Matemáticas

Résumé

The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples of Banach spaces X and Y such that the set of norm attaining operators from X to Y is not dense. He also gave isometric conditions on X for which the set of norm attaining operators from X to Y are dense in the space of all operators between these Banach spaces. If the above conclusion holds for every Y, X is said to have the property A. Also, there are known sufficient conditions on the range space Y in order to have the same denseness conditions for every X. In such a case, Y has the property B. Bourgain proved that every space satisfying the Radon-Nikodym property has the property A. For classical Banach spaces, it is known that C[0,1], lp (1 < p < ∞) and any infinite-dimensional L1(μ) do not satisfy the property B (results due to Schachermayer, Gowers and Acosta, respectively). When both X and Y are either C(K) or L1(μ), there are positive results due to Johnson and Wolfe, and Iwanik, respectively. Finet and Payá proved that there is also positive result for X = L1 and Y = L∞.

For multilinear mappings, Aron, Finet and Werner initiated the research and gave sufficient conditions on a Banach space X in order to satisfy the denseness of the set of norm attaining N-linear mappings in the set of all the N-linear mappings (Radon-Nikodym property, for instance). Choi showed that the space L1[0,1] does not satisfy the denseness of the set of norm attaining bilinear forms. Alaminos, Choi, Kim and Payá proved that for any scattered compact space K, the set of norm attaining N-linear forms on C(K) is dense in the space of all N-linear forms, and for the bilinear case no restriction on the compact is needed. Acosta, García and Maestre proved that the set of N-linear forms whose Arens extensions to the bidual attains the norm is dense in the space of all the N-linear forms on a product of N Banach spaces. For polynomials and for holomorphic mappings, there are some results along the same line, but more open problems than for the multilinear case.

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Publié le : 2006-01-01

MR MR2267397

DMLE-ID : 4140