A Lindenstrauss theorem for some classes of multilinear mappings (Q907721)

The Bishop-Phelps theorem states that for any Banach space \(X\), the set of norm attaining bounded linear functionals is dense in \(X'\) (the dual of \(X\)). \textit{J. Lindenstrauss} [Isr. J. Math. 1, 139--148 (1963; Zbl 0127.06704)] showed that for linear operators this result is not true but that a relative statement holds: if \(X\) and \(Y\) are Banach spaces then the set of bounded linear operators from \(X\) to \(Y\) whose second adjoints attain their norm is always dense in the space of all bounded linear operators \(\mathcal L(X,Y)\). The study of Lindenstrauss-type theorems in the context of multilinear mappings was initiated by \textit{M. D. Acosta} et al. in [J. Funct. Anal. 235, No. 1, 122--136 (2006; Zbl 1101.46029)] and, in the polynomial setting, by the authors of the article under review [ibid. 263, No. 7, 1809--1824 (2012; Zbl 1258.46017)]. In both frameworks, second adjoints are, as usually, changed by Arens extensions. Here, the authors search for Lindenstrauss-type theorems in the context of multilinear ideals. They show a positive result (Theorem 1.2) for a wide set of multilinear ideals (which they call stable ideals) that includes the classes of nuclear, integral, extendible, multiple \(p\)-summing and Hilbert-Schmidt multilinear mappings. Also, they consider symmetric multilinear mappings and they obtain a result (Theorem 3.1) by means of a useful integral representation formula for the duality between projective tensor products and multilinear mappings. They present some examples where the symmetric multilinear Bishop-Phelps theorem fails but their Lindenstrauss-type result holds. The last section is devoted to quantitative versions (in the Bollobás style) of Bishop-Phelps and Lindenstrauss theorems for ideals of multilinear mappings.