On the polynomial Lindenstrauss theorem

Abstract

Under certain hypotheses on the Banach space $X$, we show that the set of $N$-homogeneous polynomials from $X$ to any dual space, whose Aron-Berner extensions are norm attaining, is dense in the space of all continuous $N$-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop-Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollob\'as, of these results

A non-linear Bishop–Phelps–BollobÁs type theorem

“…In [8,Theorem B], the authors proved that if X is a Banach space whose dual is separable and has the approximation property, then the set of analytic functions whose Aron-Berner extensions attain their norm is dense in A u (B X ). On the other hand, [7,Proposition 4.4] gives us a counterexample of a space which does not satisfy a Lindenstrauss-Bollobás type theorem for multilinear forms or multilinear mappings. Here, since A wu (B X ) is isometrically isomorphic to A w * u (B X * * ) (see [4,Theorem 6.3]), we have the following consequence of Theorem 1.…”

Bounded Holomorphic Functions Attaining their Norms in the Bidual

“…On the other hand, A positive fundamental result given also in [20], the so-called Lindenstrauss theorem for linear operators, states that the set of bounded linear operators (between any two Banach spaces X and Y ) whose bitransposes are norm attaining, is dense in the space of all operators. This result was generalized by Acosta, García and Maestre in [5] for multilinear operators, where the Bishop-Phelps theorem does not hold in general even in the scalar-valued case (we refer the reader to [1,19,11] for counterexamples to the Bishop-Phelps theorem in the multilinear case). In this context, the role of the bitranspose is played by the canonical (Arens) extension to the bidual, obtained by weak-star density (see [6], [15, 1.9] and the definitions below).…”

A Lindenstrauss theorem for some classes of multilinear mappings

“…Here, we extend [21,Theorem 2.2] for the duality between tensor products and multilinear mappings. As a consequence, we obtain in Theorem 3.1 a Lindenstrauss-type theorem for the space of symmetric multilinear mappings L s ( N X; Z), whenever X has separable dual with the approximation property and Z is a dual space or a Banach space with the property (β) of Lindenstrauss.…”

“…Following [3,21] we say that U(X; Y ) has the Lindenstrauss-Bollobás property (LBp) if, with ε, η and β as above, given Φ ∈ U(X; Y ), Φ = 1 andx = (x j ) N j=1 ∈ S X satisfying Φ(x) > 1 − η(ε), there exist Ψ ∈ U(X; Y ), Ψ = 1, and a ′′ = (a…”

“…In [21] we give an integral representation formula for the duality between tensor products and polynomials on Banach spaces satisfying appropriate hypotheses. Here, we extend [21,Theorem 2.2] for the duality between tensor products and multilinear mappings.…”