On the polynomial Lindenstrauss theorem
نویسندگان
چکیده
The Bishop-Phelps theorem [1] states that for any Banach space X, the set of norm attaining linear bounded functionals is dense in X ′, the dual space of X. Since then, the study of norm attaining functions has attracted the attention of many authors. Lindenstrauss showed in [2] that there is no Bishop-Phelps theorem for linear bounded operators. Nevertheless, he proved that the set of bounded linear operators (between any two Banach spaces X and Y ) whose second adjoints attain their norm, is dense in the space of all operators. This result was later extended by Acosta, Garćıa and Maestre [3] for multilinear operators. In the context of homogeneous polynomials, where symmetry represents an additional difficulty, Aron, Garćıa and Maestre showed in [4] the density of the scalar valued 2-homogeneous polynomials whose Aron-Berner extension attain their norm. Under certain hypothesis on the space X, we show that a Lindenstrauss theorem holds for N -homogeneous polynomials from X into any dual space (and, therefore, for scalar-valued polynomials on X). For this, we present an integral representation for the elements of some tensor products. We also exhibit many situations in which there is no polynomial Bishop-Phelps theorem but our results apply. In particular, we present couples of Banach spaces which do not satisfy the polynomial Bishop-Phelps theorem for any degree N ≥ 1 but satisfy the polynomial Lindenstrauss theorem for every degree.
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