On the polynomial Lindenstrauss theorem

TL;DR

This paper extends the polynomial Lindenstrauss theorem to certain Banach spaces, showing density of norm-attaining polynomials via an integral duality formula, with applications to Lorentz spaces and quantitative versions.

Contribution

It introduces a new integral formula for duality between tensor products and polynomials, enabling the extension of the polynomial Lindenstrauss theorem to broader classes of Banach spaces.

Findings

Density of norm-attaining N-homogeneous polynomials in specific Banach spaces

Application to Lorentz sequence spaces lacking polynomial Bishop-Phelps theorem

Development of quantitative versions of the main results

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.