Concentration theorems for $2$-homogeneous polynomials and bilinear forms with applications to the Kadec-Klee properties

Abstract

We study Kadec-Klee properties in spaces of homogeneous polynomials and multilinear forms by presenting optimal results. Indeed, our main tool is a family of concentration theorems showing that, on suitable $p$-convex Banach sequence lattices, a $2$-homogeneous polynomial or a bilinear form which almost attains its norm at finitely supported vectors is uniformly close to its restriction to the corresponding finite set of coordinates. As a consequence, we obtain the weak-star uniform Kadec-Klee property for spaces of complex $2$-homogeneous polynomials and complex bilinear forms on $p$-convex Banach sequence lattices with constant one for some $p>2$. This applies, in particular, to $c_0$, $\ell_p$, Lorentz spaces $d(w,p)$, Garling spaces $g(w,p)$, among others. In the real setting, a concentration argument based on the modulus of convexity of power type yields further positive results for spaces of operators and bilinear forms, extending previous results in the literature. We also prove that the restriction $p>2$ is optimal for $2$-homogeneous polynomials on $\ell_p$, $d(w,p)$ and $g(w,p)$, and establish general failures of the sequential weak-star Kadec-Klee property for real homogeneous polynomials of degree at least two and for complex homogeneous polynomials and symmetric multilinear forms of degree at least three. These results show that our positive results are optimal respect to the degree and that the assumption $p>2$ is optimal for the classical spaces $\ell_p$, $d(w,p)$ and $g(w,p)$. As a consequence of our results, we establish the strong subdifferentiability of the associated projective and symmetric projective tensor norms.

Disclosure

“be used to obtain norm-attainment-type results for bilinear forms and 2-homogeneous polynomials (see [4, Proposition 4.4]). 7. Acknowledgments During the preparation of this manuscript, the authors used ChatGPT Go to assist with the verification of mathematical arguments, the organization of the manuscript and bibliography, language editing and the preparation of portions of the LATEX source. All AI-assisted material was subsequently reviewed, ve”

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2026-08-23T19:17:14.571708Z

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