2026/08/01 by Sheldon Dantas, Jorge Tomás Rodríguez
Mathematics · #math.FA #msc:46B20 #msc:46G25 #msc:46B28
arxiv created 2026/08/01 · arxiv updated 2026/08/04
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 c0, ℓp, 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 ℓp, 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 ℓp, 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.