On the strong subdifferentiability of homogeneous polynomials and (symmetric) tensor products
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We study the (uniform) strong subdifferentiability of norms of Banach spaces P(N X, Y ∗) of all continuous N-homogeneous polynomials and tensor products of Banach spaces, namely X⊗b π· · · ⊗b πX and ⊗b πs,NX. Among other results, we characterize when the norms of spaces P(N`p, `q), P(N lM1, lM2), and P(N d(w, p), lM2) are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results in [38, 48, 49] (in the spirit of Pitt’s compactness theorem) on the reflexivity of spaces of N-homogeneous polynomials and N-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results by considering the subsets U and Us of elementary tensors on the unit spheres of X⊗b π · · · ⊗b πX and ⊗b πs,N X, respectively. Specifically, we prove that the norms of ⊗b πs,N `2 and `2⊗b π · · · ⊗b π`2 are uniformly strongly subdifferentiable on Us and U, and that the norms of c0⊗b πs c0 and c0⊗b πc0 are strongly subdifferentiable on Us and U in the complex case.
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