In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{ P}(^N X, Y^*)$, $X \hat{ \otimes}_\pi \cdots \hat{ \otimes}_\pi X$ and $\hat{ \otimes}_{ \pi_s,N} X$. Among other results, we characterize when the norms of the spaces $\mathcal{ P}(^N \ell_p, \ell_{ q}), \mathcal{ P}(^N l_{ M_1}, l_{ M_2})$, and $\mathcal{ P}(^N d(w,p), l_{ M_2})$ 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 on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X \hat{ \otimes}_\pi \cdots \hat{ \otimes}_\pi X$ and $\hat{ \otimes}_{ \pi_s,N} X$, respectively. Specifically, we prove that $\hat{ \otimes}_{ \pi_s,N} \ell_2$ and $\ell_2 \hat{ \otimes}_\pi \cdots \hat{ \otimes}_\pi \ell_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0 \hat{ \otimes}_{ \pi_s} c_0$ and $c_0 \hat{ \otimes}_\pi c_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case.
Categories: math.FA