Slices in the unit ball of the symmetric tensor product of $\mathcal{C}(K)$ and L1(μ)

Abstract

We prove that for the cases $X=\mathcal{C}(K)$ (K infinite) and X=L₁(μ) (μ σ-finite and atomless) it holds that every slice of the unit ball of the N-fold symmetric tensor product of X has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of N-homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability.