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=C(K) (K infinite) and X=L1(m) (m s-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 Frechet differentiability.