James type results for polynomials and symmetric multilinear forms

Abstract

We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space X is reflexive if and only if it admits and equivalent norm such that there exists x0≠0 in X and a weak-*-open subset A of the dual space, satisfying that x*⊗x0 attains its numerical radius. for each x* in A.