Geometric Theory of Spaces of Integral Polynomials and Symmetric Tensor Products

We investigate the nature of extreme, (weak*-) exposed, and (weak*-) strongly exposed points of the unit ball of spaces of n-homogeneous integral polynomials on, and n-fold symmetric products of, a Banach space E. For the space of integral polynomials we show the set of extreme points is contained in the set {±φn: φ∈E′, ‖φ‖=1}. We give Šmulʹyan type theorems for spaces of n-homogeneous polynomials and n-fold symmetric tensors that characterise weak*-exposed (resp. weak*-strongly exposed) points in terms of Gâteaux (resp. Fréchet) differentiability of the norm on various spaces of tensor products and polynomials. Our study of the geometry of these spaces has many applications: When E has the Radon–Nikodým property we show that the spaces of n-homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n. When the dimensions of E and n are both at least 2 then the space of n-homogeneous polynomials on E is neither smooth nor rotund. For a certain class of reflexive Banach space the space of n-homogeneous approximable polynomials on E is either reflexive or is not isometric to a dual Banach space. We conclude with a Choquet Theorem for a space of homogeneous polynomials.