POLYNOMIAL REFLEXIVITY IN BANACH SPACES

We ask when the space of N-homogeneous analytic polynomials on a Banach space is reflexive. This turns out to be related to whether polynomials are weakly sequentially continuous, and to the geometry of spreading models. For example, if these spaces are reflexive for all N, no quotient of the dual space may have a spreading model with an upper q-estimate, and every bounded holomorphic function on the unit ball has a Taylor series made up of weakly sequentially continuous polynomials (we assume the approximation property). Alencar, Aron and Dineen [AAD] gave the first example of some properties of a polynomially reflexive space (using T*, the original Tsirelson space); we show that these properties and others are shared by all polynomially reflexive spaces.