Reflexivity of Spaces of Polynomials on Direct Sums of Banach Spaces

Abstract

Let P(nE,F) be the space of the continuous n-homogeneous polynomials from E into F and Hb(E,F) be the space of the holomorphic mappings from E into F that are bounded in the bounded subsets of E, both spaces endowed with the topology τb of uniform convergence on the bounded subsets of E. The reflexivity of P(nE,F) is studied in connection with the density of the space of the finite type n-homogeneous polynomials in P(nE,F) and in connection with the equality [P(nE,F), τb] = [P(nE,F), _τ_0]' in case E is a reflexive countable direct sum of complex Banach spaces and F is a reflexive complex Banach space. The reflexivity of Hb(E) is also considered.