Reflexivity of Spaces of Polynomials on Direct Sums of Banach Spaces

Abstract

Let P( n E, 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( n E, F) is studied in connection with the density of the space of the finite type n-homogeneous polynomials in P( n E, F) and in connection with the equality [P( n E, F) ,τ b] � =[ P( n E, 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.