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 $H_b(E,F)$ 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 $\tau_b$ 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 $[{P}(^nE,F),\tau_b ]'=[{P}(^nE,F),\tau_0]'$ in case $E$ is a reflexive countable direct sum of complex Banach spaces and $F$ is a reflexive complex Banach space. The reflexivity of $H_b(E)$ is also considered.