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 $H_b(E,F)$ 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 $P({}^{n}E,F)$ 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 $[P({}^{n}E,F),{\tau }_b{]}^{\prime }=[P({}^{n}E,F),{\tau }_0{]}^{\prime }$ 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.