*(English)*Zbl 0801.46021

For $v$ a positive continuous weight function on an open subset $G$ of ${\u2102}^{N}$, let $Hv\left(G\right)$ and $H{v}_{0}\left(G\right)$ denote the Banach spaces, under the norm $\parallel f\parallel =sup\left\{v\right(z\left)\right|f\left(z\right)|:z\in G\}$, of holomorphic functions $f$ on $G$ such that $vf$ is bounded and $vf$ vanishes at infinity on $G$, respectively. The principal question addressed in the paper under review is whether $Hv\left(G\right)$ is (isometrically isomorphic to) the bidual of $H{v}_{0}\left(G\right)$. This problem may be viewed as an analogue to weighted function spaces because of the fact that ${\ell}_{\infty}$ is the bidual of ${c}_{0}$.

An affirmative result (that $Hv\left(G\right)$ is the bidual of $H{v}_{0}\left(G\right))$ was obtained in 1970 by Rubel and Shields for the case where $G$ is the open unit disc in $\u2102$ and $v$ is a radial weight function vanishing at the boundary; that is, $v\left(z\right)=v\left(\right|z\left|\right)$ for all $z$ and ${lim}_{\left|z\right|\to {1}^{-}}v\left(z\right)=0$. This result, cited by the authors here, occurs as a special case of the main result of this paper. Specifically, Theorem 1.1 shows that $Hv\left(G\right)$ is always (isometrically isomorphic to) the dual space of a Banach space and that $Hv\left(G\right)$ is the bidual of $H{v}_{0}\left(G\right)$ if and only if the unit ball of $H{v}_{0}\left(G\right)$ is dense in the unit ball of $Hv\left(G\right)$ in the compact-open topology. This density criterion is then used to obtain the Rubel-Shields result as well as other more general examples. Theorem 2.3 shows that, if the set $G$ is balanced and if $v$ is radial, then $Hv\left(G\right)$ is the bidual of $H{v}_{0}\left(G\right)$ whenever the latter space contains all polynomials.

The final section of the paper concerns the predual of $Hv\left(G\right)$ and the structure of $Hv\left(G\right)$ when equipped with various topologies, most notably the bounded weak star topology.

##### MSC:

46E15 | Banach spaces of continuous, differentiable or analytic functions |

46A70 | Saks spaces and their duals |

46B10 | Duality and reflexivity in normed spaces |

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |